linear algebra notes 08//13/2017 lecture 5 & 6

lecture 5:

LU 分解的意义：
http://blog.csdn.net/carrierlxksuper/article/details/8487276

permutation:
P-1 = Pt
一共n!个permutation matrix

transpose 的概念

symmetric matrix：
At = A

Rt*R is always symmetric.
WHY? Take transpose!
(RtR)t = Rt*Rtt = Rt*R  （和逆矩阵的公式很像 (AB)-1 = B-1A-1）
所以这个是symmetric的


Vector space
Example: R2 = all 2 dimensional vectors

a vector space inside R2 is : a subspace of R2
一定得过零点

subspace of R2
1. all of R2 (最大子空间)
2. any lines through [0;0]
3. zero vector only （最小子空间）
子空间 补充：https://www.zhihu.com/question/48849797

column space: columns in Rn. Add their combinations form a subspace . (满足定义)


lecture 6:

Ax=b 有解 当b in A的column spaces (根据定义来的)

主列

null space (零空间) of A
all vectors x to let Ax = 0
at least contains [0;0;0]

solutions to Ax = 0 always gives a subspace.

subspace 一定会过零点