Four Fundamental Matrix Spaces
- Row space of A
- Column space of A
- Nullspace of A
- 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:
- The row vectors with the leading 1’s form a basis for the row space of R.
- 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.