the transformation states this: P projects a vector onto the line y=x.
First: I'm not sure how to set this up. my first approach is to say:
$$F\left[\begin{matrix} x\\ y \end{matrix}\right]=\left[\begin{matrix} x=y\\ \end{matrix}\right]$$
after this its a lost cause for me. I know that to find the stadard matrix you can take the transformation and insert the standard bases, but I'm not sure how this would get the standard matrix.
Best Answer
Consider a basis of the space, formed from by the vector $(1,1)$ and a vector perpendicular to it, such as $(-1,1)$.
For any $v$, you can write $v = a(1,1) + b(-1,1)$ for some $a,b$. Then, you can prove that $P(v) = a(1,1)$ (this is because you should have both $P(v) - v \perp (1,1)$ and $P(v) \parallel (-1,1)$).
From this, you can prove