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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
LINEAR ALGEBRA
Dr.N.A.Pande Associate Professor
Department of Mathematics & Statistics, Yeshwant Mahavidyalaya, Nanded – 431602
Maharashtra, INDIA
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 3rd
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 3rd
• Semester : 5th
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 3rd
• Semester : 5th
• Paper No. : 10(B.A.) / 14(B.Sc.)
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 3rd
• Semester : 5th
• Paper No. : 10(B.A.) / 14(B.Sc.)
• Syllabus Effective From : 2015-16
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 3rd
• Semester : 5th
• Paper No. : 10(B.A.) / 14(B.Sc.)
• Syllabus Effective From : 2015-16
• Paper Code : MT 302
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth
Marathwada University, Nanded, India
• Course : B.A./B.Sc.
• Subject : Mathematics
• Year : 3rd
• Semester : 5th
• Paper No. : 10(B.A.) / 14(B.Sc.)
• Syllabus Effective From : 2015-16
• Paper Code : MT 302
• Marks : 40 (University) + 10 (Internal) = 50
PAPER DETAILS
2
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus
PAPER DETAILS
3
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-I
PAPER DETAILS
3
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-I
�Vector Spaces : Elementary Basic Concepts of Vector Spaces, Linear Independence and Bases, Dual Spaces.
PAPER DETAILS
3
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-I
�Vector Spaces : Elementary Basic Concepts of Vector Spaces, Linear Independence and Bases, Dual Spaces.
• Unit-II
PAPER DETAILS
3
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-I
�Vector Spaces : Elementary Basic Concepts of Vector Spaces, Linear Independence and Bases, Dual Spaces.
• Unit-II
� Inner Product Spaces, Fields : Extension Fields (Definitions Only).
PAPER DETAILS
3
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-I
�Vector Spaces : Elementary Basic Concepts of Vector Spaces, Linear Independence and Bases, Dual Spaces.
• Unit-II
� Inner Product Spaces, Fields : Extension Fields (Definitions Only).
• Unit-III
PAPER DETAILS
3
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Syllabus • Unit-I
�Vector Spaces : Elementary Basic Concepts of Vector Spaces, Linear Independence and Bases, Dual Spaces.
• Unit-II
� Inner Product Spaces, Fields : Extension Fields (Definitions Only).
• Unit-III
�Linear Transformation : The Algebra of Linear Transformations, Characteristic Roots, Matrices.
PAPER DETAILS
3
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope
PAPER DETAILS
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
PAPER DETAILS
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Author : I. N. Herstein
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Author : I. N. Herstein
�Edition : Second Edition
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Author : I. N. Herstein
�Edition : Second Edition
�Publisher : John Wiley & Sons
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Author : I. N. Herstein
�Edition : Second Edition
�Publisher : John Wiley & Sons
� ISSN : 978-0-471-01090-6
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Author : I. N. Herstein
�Edition : Second Edition
�Publisher : John Wiley & Sons
� ISSN : 978-0-471-01090-6
• Scope :
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Author : I. N. Herstein
�Edition : Second Edition
�Publisher : John Wiley & Sons
� ISSN : 978-0-471-01090-6
• Scope :
�Unit–I : Chapter 4 : Article 4.1, 4.2, 4.3
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Author : I. N. Herstein
�Edition : Second Edition
�Publisher : John Wiley & Sons
� ISSN : 978-0-471-01090-6
• Scope :
�Unit–I : Chapter 4 : Article 4.1, 4.2, 4.3
�Unit–II : Chapter 4 : Article 4.4
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Author : I. N. Herstein
�Edition : Second Edition
�Publisher : John Wiley & Sons
� ISSN : 978-0-471-01090-6
• Scope :
�Unit–I : Chapter 4 : Article 4.1, 4.2, 4.3
�Unit–II : Chapter 4 : Article 4.4
Chapter 5 : Article 5.1 (Definitions)
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Topics in Algebra
�Author : I. N. Herstein
�Edition : Second Edition
�Publisher : John Wiley & Sons
� ISSN : 978-0-471-01090-6
• Scope :
�Unit–I : Chapter 4 : Article 4.1, 4.2, 4.3
�Unit–II : Chapter 4 : Article 4.4
Chapter 5 : Article 5.1 (Definitions)
�Unit–III : Chapter 6 : Article 6.1, 6.2, 6.3
PAPER DETAILS
4
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Reference Books
PAPER DETAILS
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Reference Books • A First Course in Abstract Algebra, By J.B. Fraleigh,
Narosa Publications
• Contemporary Abstract Algebra, By Joseph Gallion, Narosa Publications
• Linear Algebra for Undergraduates, By S.R.Mangalgiri and D.K.Daftari
• First Course in Abstract Algebra, By P.B.Bhattacharya, S.K.Jain and S.R.Nagpaul
• An Introduction to Linear Algebra , By V. Krishnamurty, V. P. Mainru, J. L. Arrora
• Linear Algebra, by L. Smith, Springer-Verlag New York.
• Matrix and Linear Algebra, by K. B. Datta, Prentice Hall of India Pvt. Ltd, New Delhi, 2000.
PAPER DETAILS
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Vector Space : A non-empty set V is said to be
vector space over a field F ⇔ V is abelian group under an operation denoted by + and ∀ α ∈ F, v ∈ V, there exists an element αv in V satisfying :
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Vector Space : A non-empty set V is said to be
vector space over a field F ⇔ V is abelian group under an operation denoted by + and ∀ α ∈ F, v ∈ V, there exists an element αv in V satisfying :
�α(u + v) = αu + αv
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Vector Space : A non-empty set V is said to be
vector space over a field F ⇔ V is abelian group under an operation denoted by + and ∀ α ∈ F, v ∈ V, there exists an element αv in V satisfying :
�α(u + v) = αu + αv
� (α + β)v = αv + βv
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Vector Space : A non-empty set V is said to be
vector space over a field F ⇔ V is abelian group under an operation denoted by + and ∀ α ∈ F, v ∈ V, there exists an element αv in V satisfying :
�α(u + v) = αu + αv
� (α + β)v = αv + βv
� (αβ)v = α(βv)
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Vector Space : A non-empty set V is said to be
vector space over a field F ⇔ V is abelian group under an operation denoted by + and ∀ α ∈ F, v ∈ V, there exists an element αv in V satisfying :
�α(u + v) = αu + αv
� (α + β)v = αv + βv
� (αβ)v = α(βv)
�1v = v
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Vector Space : A non-empty set V is said to be
vector space over a field F ⇔ V is abelian group under an operation denoted by + and ∀ α ∈ F, v ∈ V, there exists an element αv in V satisfying :
�α(u + v) = αu + αv
� (α + β)v = αv + βv
� (αβ)v = α(βv)
�1v = v
• There are 9 conditions in the definition of a vector space, 5 of abelian group and 4 listed explicitly above.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Members of vector space are called as vectors
and are denoted by smallcase Latin letters.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Members of vector space are called as vectors
and are denoted by smallcase Latin letters.
