1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted