Applying Gram-Schmidt process to a set of vectors to find first three polynomials orthogonal with respect to inner product

Question

$.$ $\langle f, g \rangle = \displaystyle\int_{-1}^{1} f(x)g(x)dx$

Apply Gram-Shmidt process to the set of vectors $:$ {1, x, $x^2$, ...} to find the first three polynomials orthogonal with respect to $:$ $\langle \cdot, \cdot \rangle $

So I am a bit confused on how to get this started. For the vector 1 do I input $:$ $\langle 1,1 \rangle$ as a inner product and solve the integral?

(which would be for the first instance of the Gram-Shmidt process).

and then for the next instance it would be $:$ $\langle$ (answer from first instance), $x\rangle$ and finally $:$ $\langle $(answer from second instance, $x^2)$?

Answer 1

Hint:

Let $v_1=1$.

Let $v_2=x-\frac{\langle x,v_1\rangle}{\langle v_1, v_1\rangle}v_1=x$ $\;\;$since $\langle x, 1\rangle=\int_{-1}^{1}x dx=0$

Let $v_3=x^2-\frac{\langle x^{2},v_2\rangle}{\langle v_2,v_2\rangle}v_2-\frac{\langle x^2, v_1\rangle}{\langle v_1, v_1\rangle}v_1=x^2-\frac{\langle x^2,1\rangle}{\langle 1,1\rangle}1$ $\;\;$since $\langle x^2,x\rangle=\int_{-1}^{1}x^3 dx=0$