人工智能教程 - 數學基礎課程1.3 - 高等代數(一)-9 線性相關性,基,維數

線性相關性,基,維數

linear independence

vectors $x_1,x_2,...,x_n$ are independent if no combination gives zero vector : $c_1x_1+c_2x_2+...+c_nx_n\neq0$

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"生成"空間(spanning a space)

vectors $v_1,v_2,...,v_l$ span a space means : The space consists of all combs of those vectors

• shorten : 矩陣的各列生成列空間

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基(basis): 一系列的向量

(a sequence of vectors)

屬性(properties)

• independent
• span a space

Ex:

space is $R^3$

One basis is $\begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix},\begin{bmatrix} 0\\ 0 \\ 1 \\ \end{bmatrix}$

矩陣的秩 = A的列空間的維數

2 = rank(A) = # pivot columns
= dimensions of C(A)