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Prove that Col(A) is a proper subset of Nul(A)

  1. Mar 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that if A is a nxn matrix such that A^2=0, then Col(A) is a proper subset of Nul(A)


    2. Relevant equations



    3. The attempt at a solution

    None, i have no idea how to start.
    Please guide me or explain what does the question means and how to approach.
     
  2. jcsd
  3. Mar 5, 2012 #2

    tiny-tim

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    hi junsugal! :smile:

    (do you mean Ker(A)? it's spelled "kernel", not "colonel" … a kernel is the soft part inside the shell of a nut! :biggrin:)

    it means you have to prove that there exists an x such that x is in Nul(A) but not Col(A)

    (i think the question should have stipulated that A is non-zero)
     
  4. Mar 5, 2012 #3
    Hi :)

    It is Col(A).
    I thought it means that Nul(A) is a subspace of Col(A)?
    well, I'm still confused though.
    I tried to solve this problem asuming that A is zero matrix.
    Because I couldn't think of any nxn matrix that will get zero after multiply by itself.
     
  5. Mar 5, 2012 #4

    tiny-tim

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    no …
    … means that Col is a subset of Nul, but is less than Nul
    no!

    A must be non-zero
     
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