# Thread: Col (AB) and Nul (AB)

1. ## Col (AB) and Nul (AB)

Hi, I'm totally stuck on this question.

Show that Col (AB) is contained in Col A, and that Nul B is contained in Nul (AB). Deduce that rank AB <= (smaller or equal) rank A and rank AB <= rank B.

I can easily do computations, but I don't know how to approach a conceptual question without any numbers...

Thanks.

2. 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.

3. 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.
Did you just recall the definitions from memory?

I have a really hard time memorizing linear algebra definitions.

4. Originally Posted by CaesarXXIV
Did you just recall the definitions from memory?

I have a really hard time memorizing linear algebra definitions.
Don't memorize. Understand.

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### col(ab) subspace col(a)

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