Need help finding a basis of a subspace $\mathbb{R_2[-1,1]}$ is the polynomial space consisting of 0 togeher with all polynomials of degree less than or equal to $\mathbb{2}$ in the interval $\mathbb{[-1,1]}$ , the scalar (dot) product is given by $\mathbb{(f|g) = \int_{-1}^1f(t)g(t)dt}$. Find a basis of the subspace U = {$\mathbb{ f ∈ {R_2}[-1,1] : (f | h) = 0 }$} where $\mathbb h(t) = 2t +1$
I'm having trouble onto where to start, some hints or initial steps would be appreciated.
 A: hint
Let $ f$ be a polynomial of degree less or equal to $2$ such that $$(f|h)=0$$
then
$$f(t)=a+bt+ct^2$$
and
$$\int_{-1}^1f(t)h(t)dt=0$$
so,
$$\int_{-1}^1(a+bt+ct^2)(2t+1)dt=0$$
$$=\int_{-1}^1\Bigl(a+(2a+b)t+(2b+c)t^2+2ct^3\Bigr)dt$$
$$=\int_{-1}^1\Bigl(a+(2b+c)t^2\Bigr)dt$$
$$=2a + \frac 23(2b+c)$$
because $$\int_{-1}^1tdt=\int_{-1}^1t^3dt=0$$
thus
$$c = -3a - 2b$$
and
$$f(t) = a + bt - (3a+2b)t^2$$
$$= a(1-3t^2) + b(t - 2t^2)$$
$$= af_1(t) + bf_2(t)$$
Therefore, $ \; U = \text{Span}(f_1,f_2)$.
Now assume that
$$\exists (\lambda,\mu)\in \Bbb R^2 \;\;: $$
$$\; (\forall t\in [-1,1]) \;\;\lambda f_1(t) + \mu f_2(t)=0$$
then, for $t=0$ we get $ \lambda=0$ and for $ t=1 $, we find that $\mu=0$
It follws that $(f_1,f_2) $ is a basis of the subspace $ U $ with $2$ as  dimension.
A: I am new to this site so please forgive my poor formatting.
$(f\mid h)$ is the inner product of $f$ and $h$ in inner product space. $(f \mid h)=0$ means that both of these functions are orthogonal if $f$ does not equal $h$. So since youre looking for $f$, what you re basically doing is finding an orthogonal polynomial of degree $2$ in the interval $-1$ to $1$. You can start by using the Gram-Schmidt process for polynomials. 
That is your hint