• Members of field are called as scalars and are denoted by smallcase Greek letters.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Members of vector space are called as vectors
and are denoted by smallcase Latin letters.
• Members of field are called as scalars and are denoted by smallcase Greek letters.
• Examples of Vector Spaces :
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Members of vector space are called as vectors
and are denoted by smallcase Latin letters.
• Members of field are called as scalars and are denoted by smallcase Greek letters.
• Examples of Vector Spaces :
�Every field is a vector space over any of its subfields
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Members of vector space are called as vectors
and are denoted by smallcase Latin letters.
• Members of field are called as scalars and are denoted by smallcase Greek letters.
• Examples of Vector Spaces :
�Every field is a vector space over any of its subfields
� If F is a field, then the set F(n) of all ordered n-tuples of members of F is a vector space over F, with respect to coordinatewise addition and coordinatewise scalar multiplication.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Examples of Vector Spaces (Continued) :
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Examples of Vector Spaces (Continued) :
� If F is a field, then the set F[x] of all polynomials in x with coefficients in F is a vector space over F, with respect to coefficientwise addition and coefficientwise scalar multiplication.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • Examples of Vector Spaces (Continued) :
� If F is a field, then the set F[x] of all polynomials in x with coefficients in F is a vector space over F, with respect to coefficientwise addition and coefficientwise scalar multiplication.
� If F is a field, then the set Fn[x] of all polynomials of degree at most n − 1 in x with coefficients in F is a vector space over F, with respect to coefficientwise addition and coefficientwise scalar multiplication.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts
UNIT-I
9
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • A non-empty subset W of a vector space V
over F is said to be subspace of V, if W itself is vector space over F with respect to same operations in V.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • A non-empty subset W of a vector space V
over F is said to be subspace of V, if W itself is vector space over F with respect to same operations in V.
• Subspace Criteria : A non-empty subset W of a vector space V over F is said to be subspace of V ⇔ ∀ w1, w2 ∈ W, ∀ α, β ∈ F, αw1 + βw2 ∈ W.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • A non-empty subset W of a vector space V
over F is said to be subspace of V, if W itself is vector space over F with respect to same operations in V.
• Subspace Criteria : A non-empty subset W of a vector space V over F is said to be subspace of V ⇔ ∀ w1, w2 ∈ W, ∀ α, β ∈ F, αw1 + βw2 ∈ W.
• Intersection of two subspaces is a subspace.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • A non-empty subset W of a vector space V
over F is said to be subspace of V, if W itself is vector space over F with respect to same operations in V.
• Subspace Criteria : A non-empty subset W of a vector space V over F is said to be subspace of V ⇔ ∀ w1, w2 ∈ W, ∀ α, β ∈ F, αw1 + βw2 ∈ W.
• Intersection of two subspaces is a subspace.
• Addition of two subspaces is a subspace.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts
UNIT-I
10
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If U and V are two vector spaces over a
common field F, then a mapping T : U → V is said to be homomorphism ⇔ ∀ u1, u2 ∈ U, ∀ α ∈ F,
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If U and V are two vector spaces over a
common field F, then a mapping T : U → V is said to be homomorphism ⇔ ∀ u1, u2 ∈ U, ∀ α ∈ F,
� (u1 + u2)T = u1T + u2T
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If U and V are two vector spaces over a
common field F, then a mapping T : U → V is said to be homomorphism ⇔ ∀ u1, u2 ∈ U, ∀ α ∈ F,
� (u1 + u2)T = u1T + u2T
� (αu1)T = α(u1T)
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If U and V are two vector spaces over a
common field F, then a mapping T : U → V is said to be homomorphism ⇔ ∀ u1, u2 ∈ U, ∀ α ∈ F,
� (u1 + u2)T = u1T + u2T
� (αu1)T = α(u1T)
• A one-to-one homomorphism is called isomorphism.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If U and V are two vector spaces over a
common field F, then a mapping T : U → V is said to be homomorphism ⇔ ∀ u1, u2 ∈ U, ∀ α ∈ F,
� (u1 + u2)T = u1T + u2T
� (αu1)T = α(u1T)
• A one-to-one homomorphism is called isomorphism.
• Two vector spaces are isomorphic ⇔ there exists an isomorphism of one onto another.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • ‘is isomorphic to’ is an equivalence relation on
the class of all vector spaces.
UNIT-I
11
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • ‘is isomorphic to’ is an equivalence relation on
the class of all vector spaces.
• If T : U → V is a homomorphism, then kernel of homomorphism is defined as K = {u ∈ U | uT = 0}.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • ‘is isomorphic to’ is an equivalence relation on
the class of all vector spaces.
• If T : U → V is a homomorphism, then kernel of homomorphism is defined as K = {u ∈ U | uT = 0}.
• Kernel of homomorphism is subspace of domain vector space.
UNIT-I
11
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • ‘is isomorphic to’ is an equivalence relation on
the class of all vector spaces.
• If T : U → V is a homomorphism, then kernel of homomorphism is defined as K = {u ∈ U | uT = 0}.
• Kernel of homomorphism is subspace of domain vector space.
• The set of all homomorphism from a vector space U to a vector space V is denoted by Hom(U, V).
UNIT-I
11
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts
UNIT-I
12
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If V is a vector space over a field F, then ∀
v ∈ V, ∀ α ∈ F,
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If V is a vector space over a field F, then ∀
v ∈ V, ∀ α ∈ F,
�α0 = 0
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If V is a vector space over a field F, then ∀
v ∈ V, ∀ α ∈ F,
�α0 = 0
�0v = 0
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If V is a vector space over a field F, then ∀
v ∈ V, ∀ α ∈ F,
�α0 = 0
�0v = 0
� (−α)v = −(αv)
UNIT-I
12
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If V is a vector space over a field F, then ∀
v ∈ V, ∀ α ∈ F,
�α0 = 0
�0v = 0
� (−α)v = −(αv)
�αv = 0 ⇒ α = 0 or v = 0
UNIT-I
12
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If V is a vector space over a field F, then ∀
v ∈ V, ∀ α ∈ F,
�α0 = 0
�0v = 0
� (−α)v = −(αv)
�αv = 0 ⇒ α = 0 or v = 0
• If V is a vector space over a field F and W is a subspace of V, then V/W = {v + W | v ∈ V}is vector space over F with respect to coset addition and scalar multiplication and is called quotient space.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If T is a homomorphism of U onto V with
kernel W, then U/W is isomorphic to V.
UNIT-I
13
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If T is a homomorphism of U onto V with
kernel W, then U/W is isomorphic to V.
