Notes of Linear Algebra Review

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Notes of Linear Algebra Review

Matrix

\(\mathbb{R} ^ {3 \times 2}\):

\[ \begin{bmatrix} a & b \newline c & d \newline e & f \newline \end{bmatrix} \]

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2
3
A = [1, 2; 3, 4; 5, 6]

[rows, columns] = size(A)

\(A_{ij}\): element in the ith row and jth column of matrix \(A\)

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A(3, 2)

Vector

\(\mathbb{R} ^ {3}\):

\[ \begin{bmatrix} x \newline y \newline z \newline \end{bmatrix} \]

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v = [1; 2; 3]

\(v_i\): element in the ith row of the vector \(v\)

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v(3)

Addition

\[ \begin{bmatrix} a & b \newline c & d \newline \end{bmatrix} + \begin{bmatrix} w & x \newline y & z \newline \end{bmatrix} = \begin{bmatrix} a + w & b + x \newline c + y & d + z \newline \end{bmatrix} \]

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2
3
4
A = [1, 2, 3; 4, 5, 6]
B = [1, 4, 7; 2, 5, 8]

A + B

Scalar Multiplication

\[ \begin{bmatrix} a & b \newline c & d \newline \end{bmatrix} \cdot x = \begin{bmatrix} a \cdot x & b \cdot x \newline c \cdot x & d \cdot x \newline \end{bmatrix} \]

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2
3
4
A = [1, 2, 3; 4, 5, 6]
s = 2

A * s

Matrix-Vector Multiplication

\[ \begin{bmatrix} a & b \newline c & d \newline e & f \newline \end{bmatrix} \cdot \begin{bmatrix} x \newline y \newline \end{bmatrix} = \begin{bmatrix} a \cdot x + b \cdot y \newline c \cdot x + d \cdot y \newline e \cdot x + f \cdot y \newline \end{bmatrix} \]

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2
3
4
A = [1, 2, 3; 4, 5, 6; 7, 8, 9]
v = [1; 1; 1]

A * v

Matrix-Matrix Multiplication

\[ \begin{bmatrix} a & b \newline c & d \newline e & f \newline \end{bmatrix} \cdot \begin{bmatrix} w & x \newline y & z \newline \end{bmatrix} = \begin{bmatrix} a \cdot w + b \cdot y & a \cdot x + b \cdot z \newline c \cdot w + d \cdot y & c \cdot x + d \cdot z \newline e \cdot w + f \cdot y & e \cdot x + f \cdot z \newline \end{bmatrix} \]

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2
3
4
A = [1, 2; 3, 4; 5, 6]
B = [1; 2]

A * B

Matrix Multiplication Properties

Not Commutative: \(A B \neq B A\) in general

Associative: \((A B) C = A (B C)\)

Identity: \(I\) (or \(I_{n \times n}\))   \(s.t.\)   \(A I = I A = A\)

\[ I_{3 \times 3} = \begin{bmatrix} 1 & 0 & 0 \newline 0 & 1 & 0 \newline 0 & 0 & 1 \newline \end{bmatrix} \]

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I = eye(3)

Inverse and Transpose

Transposition: \(A^T\)   \(s.t.\)   \(A_{ij} = (A^T)_{ji}\)

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A = [1, 2, 3; 4, 5, 6; 7, 8, 9]

A'

Inverse: \(A^{-1}\)   \(s.t.\)   \(A A^{-1} = A^{-1} A = I\)

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2
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A = [1, 2, 0; 0, 5, 6; 7, 0, 9]

inv(A)

Matrices that don't have an inverse are singular or degenerate

Pseudoinverse: \(A^{+}\)   \(s.t.\)   \(A A^{+} A = A\),   \(A^{+} A A^{+} = A^{+}\)

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2
3
A = [1, 2, 0; 0, 5, 6; 7, 0, 9]

pinv(A)