Show that for any $y \in R^p$, $Hy \in V$

Question

Assume $p \le n$ and $X \in R^{nxp}$ is a full rank matrix, and $H=X(X^T X)^{-1} X^T$. Let $V = Im(X)$. Show that for any $y \in R^p$, $Hy \in V$.

So far: To show $Hy\in V$, we can show that $span(H)$ is a subset of $V$.

Answer 1

By definition, $V$ is the set of vectors of the form $Xz$ for some $z$. So you just want to show that for any $y$, $Hy=Xz$ for some $z$. To do this, just write out the definition of $H$: $$Hy=X(X^TX)^{-1}X^Ty.$$ If you want the right-hand side to be $Xz$, what should you define $z$ to be?