Suppose that $V$ is a vector space, and $M$ is a subspace of $V$.
A transformation $P:V \rightarrow V$ is called the projection of $V$ onto $M$ if
(i) there exists a subspace $N$ such that every vector $v \in V$ can be written uniquely as $v = x + y$ for some $x \in M$ and $y \in N$; and
(ii) $P$ is given by $P(x + y) = x$, for all $x \in M$ and $y \in N$.
Suppose that $P:V \rightarrow V$ is a linear transformation. Prove that $P$ is a projection onto some subspace of $V$ if and only if $P^2 = P$.
How do I prove that the linear transformation of $P$ projects onto a subspace of $V$ iff $P^2 = P$
Best Answer
Hint:
$\Rightarrow $
$P^2(v)=P^2(x+y)=P(x)=P(v)\,,\forall v$
$\Leftarrow $
Let $M=\operatorname{im}P$ and $N=\operatorname{ker}P$.