• A vector space V over F is said to be internal
direct sum of its subspaces U1, U2, ⋯⋯⋯⋯ , Un ⇔ every vector in V can be expressed as sum of members of Ui in a unique way.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If T is a homomorphism of U onto V with
kernel W, then U/W is isomorphic to V.
• A vector space V over F is said to be internal
direct sum of its subspaces U1, U2, ⋯⋯⋯⋯ , Un ⇔ every vector in V can be expressed as sum of members of Ui in a unique way.
• If V1, V2, ⋯⋯⋯⋯ , Vn are vector spaces over a common field F, then their Cartesian product V1 × V2 × ⋯⋯⋯⋯ × Vn is a vector space over F with respect to coordinatewise addition and coordinatewise scalar multiplication and is called external direct sum of V1, V2, ⋯⋯⋯⋯ , Vn.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts
UNIT-I
14
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If a vector space V over F is internal direct
sum of its subspaces U1, U2, ⋯⋯⋯⋯ , Un ⇒ V is isomorphic to external direct sum of U1, U2, ⋯⋯⋯⋯ , Un.
UNIT-I
14
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If a vector space V over F is internal direct
sum of its subspaces U1, U2, ⋯⋯⋯⋯ , Un ⇒ V is isomorphic to external direct sum of U1, U2, ⋯⋯⋯⋯ , Un.
• In a vector space V over F, v, w ∈ V, ∀ α ∈ F, α(v − w) = αv − αw.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Vector Spaces : Basic Concepts • If a vector space V over F is internal direct
sum of its subspaces U1, U2, ⋯⋯⋯⋯ , Un ⇒ V is isomorphic to external direct sum of U1, U2, ⋯⋯⋯⋯ , Un.
• In a vector space V over F, v, w ∈ V, ∀ α ∈ F, α(v − w) = αv − αw.
• If A and B are subspaces of a vector space V over F, then (A + B)/B is isomorphic to A/(A ∩ B).
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • If V is a vector space over F and if
v1, v2, ⋯⋯⋯⋯ , vn are in V, then their linear combination is a vector of form α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn, where αi ∈ F.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • If V is a vector space over F and if
v1, v2, ⋯⋯⋯⋯ , vn are in V, then their linear combination is a vector of form α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn, where αi ∈ F.
• If S is a non-empty subset of a vector space V over F, then linear span of S, denoted by L(S) is the set of all linear combinations of finite sets of elements of S.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Properties of L(S) :
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Properties of L(S) :
�L(S) is a subspace of V.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Properties of L(S) :
�L(S) is a subspace of V.
�S ⊂ T ⇒ L(S) ⊂ L(T)
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Properties of L(S) :
�L(S) is a subspace of V.
�S ⊂ T ⇒ L(S) ⊂ L(T)
�L(S ∪ T) = L(S) + L(T)
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Properties of L(S) :
�L(S) is a subspace of V.
�S ⊂ T ⇒ L(S) ⊂ L(T)
�L(S ∪ T) = L(S) + L(T)
�L(L(S)) = L(S)
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Properties of L(S) :
�L(S) is a subspace of V.
�S ⊂ T ⇒ L(S) ⊂ L(T)
�L(S ∪ T) = L(S) + L(T)
�L(L(S)) = L(S)
• A vector space is said to be finite dimensional ⇔ ∃ a finite subset S of V such that L(S) = V.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Properties of L(S) :
�L(S) is a subspace of V.
�S ⊂ T ⇒ L(S) ⊂ L(T)
�L(S ∪ T) = L(S) + L(T)
�L(L(S)) = L(S)
• A vector space is said to be finite dimensional ⇔ ∃ a finite subset S of V such that L(S) = V.
• A vector space is said to be infinite dimensional ⇔ it is not finite dimensional.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • In a vector space V over F, v1, v2, ⋯⋯⋯⋯ , vn in V
are said to be linearly dependent ⇔ ∃ scalars α1, α2, ⋯⋯⋯⋯ , αn in F, not all zero such that α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn = 0.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • In a vector space V over F, v1, v2, ⋯⋯⋯⋯ , vn in V
are said to be linearly dependent ⇔ ∃ scalars α1, α2, ⋯⋯⋯⋯ , αn in F, not all zero such that α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn = 0.
• In a vector space V over F, v1, v2, ⋯⋯⋯⋯ , vn in V are said to be linearly independent ⇔ whenever α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn = 0, each αi = 0.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • In a vector space V over F, v1, v2, ⋯⋯⋯⋯ , vn in V
are said to be linearly dependent ⇔ ∃ scalars α1, α2, ⋯⋯⋯⋯ , αn in F, not all zero such that α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn = 0.
• In a vector space V over F, v1, v2, ⋯⋯⋯⋯ , vn in V are said to be linearly independent ⇔ whenever α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn = 0, each αi = 0.
• Linear dependence or independence of vectors depends upon vectors as well as field over which V is the vector space.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • v1, v2, ⋯⋯⋯⋯ , vn in V are linearly independent ⇒
every vector in their linear span as unique representation of the form α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • v1, v2, ⋯⋯⋯⋯ , vn in V are linearly independent ⇒
every vector in their linear span as unique representation of the form α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn.
• v1, v2, ⋯⋯⋯⋯ , vn in V are linearly dependent ⇒ some vk is linear combination of preceding ones.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • v1, v2, ⋯⋯⋯⋯ , vn in V are linearly independent ⇒
every vector in their linear span as unique representation of the form α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn.
• v1, v2, ⋯⋯⋯⋯ , vn in V are linearly dependent ⇒ some vk is linear combination of preceding ones.
• v1, v2, ⋯⋯⋯⋯ , vn in V are linearly dependent with linear span W ⇒ there exists a subset of it which is linearly independent whose linear span is still W.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • A subset S of a vector space is called basis of
V ⇔ S is linearly independent and L(S) = V.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • A subset S of a vector space is called basis of
V ⇔ S is linearly independent and L(S) = V.
• Basis is the largest linearly independent set.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • A subset S of a vector space is called basis of
V ⇔ S is linearly independent and L(S) = V.
• Basis is the largest linearly independent set.
• If V is a finite-dimensional vector space over F, then any two bases of V have the same number of elements.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • A subset S of a vector space is called basis of
V ⇔ S is linearly independent and L(S) = V.
• Basis is the largest linearly independent set.
• If V is a finite-dimensional vector space over F, then any two bases of V have the same number of elements.
• F(n) is isomorphic to F(m) ⇔ m = n.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • A subset S of a vector space is called basis of
V ⇔ S is linearly independent and L(S) = V.
• Basis is the largest linearly independent set.
• If V is a finite-dimensional vector space over F, then any two bases of V have the same number of elements.
• F(n) is isomorphic to F(m) ⇔ m = n.
• Each finite-dimensional vector space is isomorphic to a unique F(n), for which n is called dimension of V and is denoted by dim V.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Any two finite dimension vector spaces having
same dimension are isomorphic.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Any two finite dimension vector spaces having
same dimension are isomorphic.
