Finding the orthogonal projection of two polynomial vectors

Question

I have two polynomial vectors:

$p$ = $x^2$ - $x$

$q$ = $-2x^2$ + $2x$ + $1$

The inner product of this vector space is defined as:

$$(p, q) = \int_{-1}^1 pq \,dx$$

I need to find the orthogonal projection of $p$ on $q$. I know how to do this with vectors containing numerical/integer values, but not sure how to do it with polynomial vectors. How do I go about doing this?

Answer 1

The orthogonal projection of $p$ onto $q$ is the unique vector $tq$ such that $$ \langle p-tq,q\rangle = 0 \\ \langle p,q\rangle-t\langle q,q\rangle = 0 \\ t=\frac{\langle p,q\rangle}{\langle q,q\rangle}. $$ So the orthogonal projection of $p$ onto $q$ is $$ \frac{\langle p,q\rangle}{\langle q,q\rangle}q $$