If A is a square matrix, and A² = A, prove that the column space C(A) = { x ∊ Rⁿ | x = Ax }

Question

Here is my attempt at the proof.

See that A can be written out as

                             [a₁₁ a₁₂....a₁n]
                             [a₂₁........a₂n]
                             [       .      ]
                             [       .      ]
                             [       .      ]         
                             [an₁........ann]

Because A² = A, see that rᵢcᵤ = aᵢᵤ ; where rᵢ and cᵤ are the rows and columns of A; i, u ∊ N⁺ and i, u <= n

(This is the part where I'm confused, I should be letting x ∊ Rⁿ | x = Ax and then proving that x ∊ C(A) right?... anyway what I did was let x ∊ C(A) and then proved that x ∊ Rⁿ | x = Ax)

Let x ∊ C(A) where x = cᵤ ; some column of A. See that A*x will leave x unchanged thus leaving you with Ax = x

Thus C(A) = {x ∊ Rⁿ | x = Ax}.

Note: Writing this proof out has made me even more sure that I'm wrong, but please if you have any input please post it. Thanks to everyone who reads this.

Answer 1

One thing you should not even attempt is using the matrix entries explicitly.

Let $U=\{x:x=Ax\}$, for simplicity. Then, obviously, $U\subseteq C(A)$, because any vector of the form $Av$ belongs to $C(A)$.

Now, suppose $x=Av\in C(A)$. Then $Ax=A^2v=\dotsb$