• Any two vector isomorphic vector spaces have same dimension.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Any two finite dimension vector spaces having
same dimension are isomorphic.
• Any two vector isomorphic vector spaces have same dimension.
• If W is a subspace of a finite-dimensional vector space V over F, then dim W ≤ dim V and dim V/W = dim V − dim W.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Independence and Bases • Any two finite dimension vector spaces having
same dimension are isomorphic.
• Any two vector isomorphic vector spaces have same dimension.
• If W is a subspace of a finite-dimensional vector space V over F, then dim W ≤ dim V and dim V/W = dim V − dim W.
• If A and B are finite-dimensional subspaces of a vector space V over F, then dim(A + B) = dim A + dim B −dim(A ∩ B).
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V and W are vector spaces over same field
F, then the set of all homomorphisms from V to W is denoted by Hom(V, W) and is itself a vector space over F.
UNIT-I
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V and W are vector spaces over same field
F, then the set of all homomorphisms from V to W is denoted by Hom(V, W) and is itself a vector space over F.
• dim Hom(V, W) = dim V × dim W
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V and W are vector spaces over same field
F, then the set of all homomorphisms from V to W is denoted by Hom(V, W) and is itself a vector space over F.
• dim Hom(V, W) = dim V × dim W
• The dimension of a field F as a vector space over itself is 1.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V and W are vector spaces over same field
F, then the set of all homomorphisms from V to W is denoted by Hom(V, W) and is itself a vector space over F.
• dim Hom(V, W) = dim V × dim W
• The dimension of a field F as a vector space over itself is 1.
• dim Hom(V, V) = dim V2
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V and W are vector spaces over same field
F, then the set of all homomorphisms from V to W is denoted by Hom(V, W) and is itself a vector space over F.
• dim Hom(V, W) = dim V × dim W
• The dimension of a field F as a vector space over itself is 1.
• dim Hom(V, V) = dim V2
• dim Hom(V, F) = dim V
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V is a vector space over F, then the vector
space Hom(V, F) is called dual space of V and is denoted by v.̂
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V is a vector space over F, then the vector
space Hom(V, F) is called dual space of V and is denoted by v.̂
• Members of v ̂are scalar valued functions of vectors.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V is a vector space over F, then the vector
space Hom(V, F) is called dual space of V and is denoted by v.̂
• Members of v ̂are scalar valued functions of vectors.
• dim v ̂= dim V.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V is a vector space over F, then the vector
space Hom(V, F) is called dual space of V and is denoted by v.̂
• Members of v ̂are scalar valued functions of vectors.
• dim v ̂= dim V.
• Members of v ̂are called functionals.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V is a vector space over F, then the vector
space Hom(V, F) is called dual space of V and is denoted by v.̂
• Members of v ̂are scalar valued functions of vectors.
• dim v ̂= dim V.
• Members of v ̂are called functionals.
• If V is finite dimensional vector space over F, then ∀ 0 ≠ v ∈ V, ∃ f ∈ v ̂such that f(v) ≠ 0.
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Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • If V is a vector space over F, then the vector
space Hom(V, F) is called dual space of V and is denoted by v.̂
• Members of v ̂are scalar valued functions of vectors.
• dim v ̂= dim V.
• Members of v ̂are called functionals.
• If V is finite dimensional vector space over F, then ∀ 0 ≠ v ∈ V, ∃ f ∈ v ̂such that f(v) ≠ 0.
• Every finite-dimensional vector space V is isomorphic to its dual space v.̂
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22
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces
UNIT-I
23
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • Annihilator A(W) of a subspace W of a
vector space V is A(W) = {f ∈ v ̂| f(w) = 0 ∀ w ∈ W}
UNIT-I
23
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • Annihilator A(W) of a subspace W of a
vector space V is A(W) = {f ∈ v ̂| f(w) = 0 ∀ w ∈ W}
• Annihilator of a subspace is itself a subspace.
UNIT-I
23
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • Annihilator A(W) of a subspace W of a
vector space V is A(W) = {f ∈ v ̂| f(w) = 0 ∀ w ∈ W}
• Annihilator of a subspace is itself a subspace.
• dim A(W) = dim V − dim W.
UNIT-I
23
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • Annihilator A(W) of a subspace W of a
vector space V is A(W) = {f ∈ v ̂| f(w) = 0 ∀ w ∈ W}
• Annihilator of a subspace is itself a subspace.
• dim A(W) = dim V − dim W.
• If V is a vector space over F and U & W are subspaces of V such that U ⊂ W, then A(U) ⊃ A(W).
UNIT-I
23
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • Annihilator A(W) of a subspace W of a
vector space V is A(W) = {f ∈ v ̂| f(w) = 0 ∀ w ∈ W}
• Annihilator of a subspace is itself a subspace.
• dim A(W) = dim V − dim W.
• If V is a vector space over F and U & W are subspaces of V such that U ⊂ W, then A(U) ⊃ A(W).
• If W is a subspace of a vector space V over F, then A(A(W)) = W.
UNIT-I
23
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Dual Spaces • Annihilator A(W) of a subspace W of a
vector space V is A(W) = {f ∈ v ̂| f(w) = 0 ∀ w ∈ W}
• Annihilator of a subspace is itself a subspace.
• dim A(W) = dim V − dim W.
• If V is a vector space over F and U & W are subspaces of V such that U ⊂ W, then A(U) ⊃ A(W).
• If W is a subspace of a vector space V over F, then A(A(W)) = W.
• For any subset S of V, then A(S) = A(L(S)).
