- #1
wjv4
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Homework Statement
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
Homework 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]}
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.