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Math Help - Col (AB) and Nul (AB)

  1. #1
    Newbie CaesarXXIV's Avatar
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    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.
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  2. #2
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    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.
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  3. #3
    Newbie CaesarXXIV's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    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.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by CaesarXXIV View Post
    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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