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Linear Algebra
Prof. Seungchul Lee
Industrial AI Lab.
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Linear Equations
• Two linear equations
• In a vector form
• Solution using inverse
1 2
1 2
4 5 13
2 3 9
x x
x x
1
2
4 5 13, ,
2 3 9
xA x b
x
1 1
1
Ax b
A Ax A b
x A b
2
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System of Linear Equations
• Consider a system of linear equations
• Can be written in a matrix form as 𝑦 = 𝐴𝑥, where
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System of Linear Equations
1) Well-determined linear systems
2) Under-determined linear systems
3) Over-determined linear systems
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1) Well-Determined Linear Systems
• System of linear equations
• Geometric point of view
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1) Well-Determined Linear Systems
• System of linear equations
• Matrix form
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2) Under-Determined Linear Systems
• System of linear equations
• Geometric point of view
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2) Under-Determined Linear Systems
• System of linear equations
• Matrix form
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3) Over-Determined Linear Systems
• System of linear equations
• Geometric point of view
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3) Over-Determined Linear Systems
• System of linear equations
• Matrix form
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Vector-Vector Products
• Inner product: 𝑥, 𝑦 ∈ ℝ𝑛
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Norm
• 𝑙2 norm
• 𝑥 Measures length of vector (from the origin)
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Orthogonality
• Two vectors 𝑥, 𝑦 ∈ ℝ𝑛 are orthogonal if
• They are orthonormal if
and
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Angle between Vectors
• (unsigned) angle between vectors in ℝ𝑛 defined as
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Half Space
• ȁ𝑥 𝑥𝑇𝑦 ≤ 0 defines a half space with outward normal vector 𝑦, and boundary passing through 0
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