Linear operator image subspace chain

Question

How to prove the proposition: $A: V \to V$ is a linear operator on a finite-dimensional vector space $V $, if $Im(A^p)=Im(A^{p+1})$,then $Im(A^{p+1})=Im(A^{p+2})$

The "Kernel" version is simple but I am stuck at the Image version of this proposition.

Answer 1

First direction:

$A^{p+2} \subseteq A^{p+1}$:

If $v = A^{p+2}(w)$ for some $w \in V$, then for some $w' \in V$

$$ v = A(A^{p+1}(w)) = A(A^p(w')) = A^{p+1}(w') \in Im(A^{p+1})$$

Other direction:

$A^{p+1} \subseteq A^{p+2}$:

If $ v = A^{p+1}(w)$ for some $w \in V$, then for some $w' \in V$

$$v = A(A^p(w)) = A(A^{p+1}(w')) = A^{p+2}(w') \in Im(A^{p+2}) $$

Shorter version

The above is instructive, but there's a one-liner to this, using the hint that states $Im(A\circ B) = A(Im(B))$:

$$ Im(A^{p+2}) = Im(A \circ A^{p+1}) = A(Im(A^{p+1})) = A(Im(A^p)) = Im(A^{p+1}) $$