Projection definition in Linear Algebra Why is a mathematical projection defined as $P^2 = P$? I understand that it might be because once a vector has been projected onto a space, if projected again, it should give the same thing. Is there anything more to it?

There's little more. If $P$ is a projection and $w\in\operatorname{Im}P$, we want to have $P(w)=w$. This is equivalent to the assertion:$$(\forall v\in V):P\bigl(P(v)\bigr)=P(v).$$And this, in turn, is equivalent to $P^2=P$. $\phantom{}$