人工智能教程 - 数学基础课程1.3 - 高等代数(一)-10,11

四个基本子空间

(4 subspaces) A is

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

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矩阵空间，秩1矩阵和小世界图

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

$\frac{d^2y}{dx^2}+y=0$dx2d2y​+y=0

y = cosx, sinx (2个basis)

Complete solution :

$y = c_1.cosx+c_2.sinx$y=c1​.cosx+c2​.sinx

In $R^4 ,\ \ \ v=\begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \\ \end{bmatrix}$R4,   v=⎣⎢⎢⎡​v1​v2​v3​v4​​⎦⎥⎥⎤​

S = all v in $R^4$R4 with $v_1+v_2+v_3+v_4=0$v1​+v2​+v3​+v4​=0 是subspace
S = nullspace of A [ 1 1 1 1 ] r=1 dimN(A)=d-r = 3 所以是3维

$\begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \\ \end{bmatrix}，\begin{bmatrix} -1 \\ 0 \\ 1 \\ 0 \\ \end{bmatrix}，\begin{bmatrix} -1 \\ 0 \\ 0 \\ 1 \\ \end{bmatrix}$⎣⎢⎢⎡​−1100​⎦⎥⎥⎤​，⎣⎢⎢⎡​−1010​⎦⎥⎥⎤​，⎣⎢⎢⎡​−1001​⎦⎥⎥⎤​

Graphs ={nodes, edges}