I've been brushing up on some linear algebra this summer, and came across this problem in a practice book. Any help on this?
Let A be an m x n matrix. Prove that if c is any nonzero scalar, then rank(cA) = rank(A).
Any m by n matrix is a linear transformation from to and, if its rank is k, its image is k dimensional subspace of . If you know the image of A, what can you say about the image of cA? In particular, if Ax= y, what is (cA)x?
Another way to thin about it is that the rank of a matrix is the number of linearly independent rows (or columns, which ever is smaller) there are in a matrix. So what happens when you multiply everything by some scalar?