I've come to the conclusion that a vector equals its orthogonal projection onto a subspace by the following:
Since by orthogonal decomposition a vector "y" can be written uniquely as the sum of its projection "p" onto a subspace, and a vector (y - p) orthogonal to the subspace. I have y = p + (y - p). So then I can say the inner product of p and (y - p) is the inner product of p & y minus the inner product of p & p, which is equal to zero. So inner product of p & y equals inner product of p & p, and y = p. I'm a student and I cannot be sure that this is correct with my knowledge of linear algebra, and I was interested in finding out if what I just proved is true.
$\langle p, y\rangle = \langle p, p \rangle$ does not imply $y=p$. For example, $p=(1,0)$ and $y=(1,1)$.