## 2016-12-04

### 如何理解四个基本子空间

Four Fundamental Matrix Spaces

1. Row space of A

2. Column space of A

3. Nullspace of A

4. Nullspace of A’

A system of linear equations Ax=b is consistent if and only if b is in the column space of A.

We already known that elementary row operations do not change the row space of a matrix.

And elementary row operations do not change the nullspace of matrix.

However this result dost not apply to the column space. Therefore only its row space is preserved under elementary operations.

If a matrix R is in row-echelon form then:

1. The row vectors with the leading 1’s form a basis for the row space of R.

2. The column vector with the leading 1’s of the row vectors form a basis for the column space.

Therefor we put it in row-echelon form and extract the row vectors with a leading 1 to find a basis for the row space of a matrix.

Relationships between four vector space

1. If A is any matrix then row space and column space of A have the same dimension.

2. The common dimension of the row space and column space of a matrix A is called the rank of A and is denoted by rank(A);

3. The dimension of the nullspace of A is called the nullity of A and is denoted by nullity(A)

4. Rank(A)=rank(A’)

5. Rank(A)+nullity(A)=n (A is a matrix with n column)

6. Rank(A)=the number of leading variable in the solution of Ax=0.

7. Nullity(A)=the number of parameters in the general solution of Ax=0.

PPT: http://slideplayer.com/slide/8413041/

1. 我是来测试的

2. 好，我知道了