The column space of a matrix
is a subspace of
Suppose the matrix
are the columns of
. Then the column space of
is the set of vectors in
which forms a linear combination of the columns of
To see if a vector
is in the column space of
, we need to see if
is a linear combination of the columns of
. In other words,
are solutions to the linear equation.
To find a basis for the column space of a matrix A, we:
1) Row reduce the matrix to echelon form.
2) Circle the columns with pivots in the row-reduced matrix.
3) Go back to the original matrix and circle the columns with the same positions.
4) Use those columns to write out the basis for
Note that the vectors in the basis are linearly independent.