Proof of p^2=p for projections [closed]
Sorry, it should be really simple, but i forgot how do prove it. If we define a projection $p$ onto subspace $U$ of a finite-dimensional space $V$ by these 2 properties: Image($p$)=$U$ and $p_U$ is identity, then why do we have $p^2=p$? Thanks in advance.
Since the image of the transformation is $U$ this mean that for all $v$ in $V$ we have $p(v)=u$ for some $u$ in $U$. Since $p$ restricted to $U$ is the identity we have $p(u)=u$. Putting these together we have $p(p(v))=p(u)=u$ so $p^2=p$.