MIT线性代数笔记-第六讲

Vector Space requirements

v + w and cv are in the space

all combs of cv + dw are in the space

intersection of S and T is a subspace

Column Space

definition:

linear combination of columns of A

Does Ax = b have a solution for every b?

which b’s allow this system to be solved?

I can solve Ax = b when b is a vector in the column space of A,C(A)

Null Space

⎡⎣⎢⎢⎢123411112345⎤⎦⎥⎥⎥⎡⎣⎢x1x2x3⎤⎦⎥=⎡⎣⎢000⎤⎦⎥[112213314415][x1x2x3]=[000]

满足这个等式的x即为A的null Space

为什么说Null Space 是一个Space?

回想下vector space的性质.

在vector space中的任两个向量的线性组合仍然在vector space中

假设Av = 0,Aw = 0,那么A(v + W) = 0,得证