Proof of projection matrix multiplication with any vector

Question

Im studying Projection matrix (linear algebra) and I understood that projection matrix onto line a can be calculated as:

$\cfrac{a a^T}{a^T a}$

I know that multiplying any vector to this matrix calculates the projection onto the line a, but I want to know why.

I tried to search, but i couldn't find any help me plz

Answer 1

Hint:

If $x=v+w$ with $v$ parallel to $a$ and $w$ perpendicular to $a$, then compute what your matrix does to $x$. Use that $a^T w=0$ and that $v=ka$ for some $k$.

Edit: The proof you need depends heavily on your base assumptions of what projection is.