MIT线性代数笔记-第五讲

在没有进行row exchange的情况下A = LU
那么有row exchange的情况呢?

PA = LU
P为对A进行行交换的矩阵
例如:

⎡⎣⎢010100001⎤⎦⎥ [ 0 1 0 1 0 0 0 0 1 ]

这是一个对A中的第一行和第二行进行交换的矩阵

P的个数与满排列的个数一致，即n!

P满足如下性质；
P−1=PT P − 1 = P T , PTP=I P T P = I

矩阵的转秩
(AT)ij=Aji ( A T ) i j = A j i

RTR R T R is always symmetric,Why?
(RTR)T=RT(RT)T=RTR ( R T R ) T = R T ( R T ) T = R T R

向量空间
定义:The space Rn R n consists of all column vectors with n component
R2 R 2 = all 2 dim real vectors
Rn R n = all vectors with n real component

注意:We can add any vectors in Rn R n , and we can multiply any vector v by any scalar c,and the result is in the space

子空间(subspace)
定义

subspace of R2 R 2
1. R2 R 2
2.any line through

[00] [ 0 0 ]

3.zero vector only Z

subspace of R3 R 3
1. R3 R 3
2.plane through zero vector
3.line through zero vector
4.zero vector

The column space of A
定义:
The column space consists of all linear combinations of the columns.The combinations are all possible vectors Ax.They fill the column space C(Aa)

This column space is crucial to the whole book, and here is why. To solve Ax = b isto express b as a combination of the columns. The right side b has to be in the column space produced by A on the left side, or no solution!
如果想解出Ax = b,那么b必为A的列的线性组合中的一个，即b为A的列空间中的一个列向量

The system Ax = b is solvable if and only if b is in the column space of Aa