Let $V$ be the vector space of polynomials of degree $≤2$ with inner product given by:
$\left \langle f,g \right \rangle=\int_{0}^{1}f(t)g(t)dt$
Let $f(t)=t+2$, $g(t)=t^{2}-2t-3$, and $h(t)=2t+1$. Find a basis for the subspace $W$ orthogonal to $h(t)$.
Picture can be found here! 1 I was able to do (a) and (b) and reckon I'd need to do (c) properly to make sure that my answer for (d) is correct. Thanks so much!
Assuming you already have seen the Gram-Schmidt procedure, I'll give you a way to solve this problem. We know that the space of polynomials of degree $\leq 2$ is a three dimensional vector space over $\mathbb{R}$. We construct a basis for this space by taking your polynomial $h(t)$, and adding the polynomials $x^2$ and $1$, these polynomials are independent and thus gives a basis for the space of polynomials.
We are going to solve problem c) and d) at once by orthonormalizing this set of polynomials $\left\{2t+1,t^2,1\right\}$ using the GS procedure, lets call the resulting set $\left\{p_1,p_2,p_3\right\}$. By the theorem of GS, $p_1$ spans the same space as $h(t)$, and thus we have that the space generated by $p_2$ and $p_3$ is orthogonally to the space generated by $h(t)$ because of the dimension theorem, which says that $$\dim(V)=\dim(W)+\dim(W^{\perp})$$ Hope this helps!