人工智能教程 - 數學基礎課程1.3 - 高等代數(一)-10,11 四個基本子空間,小世界圖

四個基本子空間

(4 subspaces) A is

• column space C(A) in $R ^m$
• null space N(A)
• row space = all combs of rows
  =all combs of columns of A = C($A^T$) in $R ^n$
• null space of $A^T=N(A^T) in R^m$

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矩陣空間，秩1矩陣和小世界圖

Bases of new vectors spaces
Rank one matrices Small world graphs
M = all 3 by 3’s

$\frac{d^2y}{dx^2}+y=0$

y = cosx, sinx (2個basis)

Complete solution :

$y = c_1.cosx+c_2.sinx$

In $R^4 ,\ \ \ v=\begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \\ \end{bmatrix}$

S = all v in $R^4$ with $v_1+v_2+v_3+v_4=0$ 是subspace
S = nullspace of A [ 1 1 1 1 ] r=1 dimN(A)=d-r = 3 所以是3維

$\begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \\ \end{bmatrix}，\begin{bmatrix} -1 \\ 0 \\ 1 \\ 0 \\ \end{bmatrix}，\begin{bmatrix} -1 \\ 0 \\ 0 \\ 1 \\ \end{bmatrix}$

Graphs ={nodes, edges}