UNIT-I
23
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces
UNIT-II
24
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • An inner product space is a vector space V
over F in which there is defined a function of two vectors giving a scalar denoted by (u, v) ∈ F ∀ u, v ∈ V such that ∀ u, v, w ∈ V and ∀ α, β ∈ F,
UNIT-II
24
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • An inner product space is a vector space V
over F in which there is defined a function of two vectors giving a scalar denoted by (u, v) ∈ F ∀ u, v ∈ V such that ∀ u, v, w ∈ V and ∀ α, β ∈ F,
� (u, v) =
UNIT-II
24
),( uv
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • An inner product space is a vector space V
over F in which there is defined a function of two vectors giving a scalar denoted by (u, v) ∈ F ∀ u, v ∈ V such that ∀ u, v, w ∈ V and ∀ α, β ∈ F,
� (u, v) =
� (u, u) ≥ 0 ∀ u ∈ V and (u, u) = 0 ⇔ u = 0
UNIT-II
24
),( uv
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • An inner product space is a vector space V
over F in which there is defined a function of two vectors giving a scalar denoted by (u, v) ∈ F ∀ u, v ∈ V such that ∀ u, v, w ∈ V and ∀ α, β ∈ F,
� (u, v) =
� (u, u) ≥ 0 ∀ u ∈ V and (u, u) = 0 ⇔ u = 0
� (αu + βv, w) = α(u, w) + β(v, w)
UNIT-II
24
),( uv
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • An inner product space is a vector space V
over F in which there is defined a function of two vectors giving a scalar denoted by (u, v) ∈ F ∀ u, v ∈ V such that ∀ u, v, w ∈ V and ∀ α, β ∈ F,
� (u, v) =
� (u, u) ≥ 0 ∀ u ∈ V and (u, u) = 0 ⇔ u = 0
� (αu + βv, w) = α(u, w) + β(v, w)
• In an inner product space V over F, for all scalars and vectors,
UNIT-II
24
),( uv
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • An inner product space is a vector space V
over F in which there is defined a function of two vectors giving a scalar denoted by (u, v) ∈ F ∀ u, v ∈ V such that ∀ u, v, w ∈ V and ∀ α, β ∈ F,
� (u, v) =
� (u, u) ≥ 0 ∀ u ∈ V and (u, u) = 0 ⇔ u = 0
� (αu + βv, w) = α(u, w) + β(v, w)
• In an inner product space V over F, for all scalars and vectors,
UNIT-II
24
),( uv
),(+),(=)+,( wuvuwvu βαβα
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces
UNIT-II
25
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • Length or norm of a vector v is given by
UNIT-II
25
( )vvv ,=||||
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • Length or norm of a vector v is given by
• || αv || = |α|⋅⋅⋅⋅||v||
UNIT-II
25
( )vvv ,=||||
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • Length or norm of a vector v is given by
• || αv || = |α|⋅⋅⋅⋅||v||
• If a, b, c are real numbers such that aλ2 + 2bλ + c ≥ 0 ∀ real λ, then b2 ≤ ac.
UNIT-II
25
( )vvv ,=||||
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • Length or norm of a vector v is given by
• || αv || = |α|⋅⋅⋅⋅||v||
• If a, b, c are real numbers such that aλ2 + 2bλ + c ≥ 0 ∀ real λ, then b2 ≤ ac.
• Schwarz’s Inequality : |(u, v)| ≤ ||u||⋅⋅⋅⋅||v||
UNIT-II
25
( )vvv ,=||||
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • Length or norm of a vector v is given by
• || αv || = |α|⋅⋅⋅⋅||v||
• If a, b, c are real numbers such that aλ2 + 2bλ + c ≥ 0 ∀ real λ, then b2 ≤ ac.
• Schwarz’s Inequality : |(u, v)| ≤ ||u||⋅⋅⋅⋅||v||
• u is said to be orthogonal to v ⇔ (u, v) = 0.
UNIT-II
25
( )vvv ,=||||
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • Length or norm of a vector v is given by
• || αv || = |α|⋅⋅⋅⋅||v||
• If a, b, c are real numbers such that aλ2 + 2bλ + c ≥ 0 ∀ real λ, then b2 ≤ ac.
• Schwarz’s Inequality : |(u, v)| ≤ ||u||⋅⋅⋅⋅||v||
• u is said to be orthogonal to v ⇔ (u, v) = 0.
• Orthogonal complement of a subspace W of an inner product space V is given by W⊥ = {v ∈ V | (v, w) = 0 ∀ w ∈ W}
UNIT-II
25
( )vvv ,=||||
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces
UNIT-II
26
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • W⊥ is a subspace of V.
UNIT-II
26
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • W⊥ is a subspace of V.
• W ∩ W⊥ = {0}
UNIT-II
26
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • W⊥ is a subspace of V.
• W ∩ W⊥ = {0}
• A set of vectors {vi} is said to be orthonormal ⇔
UNIT-II
26
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • W⊥ is a subspace of V.
• W ∩ W⊥ = {0}
• A set of vectors {vi} is said to be orthonormal ⇔
� (vi, vi) = 1 ∀ I all vectors are of unit length)
UNIT-II
26
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • W⊥ is a subspace of V.
• W ∩ W⊥ = {0}
• A set of vectors {vi} is said to be orthonormal ⇔
� (vi, vi) = 1 ∀ I all vectors are of unit length)
� (vi, vj) = 0 ∀ i ≠ j (distinct vectors are mutually⊥)
UNIT-II
26
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • W⊥ is a subspace of V.
• W ∩ W⊥ = {0}
• A set of vectors {vi} is said to be orthonormal ⇔
� (vi, vi) = 1 ∀ I all vectors are of unit length)
� (vi, vj) = 0 ∀ i ≠ j (distinct vectors are mutually⊥)
• An orthonormal set is linearly independent.
UNIT-II
26
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • W⊥ is a subspace of V.
• W ∩ W⊥ = {0}
• A set of vectors {vi} is said to be orthonormal ⇔
� (vi, vi) = 1 ∀ I all vectors are of unit length)
� (vi, vj) = 0 ∀ i ≠ j (distinct vectors are mutually⊥)
• An orthonormal set is linearly independent.
• {vi} is said to be orthonormal ⇒ for w = α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn, (w, vi) = αi.
UNIT-II
26
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces
UNIT-II
27
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • {vi} is said to be orthonormal ⇒ ∀ w ∈ V, if
u = w − (w, v1)v1 − (w, v2)v2 −⋯⋯⋯⋯− (w, vn)vn, then (u, vi) = 0 ∀ i.
UNIT-II
27
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • {vi} is said to be orthonormal ⇒ ∀ w ∈ V, if
u = w − (w, v1)v1 − (w, v2)v2 −⋯⋯⋯⋯− (w, vn)vn, then (u, vi) = 0 ∀ i.
• Gram-Schmidt Orthogonalization Process : A finite-dimensional inner product space has an orthonormal basis. (in fact, the process is of converting any basis {vi} into orthonormal by using formulae :
UNIT-II
27
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • {vi} is said to be orthonormal ⇒ ∀ w ∈ V, if
u = w − (w, v1)v1 − (w, v2)v2 −⋯⋯⋯⋯− (w, vn)vn, then (u, vi) = 0 ∀ i.
• Gram-Schmidt Orthogonalization Process : A finite-dimensional inner product space has an orthonormal basis. (in fact, the process is of converting any basis {vi} into orthonormal by using formulae :
�w1 = v1/|| v1 ||,
UNIT-II
27
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • {vi} is said to be orthonormal ⇒ ∀ w ∈ V, if
u = w − (w, v1)v1 − (w, v2)v2 −⋯⋯⋯⋯− (w, vn)vn, then (u, vi) = 0 ∀ i.
• Gram-Schmidt Orthogonalization Process : A finite-dimensional inner product space has an orthonormal basis. (in fact, the process is of converting any basis {vi} into orthonormal by using formulae :
�w1 = v1/|| v1 ||,
� .
UNIT-II
27
iiiiii
iiiiiii
vwwvwwvwwv
vwwvwwvwwvw
+),(),(),(
+),(),(),(=
112211
112211
--
--
--------
⋯
⋯
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces
UNIT-II
28
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • If V is a finite-dimensional inner product space
then for any subspace W of V, V = W + W⊥.
