Row reduce in order to find the pivotal columns:
At this point, the REF form tells us that the rank of the matrix is 3. Thus the column space (image) of the matrix is given by the first three original columns:
The pivotal columns of matrix
are the pivotal rows of
so the rows with pivots in the original matrix form the basis of the row space (coimage):
In order to find the nullspace we could either do back-substitution at this point with the REF matrix or take the matrix all the way to RREF:
Converting back to equation form with
gives the following:
Thus, we can write the basis for the null space (kernel) as the following:
