Let be a field and let be a linear transformation. Fix the standard bases and . Prove that the image of is precisely the span of the columns of , considered as elements of . Deduce that the rank of as a linear transformation is the column rank of .
Suppose , and say . Now . (Note that the are precisely the columns of .) Conversely, we can see that every linear combination of the columns of is in .
Recall that , and that is the maximal number of linearly independent columns of . Because the columns of are a generating set for , they contain a basis for . The size of this basis is precisely , so that in particular, all such subsets of the columns of have the same size and are maximal linearly independent subsets. Thus .