UNIT-II
28
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • If V is a finite-dimensional inner product space
then for any subspace W of V, V = W + W⊥.
• If V is a finite-dimensional inner product space over a field F, then for any subspace W of V, (W⊥)⊥ = W.
UNIT-II
28
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • If V is a finite-dimensional inner product space
then for any subspace W of V, V = W + W⊥.
• If V is a finite-dimensional inner product space over a field F, then for any subspace W of V, (W⊥)⊥ = W.
• Bessel’s Inequality : If {w1, w2, ⋯⋯⋯⋯ , wn} is an orthonormal set of vectors in an inner product space V over F, then ∀ v ∈ V,
UNIT-II
28
∑ ≤n
i
i vwv1=
22),(
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inner Product Spaces • If V is a finite-dimensional inner product space
then for any subspace W of V, V = W + W⊥.
• If V is a finite-dimensional inner product space over a field F, then for any subspace W of V, (W⊥)⊥ = W.
• Bessel’s Inequality : If {w1, w2, ⋯⋯⋯⋯ , wn} is an orthonormal set of vectors in an inner product space V over F, then ∀ v ∈ V,
• Parallelogram Law :
|| u + v ||2 + || u − v ||2 = 2(|| u ||2 + || v ||2 )
UNIT-II
28
∑ ≤n
i
i vwv1=
22),(
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions
UNIT-II
29
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is a field that contains a field F, then K is
called extension of F.
UNIT-II
29
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is a field that contains a field F, then K is
called extension of F.
• Extension is superfield.
UNIT-II
29
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is a field that contains a field F, then K is
called extension of F.
• Extension is superfield.
• If K is extension of a field F, the degree of K
over F, denoted by [K : F], the dimension of K as a vector space over F.
UNIT-II
29
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is a field that contains a field F, then K is
called extension of F.
• Extension is superfield.
• If K is extension of a field F, the degree of K
over F, denoted by [K : F], the dimension of K as a vector space over F.
• If L is extension of K and K is extension of F, then [L : F] = [L : K][K : F]
UNIT-II
29
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is a field that contains a field F, then K is
called extension of F.
• Extension is superfield.
• If K is extension of a field F, the degree of K
over F, denoted by [K : F], the dimension of K as a vector space over F.
• If L is extension of K and K is extension of F, then [L : F] = [L : K][K : F]
• If L is extension of K and K is extension of F, then [K : F] | [L : F].
UNIT-II
29
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions
UNIT-II
30
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is extension of a field F, then an element
a ∈ K is said to be algebraic element over F ⇔ a satisfies a non-zero polynomial with coefficients in F.
UNIT-II
30
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is extension of a field F, then an element
a ∈ K is said to be algebraic element over F ⇔ a satisfies a non-zero polynomial with coefficients in F.
• If K is extension of a field F and a ∈ K, the F(a) is the smallest subfield of K containing F and a.
UNIT-II
30
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is extension of a field F, then an element
a ∈ K is said to be algebraic element over F ⇔ a satisfies a non-zero polynomial with coefficients in F.
• If K is extension of a field F and a ∈ K, the F(a) is the smallest subfield of K containing F and a.
• If K is extension of a field F, then an element a ∈ K is algebraic element over F ⇔ F(a) is a finite extension of F.
UNIT-II
30
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions
UNIT-II
31
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is extension of a field F, then for a ∈ K
the minimal polynomial of a is the non-zero polynomial of the smallest degree satisfied by a.
UNIT-II
31
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is extension of a field F, then for a ∈ K
the minimal polynomial of a is the non-zero polynomial of the smallest degree satisfied by a.
• If K is extension of a field F, then a ∈ K is said to be algebraic of degree n over F ⇔ a satisfies a non-zero polynomial of degee n over F and no smaller degree polynomial.
UNIT-II
31
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If K is extension of a field F, then for a ∈ K
the minimal polynomial of a is the non-zero polynomial of the smallest degree satisfied by a.
• If K is extension of a field F, then a ∈ K is said to be algebraic of degree n over F ⇔ a satisfies a non-zero polynomial of degee n over F and no smaller degree polynomial.
• a ∈ K is algebraic of degree n over F ⇒ [F(a) : F] = n.
UNIT-II
31
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions
UNIT-II
32
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If a, b ∈ K are algebraic over F ⇒ a ± b, ab
and a/b (provided b ≠ 0) are all algebraic over F. So, the set of elements of K which are algebraic over F form a subfield of K.
UNIT-II
32
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If a, b ∈ K are algebraic over F ⇒ a ± b, ab
and a/b (provided b ≠ 0) are all algebraic over F. So, the set of elements of K which are algebraic over F form a subfield of K.
• If a, b ∈ K are algebraic of degrees m and n, respectively, over F ⇒ a ± b, ab and a/b (provided b ≠ 0) are all algebraic of degree at most mn, over F.
UNIT-II
32
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If a, b ∈ K are algebraic over F ⇒ a ± b, ab
and a/b (provided b ≠ 0) are all algebraic over F. So, the set of elements of K which are algebraic over F form a subfield of K.
• If a, b ∈ K are algebraic of degrees m and n, respectively, over F ⇒ a ± b, ab and a/b (provided b ≠ 0) are all algebraic of degree at most mn, over F.
• If K is extension of a field F, such that every element of K is algebraic over F, then K is called algebraic extension of F.
UNIT-II
32
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions
UNIT-II
33
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If L is algebraic extension of K and if K is
algebraic extension of F, then L is algebraic extension of F.
UNIT-II
33
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If L is algebraic extension of K and if K is
algebraic extension of F, then L is algebraic extension of F.
• A complex number is said to be algebraic
number ⇔ it is algebraic over the field of rationals.
UNIT-II
33
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If L is algebraic extension of K and if K is
algebraic extension of F, then L is algebraic extension of F.
• A complex number is said to be algebraic
number ⇔ it is algebraic over the field of rationals.
• A complex number which is not algebraic is called transcendental number.
UNIT-II
33
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Fields : Extensions • If L is algebraic extension of K and if K is
algebraic extension of F, then L is algebraic extension of F.
• A complex number is said to be algebraic
number ⇔ it is algebraic over the field of rationals.
• A complex number which is not algebraic is called transcendental number.
• π is a transcendental number.
UNIT-II
33
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations
UNIT-III
34
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • An algebra A over a field F is an associative
ring which also a vector space over F such that ∀ a, b ∈ A, and ∀ α ∈ F, α(ab) = (αa)b = a(αb).
UNIT-III
34
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • An algebra A over a field F is an associative
ring which also a vector space over F such that ∀ a, b ∈ A, and ∀ α ∈ F, α(ab) = (αa)b = a(αb).
