Linear Algebra

Prof. Seungchul Lee

Industrial AI Lab.

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

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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