Originally Posted by

**HallsofIvy** I strongly suspect that means you need to learn and use the definitions. In mathematics definitions are "working definitions". You use the specific words of definitions in proofs.

Suppose B is a linear transformation from U to V and A a linear transformation from V to W. Then AB is a linear transformation from U to W.

"col(A)", the column space of A, is defined as the subspace of W consisting of all vectors, w in W, such that w= Av for some v in V. Similarly, "col(AB)" is defined as the set of all vectors, w' in W, such that w'= ABu for some u in U.

Let v= Bu.

"nul(B), the null space of B, is defined as the subspace of U consisting of all vectors u such that Bu= 0. Similarly, "nul(AB)" is the subspace of U consisting of all vectors u such that ABu= A(Bu)= 0.

Use the fact that, for any linear transformation, A, A(0)= 0.