• If V is a vector space over a field F, then Hom(V, V) is an algebra with unit element over F.
UNIT-III
34
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • An algebra A over a field F is an associative
ring which also a vector space over F such that ∀ a, b ∈ A, and ∀ α ∈ F, α(ab) = (αa)b = a(αb).
• If V is a vector space over a field F, then Hom(V, V) is an algebra with unit element over F.
• Hom(V, V) = AF(V) = A(V).
UNIT-III
34
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • An algebra A over a field F is an associative
ring which also a vector space over F such that ∀ a, b ∈ A, and ∀ α ∈ F, α(ab) = (αa)b = a(αb).
• If V is a vector space over a field F, then Hom(V, V) is an algebra with unit element over F.
• Hom(V, V) = AF(V) = A(V).
• Members of A(V) are called as linear transformations.
UNIT-III
34
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations
UNIT-III
35
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If A is an algebra with unit element over F,
then A is isomorphic to a subalgebra of A(V) for some appropriate vector space V over F.
UNIT-III
35
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If A is an algebra with unit element over F,
then A is isomorphic to a subalgebra of A(V) for some appropriate vector space V over F.
• If A is an n-dimensional algebra with unit element over F, then every element of A satisfies a non-trivial polynomial of degree at most n.
UNIT-III
35
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If A is an algebra with unit element over F,
then A is isomorphic to a subalgebra of A(V) for some appropriate vector space V over F.
• If A is an n-dimensional algebra with unit element over F, then every element of A satisfies a non-trivial polynomial of degree at most n.
• If A is an algebra with unit element over F, then for a ∈ A, the minimal polynomial of a is the non-zero polynomial of smallest degree satisfied by a.
UNIT-III
35
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations
UNIT-III
36
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • T ∈ A(V) said to be left invertible ⇔ ∃
S ∈ A(V) such that ST = 1.
UNIT-III
36
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • T ∈ A(V) said to be left invertible ⇔ ∃
S ∈ A(V) such that ST = 1.
• T ∈ A(V) said to be right invertible ⇔ ∃ S ∈ A(V) such that TS = 1.
UNIT-III
36
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • T ∈ A(V) said to be left invertible ⇔ ∃
S ∈ A(V) such that ST = 1.
• T ∈ A(V) said to be right invertible ⇔ ∃ S ∈ A(V) such that TS = 1.
• T ∈ A(V) said to be invertible/regular ⇔ ∃ S ∈ A(V) such that ST = TS = 1.
UNIT-III
36
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • T ∈ A(V) said to be left invertible ⇔ ∃
S ∈ A(V) such that ST = 1.
• T ∈ A(V) said to be right invertible ⇔ ∃ S ∈ A(V) such that TS = 1.
• T ∈ A(V) said to be invertible/regular ⇔ ∃ S ∈ A(V) such that ST = TS = 1.
• T ∈ A(V) said to be singular ⇔ it is not invertible.
UNIT-III
36
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • T ∈ A(V) said to be left invertible ⇔ ∃
S ∈ A(V) such that ST = 1.
• T ∈ A(V) said to be right invertible ⇔ ∃ S ∈ A(V) such that TS = 1.
• T ∈ A(V) said to be invertible/regular ⇔ ∃ S ∈ A(V) such that ST = TS = 1.
• T ∈ A(V) said to be singular ⇔ it is not invertible.
• If V is a finite dimensional vector space over F, then T ∈ A(V) is invertible ⇔ the constant term in its minimal polynomial is not zero.
UNIT-III
36
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations
UNIT-III
37
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If V is a finite dimensional vector space over F
and T ∈ A(V) is invertible ⇒ T−1 is a polynomial expression in T over F.
UNIT-III
37
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If V is a finite dimensional vector space over F
and T ∈ A(V) is invertible ⇒ T−1 is a polynomial expression in T over F.
• If V is a finite dimensional vector space over F, then T ∈ A(V) is singular ⇔ ∃ 0 ≠ S ∈ A(V) such that TS = TS = 0.
UNIT-III
37
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If V is a finite dimensional vector space over F
and T ∈ A(V) is invertible ⇒ T−1 is a polynomial expression in T over F.
• If V is a finite dimensional vector space over F, then T ∈ A(V) is singular ⇔ ∃ 0 ≠ S ∈ A(V) such that TS = TS = 0.
• If V is a finite dimensional vector space over F and T ∈ A(V) is right invertible ⇒ T is invertible. (but only for finite dimensional vector space)
UNIT-III
37
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations
UNIT-III
38
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If V is a finite dimensional vector space over F,
then T ∈ A(V) is singular ⇔ ∃ 0 ≠ v ∈ V such that vT = 0.
UNIT-III
38
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If V is a finite dimensional vector space over F,
then T ∈ A(V) is singular ⇔ ∃ 0 ≠ v ∈ V such that vT = 0.
• If V is a vector space over F, then range of T is VT = {vT | v ∈ V}.
UNIT-III
38
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If V is a finite dimensional vector space over F,
then T ∈ A(V) is singular ⇔ ∃ 0 ≠ v ∈ V such that vT = 0.
• If V is a vector space over F, then range of T is VT = {vT | v ∈ V}.
• If V is a vector space over F, then range of T is VT is a subspace of V.
UNIT-III
38
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If V is a finite dimensional vector space over F,
then T ∈ A(V) is singular ⇔ ∃ 0 ≠ v ∈ V such that vT = 0.
• If V is a vector space over F, then range of T is VT = {vT | v ∈ V}.
• If V is a vector space over F, then range of T is VT is a subspace of V.
• If V is a vector space over F, then T ∈ A(V) is invertible ⇔ VT = V, i.e., T is onto.
UNIT-III
38
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • If V is a finite dimensional vector space over F,
then T ∈ A(V) is singular ⇔ ∃ 0 ≠ v ∈ V such that vT = 0.
• If V is a vector space over F, then range of T is VT = {vT | v ∈ V}.
• If V is a vector space over F, then range of T is VT is a subspace of V.
• If V is a vector space over F, then T ∈ A(V) is invertible ⇔ VT = V, i.e., T is onto.
