# Linear Algebra Four fundamental subspaces small proof.

1. Oct 14, 2012

### wjv4

1. The problem statement, all variables and given/known data

Given A$\in$ $M$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) = {$\vec{x}$$\in$ ℝn such that $\vec{x}$ = $\vec{u}$-A$\vec{u}$ for some $\vec{u}$ $\in$ℝn}

N(A) = {$\vec{x}$$\in$ℝn such that $\vec{x}$ = A$\vec{x}$}

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.

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