Four fundamental subspaces( for matrix A)

 


if A is m by n matrix:

Column space  C(A) in Rm (列空间在m维实空间中)

Null space N(A) in Rn

Row space C(A^)(^代表转置)in Rn (all combinations of rows=all columns of A^)

Null space of A^ N(A^) in Rm  (left null space of A   左零空间)

C(R) ≠ C(A)

different column space, same row space

“行变换不会对行空间产生影响,但“列空间”发生了变化

Basis for row space is first r rows of R

It’s called 基的最简形式

考虑:N(A^)

观察以上形式,所以称之为左零空间

由Gauss-Jordan 方法可得:

我们在矩阵右侧增广矩阵,并进行相同的变换

let’s check

求矩阵的左零空间,试着找一个产生零行向量的行组合

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