Linear Algebra review (optional) 线代复习（选学）

Matrix

Definition: Rectangle array of numbers. 矩阵是矩形数字数组
Dimension of matrix: number of rows ( imes) number of columns.

矩阵的维度：行数 ( imes) 列数，如2 ( imes) 3的矩阵，4 ( imes) 2的矩阵…
Matrix Element(entries if matrix): (A_{ij}) = "(i, j) entry" in the (i^{th}) row, (j^{th}) column.

Vector

Definition: An n ( imes) 1 matrix. 向量是一个n ( imes) 1的矩阵
Also called: n-dimensional vector. 也叫作n维向量
Vector Element: (y_i = i^{th}) element.

Addition and Scalar Multiplication 加法和标量乘法运算

Matrix Addition

Scalar Multiplication

Combination of Operands

Matrix Vector Multiplication 矩阵和向量的乘法

Matrix Matrix Multiplication 矩阵和矩阵的乘法

House sizes:

[2104, 1416, 1534, 852]

Have 3 competing hypotheses:

(h_{ heta}(x) = -40 + 0.25 imes x)
(h_{ heta}(x) = 200 + 0.1 imes x)
(h_{ heta}(x) = -150 + 0.4 imes x)

Matrix:

( left[ egin{matrix} 1 & 2104 \ 1 & 1416 \ 1 & 1534 \ 1 & 852 end{matrix} ight] ) ( imes) ( left[ egin{matrix} -40 & 200 & -150 \ 0.25 & 0.1 & 0.4 \ end{matrix} ight] ) = ( left[ egin{matrix} 486 & 410 & 692 \ 314 & 342 & 416 \ 344 & 353 & 464 \ 173 & 285 & 191 end{matrix} ight] )

Matrix Multiplication Properties 矩阵乘法的特性

不满足交换律：A ( imes) B ( eq) B ( imes) A （单位矩阵除外）

满足结合律：A ( imes) (B ( imes) C) = (A ( imes) B) ( imes) C

Inverse and Transpose 矩阵的逆运算和转置运算

Matrix inverse: 矩阵的逆

If A is an m ( imes) m matrix, and if it has an inverse, (AA^{-1} = A^{-1}A = E).

Matrix Transpose: 矩阵的转置

Let A be an m ( imes) n matrix, and let (B =) (A^T).

Then B is an n ( imes) m matrix, and (B_{ij} = A_{ji}).

Example:

(A =) ( left[ egin{matrix} 1 & 2 & 0 \ 3 & 5& 9 \ end{matrix} ight] )， $B = A^T = $ ( left[ egin{matrix} 1 & 3 \ 2 & 5 \ 0 & 9 \ end{matrix} ight] )