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Linear Algebra Four fundamental subspaces small proof.

  1. Oct 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Given A[itex]\in[/itex] [itex]M[/itex]nxn and A = A2, show that C(A) +N(A) = ℝn.

    note: C(A) means the column space of A.
    N(A) means the null space of A




    2. Relevant equations

    These equations were proved in earlier parts of the problem...

    C(A) = {[itex]\vec{x}[/itex][itex]\in[/itex] ℝn such that [itex]\vec{x}[/itex] = [itex]\vec{u}[/itex]-A[itex]\vec{u}[/itex] for some [itex]\vec{u}[/itex] [itex]\in[/itex]ℝn}

    N(A) = {[itex]\vec{x}[/itex][itex]\in[/itex]ℝn such that [itex]\vec{x}[/itex] = A[itex]\vec{x}[/itex]}



    3. The attempt at a solution

    I feel that my attempt is logical and it works, but I'm not sure if the last step I took works, but if anyone could prove me wrong, confirm that I am right, or offer an alternative, that would be cool! My soln is attached as a picture.
     

    Attached Files:

  2. jcsd
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