math 22 linear algebra and its applications - lecture 7
TRANSCRIPT
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Math 22 –
Linear Algebra and its
applications
- Lecture 7 -
Instructor: Bjoern Muetzel
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GENERAL INFORMATION
▪ Office hours: Tu 1-3 pm, Th, Sun 2-4 pm in KH 229
▪ Tutorial: Tu, Th, Sun 7-9 pm in KH 105
▪ Homework: Homework 2 due Wednesday at 4 pm in the boxes
outside Kemeny 008. Separate your homework into
part A, part B, part C and part D and staple it.
▪ Midterm 1: Monday Oct 7 from 4-6 pm in Carpenter 013
Topics: till this Thursday (included)
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1
1.8
Linear Equations
in Linear Algebra
INTRODUCTION TO LINEAR
TRANSFORMATIONS
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LINEAR TRANSFORMATIONS
▪ Summary: If a transformation T from Rn to Rm is
linear, then it maps Euclidean subspaces to Euclidean
subspaces.
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GEOMETRIC INTERPRETATION
▪ Example:
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GEOMETRIC INTERPRETATION
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Slide 1.8- 9
TRANSFORMATIONS
▪
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Slide 1.8- 10
TRANSFORMATIONS
▪
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Slide 1.8- 11
MATRIX TRANSFORMATIONS
▪ Definition: Let A be an matrix. A matrix transformation
is the associated map given by T(x) = Ax, for any x in Rn
▪ For simplicity, we denote such a matrix transformation often by
.
▪ As each column of A has m columns, the domain of T is Rn and
the codomain of T is Rm.
m n
x xA
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Slide 1.8- 12
LINEAR TRANSFORMATIONS
▪
(u v) (u) (v)T T T+ = +
( u) (u)T c cT=
x xA
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LINEAR TRANSFORMATIONS
▪ Consequences:
▪ As and we have:
iii. T(0) = 0
iv.
▪ If v1, …, vp in Rn are vectors then iv. implies that
T(Span{v1, …, vp }) = Span{T (v1), …, T (vp)}
Proof:
1 1 1 1( v ... v ) (v ) ... (v )
p p p pT c c cT c T+ + = + +
(u v) (u) (v)T T T+ = + ( u) (u)T c cT=
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Slide 1.8- 15
LINEAR TRANSFORMATIONS
▪ Consequences:
iv.
▪ In engineering and physics, iv. is referred to as a superposition
principle.
▪ Think of v1, …, vp as signals that go into a system and
T (v1), …, T (vp) as the responses of that system to the signals.
▪ The system satisfies the superposition principle if whenever an
input is expressed as a linear combination of such signals, the
system’s response is the same linear combination of the
responses to the individual signals.
1 1 1 1( v ... v ) (v ) ... (v )
p p p pT c c cT c T+ + = + +
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Slide 1.8- 16
MATRIX TRANSFORMATIONS
▪1 3
3 5
1 7
A
− = −
2u
1
= −
(x) xT A=
a. Find T (u), the image of u under the transformation T.
b. Find an x in Rn whose image under T is b.
c. Is there more than one x whose image under T is b?
d. Determine if c is in the range of the transformation T.
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MATRIX TRANSFORMATIONS
▪ Solution:
.
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MATRIX TRANSFORMATIONS
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SHEAR TRANSFORMATION
▪ 1 3
0 1A
=
(x) xT A=
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SCALAR MULTIPLICATION
▪(x) xT r=
0 1r
1r
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ROTATION
▪