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.

example_row_space

example_column_space

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/


要求最佳拟合直线,即求误差向量最小的时候。只要将被拟合曲线投影到列空间,此时误差向量最小。同时误差向量落在左零空间。也正因为误差向量落在左零空间,所以误差向量与落在列向量空间的投影向量垂直,反过来,也因为误差向量与投影向量垂直,所以此时误差最小。

所以方程Ax=b有精确解的时候,b就必须落在A的列向量空间中。而当b不落在A的列向量空间的时候,b有最小误差解。而要找到最小误差解,则只需要将b投影到A的列空间中,此时误差向量垂直与投影向量指向b,根据勾股定理此时误差最小。同时,值得注意的是误差向量落在了A的左零空间中。

从另一个角度思考,则可以看作是向量b在空间中的分解。

column_space_left_nullspace

从此也可以看出,列向量空间与左零空间正交,且互为补集。他们共同构成了b所在的空间。

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