Linear Algebra Column and Null Spaces

Question

How would you do a double containment proof of $\mathrm{Nul}(A)=\mathrm{Col}(A-I)$, where $I$ is an identity matrix and $A$ is a square matrix, such that $A^2=A$?

Answer 1

"$\supseteq$": Take a vector $v \in Col(A-I)$. You want to show $Av =0$. Since $v \in Col(A-I)$ there exists a vector $x$ such that $(A - I)x = v$. This gives you $v = Ax - x$. What is $Av$ ?

"$\subseteq$": Take a vector $v \in Nul(A)$. You want to show there is a vector $x$ such that $(A-I)x = v$. Since $v \in Nul(A)$, we have $Av =0$ and adding $-v$ to both sides results in $$(A-I)v = Av - v = 0 - v = -v$$ Can you see what $x$ should be ?

Answer 2

$x \in Null(A) \iff Ax=0 \iff (A-I)x=-x \iff -x \in Col(A-I) \iff x \in Col(A-I) $.