Given $\langle p,q\rangle = \int_{-1}^{1} {t^2 p(t)q(t)dt}$, find an orthogonal basis for $W$ subspace

Question

Given $\langle p,q\rangle = \int_{-1}^1 t^2 p(t)q(t)dt$ in $V = F[x]_{\leq 3}$, find an orthogonal basis for $W = span\{1,t\}$ subspace

There was a hint: Lagrange.

But I solved it the regular way, I just applied gram-schmidt and got $B = \{ \frac{\sqrt{3}}{\sqrt{2}}, t \cdot \frac{\sqrt{5}}{\sqrt{2}} \}$. Is that wrong? If not, then what is the use of Lagrange here?

Answer 1

Your answer is correct; in fact, you have found an orthonormal basis for W.
Since $\int^1_{-1} t^2(1)(t)\,dt=0$, the vectors $1$ and $t$ already form an orthogonal basis for W.