I'm tyring to proof this theorem regarding projections:
Given a projection (linear transformation) $P:V\rightarrow V$ over $W$ and a given vector $v \in V$
i need to proof $v \in W \Leftrightarrow P(v)=v $
I tried using $P^2=P$ and also the face that $Im(P)=W$ but got stuck,
thank you for your help.
Best Answer
Since $P$ is the projection over $W$ the range of $P$ is $W$.
If $y$ belongs to $W$ then $y=Px$ for some $x$ so $Py=P^{2}x=Px=y$.
Conversely if $Py=y$ the $y$ belongs to the range of $P$ so $y \in W$.