Let $V$ be a finite dimensional vector space and let $T: V\to V$ be a linear transformation. Assume that $T$ is a projection - i.e.,
$T^2(v) = T(v)$ for every $v \in V$.
Assuming that $v \in \mathrm{range}(T)$, how do we show that $T(v) = v$?
I thought that it would be straight-forward to say that
$v \in \mathrm{range}(T)$ which equals $\{w \in V : w = T(v), v \in V\}$
so $v$ obviously $= T(v)$ since it must fit the condition $w = T(v)$. However, I was told that we must show this:
$T(v) = T^2(w) = T(w) = v$ where $v = T(w)$, $w \in V$.
But I don't understand why. Help appreciated! (I'm sure the explanation is something quick I'm missing)