• If V is a vector space over F, then for T ∈ A(V) rank of T, denoted by r(T) = dim VT
UNIT-III
38
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations
UNIT-III
39
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • Properties of Rank of rank of linear
transformation :
UNIT-III
39
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • Properties of Rank of rank of linear
transformation :
�r(ST) ≤ r(T)
UNIT-III
39
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • Properties of Rank of rank of linear
transformation :
�r(ST) ≤ r(T)
�r(TS) ≤ r(T)
UNIT-III
39
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • Properties of Rank of rank of linear
transformation :
�r(ST) ≤ r(T)
�r(TS) ≤ r(T)
�r(ST) ≤ min{r(S), r(T)}
UNIT-III
39
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • Properties of Rank of rank of linear
transformation :
�r(ST) ≤ r(T)
�r(TS) ≤ r(T)
�r(ST) ≤ min{r(S), r(T)}
� If S is invertible, then r(ST) = r(TS) = r(T)
UNIT-III
39
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • Properties of Rank of rank of linear
transformation :
�r(ST) ≤ r(T)
�r(TS) ≤ r(T)
�r(ST) ≤ min{r(S), r(T)}
� If S is invertible, then r(ST) = r(TS) = r(T)
� If S is invertible, then r(STS−1) = r(T)
UNIT-III
39
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Algebra of Linear Transformations • Properties of Rank of rank of linear
transformation :
�r(ST) ≤ r(T)
�r(TS) ≤ r(T)
�r(ST) ≤ min{r(S), r(T)}
� If S is invertible, then r(ST) = r(TS) = r(T)
� If S is invertible, then r(STS−1) = r(T)
• If V is a vector space over F, then T ∈ A(V) is invertible ⇔ image under T of linearly independent set is linearly independent set.
UNIT-III
39
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots
UNIT-III
40
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • λ ∈ F is said to be characteristic root of
T ∈ A(V) ⇔ λ − T is singular.
UNIT-III
40
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • λ ∈ F is said to be characteristic root of
T ∈ A(V) ⇔ λ − T is singular.
• λ ∈ F is said to be characteristic root of T ∈ A(V) ⇔ ∃ 0 ≠ v ∈ V such that vT = λv.
UNIT-III
40
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • λ ∈ F is said to be characteristic root of
T ∈ A(V) ⇔ λ − T is singular.
• λ ∈ F is said to be characteristic root of T ∈ A(V) ⇔ ∃ 0 ≠ v ∈ V such that vT = λv.
• λ ∈ F is characteristic root of T ∈ A(V) ⇒ for any polynomial q(x) with coefficients in F, q(λ) is characteristic root of q(T).
UNIT-III
40
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • λ ∈ F is said to be characteristic root of
T ∈ A(V) ⇔ λ − T is singular.
• λ ∈ F is said to be characteristic root of T ∈ A(V) ⇔ ∃ 0 ≠ v ∈ V such that vT = λv.
• λ ∈ F is characteristic root of T ∈ A(V) ⇒ for any polynomial q(x) with coefficients in F, q(λ) is characteristic root of q(T).
• Every characteristic root of T is a root of minimal polynomial of T.
UNIT-III
40
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • λ ∈ F is said to be characteristic root of
T ∈ A(V) ⇔ λ − T is singular.
• λ ∈ F is said to be characteristic root of T ∈ A(V) ⇔ ∃ 0 ≠ v ∈ V such that vT = λv.
• λ ∈ F is characteristic root of T ∈ A(V) ⇒ for any polynomial q(x) with coefficients in F, q(λ) is characteristic root of q(T).
• Every characteristic root of T is a root of minimal polynomial of T.
• If S ∈ A(V) is regular, the T ∈ A(V) and STS−1 have the same minimal polynomial.
UNIT-III
40
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots
UNIT-III
41
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • λ ∈ F is characteristic root of T ∈ A(V) ⇒ a
non-zero vector v for which vT = λv is called characteristic vector T belonging to characteristic root λ.
UNIT-III
41
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • λ ∈ F is characteristic root of T ∈ A(V) ⇒ a
non-zero vector v for which vT = λv is called characteristic vector T belonging to characteristic root λ.
• λ1, λ2, ⋯⋯⋯⋯ , λn ∈ F are distinct characteristic root of T ∈ A(V) and v1, v2, ⋯⋯⋯⋯ , vn are characteristic vectors of T belonging to λ1, λ2, ⋯⋯⋯⋯ , λn, then v1, v2, ⋯⋯⋯⋯ , vn are linearly independent.
UNIT-III
41
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Characteristic Roots • λ ∈ F is characteristic root of T ∈ A(V) ⇒ a
non-zero vector v for which vT = λv is called characteristic vector T belonging to characteristic root λ.
• λ1, λ2, ⋯⋯⋯⋯ , λn ∈ F are distinct characteristic root of T ∈ A(V) and v1, v2, ⋯⋯⋯⋯ , vn are characteristic vectors of T belonging to λ1, λ2, ⋯⋯⋯⋯ , λn, then v1, v2, ⋯⋯⋯⋯ , vn are linearly independent.
• If dim V = n and some T ∈ A(V) has n distinct characteristic roots, then V has a basis consisting only if characteristic vectors of T.
UNIT-III
41
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices
UNIT-III
42
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices • If V is a n-dimensional vector space over F, v1,
v2, ⋯⋯⋯⋯ , vn is a basis of V, then for a T ∈ A(V), matrix of T in basis is (αij) where
UNIT-III
42
∑αn
i
jiji vTv1=
=
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices • If V is a n-dimensional vector space over F, v1,
v2, ⋯⋯⋯⋯ , vn is a basis of V, then for a T ∈ A(V), matrix of T in basis is (αij) where
• For a field F, the set Fn of all n × n matrices over F, is an algebra with unit element over F.
UNIT-III
42
∑αn
i
jiji vTv1=
=
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices • If V is a n-dimensional vector space over F, v1,
v2, ⋯⋯⋯⋯ , vn is a basis of V, then for a T ∈ A(V), matrix of T in basis is (αij) where
• For a field F, the set Fn of all n × n matrices over F, is an algebra with unit element over F.
• If V is a n-dimensional vector space over F, then V is isomorphic to Fn.
UNIT-III
42
∑αn
i
jiji vTv1=
=
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices
UNIT-III
43
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices • If V is a n-dimensional vector space over F,
T ∈ A(V) has a matrix m1(T) in a basis v1, v2, ⋯⋯⋯⋯ , vn, and a matrix m2(T) in a basis w1, w2, ⋯⋯⋯⋯ , wn, then there exists a matrix C such that m2(T) = Cm1(T)C−1, in fact, one of the values of C can be chosen to be m1(S) where S is linear transformation mapping v1, v2, ⋯⋯⋯⋯ , vn, to w1, w2, ⋯⋯⋯⋯ , wn.
UNIT-III
43
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Matrices • If V is a n-dimensional vector space over F,
T ∈ A(V) has a matrix m1(T) in a basis v1, v2, ⋯⋯⋯⋯ , vn, and a matrix m2(T) in a basis w1, w2, ⋯⋯⋯⋯ , wn, then there exists a matrix C such that m2(T) = Cm1(T)C−1, in fact, one of the values of C can be chosen to be m1(S) where S is linear transformation mapping v1, v2, ⋯⋯⋯⋯ , vn, to w1, w2, ⋯⋯⋯⋯ , wn.
• Using this result, we can obtain different matrices of same linear transformation in different bases of the vector space.
UNIT-III
43
Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.
॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Linear Algebra LINEAR ALGEBRA
44