Direct sum of the column and nullspace of A

Question

Suppose that $A^2=A$ and $x \in \mathbb{F}^{n \times 1}$. What can be said of $x-Ax$? Show that $\mathbb{F}^{n \times 1} = C(A) \oplus N(A)$.

To show that $\mathbb{F}^{n \times 1} = C(A) \oplus N(A)$, we need to verify that

1. $\mathbb{F}^{n \times 1} = C(A) + N(A)$

Can I use the rank-nullity theorem here?

dim $\mathbb{F}^{n \times 1} = \ $dim$ \ $C(A)$ \ + \ $dim$\ $$N(A)$

dim $\mathbb{F}^{n \times 1}$ = rank A + nullity A

I don't know how to proceed from here.

2. $v \in C(A) \cap N(A)$ = 0

but how to do that is I don't know.

Can someone please help me.

Answer 1

A quick thought: $x-Ax$ is an eigenvector of $A$ with the eigenvalue $\lambda=0$, since $A(x-Ax)=Ax-A^2x=Ax-Ax=0x.$