Answer to Question #24294 in Linear Algebra for Matthew Lind

Question

Let W = {w^1,....,w^N} be a spanning set for a subspace W in R^K, and U = {u^1,....,u^M} a linearly independent subset of W. Let A be the K by M matrix whose columns are the vectors u^m, and B the K by N matrix whose column vectors are w^n. Then there is an N by M matrix D such that A = BD. Why??

(hint: prove by showing that, if M>N, then there is A non-zero vector x with Dx = 0.)