I have a linear algebra problem that I have been struggling with.
If $V$ is a vector space $C[0, 1$] with inner product defined by $\int_{0}^{1} fg dx$ for $f, g$ , find an orthogonal basis for Span{$1, x, x^2$}.
I recall having to do something with dot products, and from the inner product definition provided, I think I should integrate the functions from the Span. However, I am not really sure on how to proceed.
I have looked at a few examples on Stack Exchange, but none of which are very similar to this; I am still having difficulty understanding what to do.
Following the Gram-Schmidt Algorithm:
$v_1= \textbf{1}$
$v_2 = x - \frac{\int_{0}^{1} x dx}{\int_{0}^{1} 1 dx}$ = $\mathbf{x - 0.5}$
$v_3 = x^2 - \frac{\int_{0}^{1} x^2 dx}{\int_{0}^{1} 1 dx} - \frac{\int_{0}^{1} (x^3)(x - 0.5)}{\int_{0}^{1} (x - 0.5)^2} * (x - 0.5)$ = $\mathbf{x^2 - x + 1/6}$.
Normalizing the vectors:
$v_1 = \frac{1}{\int_{0}^{1} 1 * 1 dx} = 1/2$
$v_2 = \frac{1}{\sqrt {\int_{0}^{1} (x - 0.5)^2} dx} * (x - 0.5)$ = $ 2\sqrt{3}(x - 0.5)$
$v_2 = \frac{1}{\sqrt {\int_{0}^{1} (x^2 - x + \frac{1}{6}) } dx} * (x^2 - x + \frac{1}{6})$ = $6\sqrt{5}(x^2 - x + \frac{1}{6})$
Hint: Gram-Schimidt orthonormalization process can still be applied, provided you change the definitions of the inner product and the corresponding norm. For example, the inner product between $x$ and $x^2$ becomes:
$$\langle x,x^2 \rangle=\int_0^1 (x\cdot x^2)dx=\int_0^1x^3dx$$
And the norm of, say, $x$, becomes: $$||x||=\sqrt{\langle x,x \rangle}=\sqrt{\int_0^1(x\cdot x) dx}=\sqrt{\int_0^1x^2dx}$$
Edit: Outlining the process:
Let $\{ u_1, u_2, u_3 \}$ be your non-normalized basis. You can begin with any power of $x$, but let's stick with $u_1=1$. Then we proceed to the second element using, say, the polynomial $x$: $$u_2=x-\frac{\langle x,1\rangle}{\langle 1,1\rangle}1$$
Then we work for $u_3$: $$u_3=x^2-\frac{\langle x^2,1\rangle}{\langle 1,1\rangle}1-\frac{\langle x^2,x\rangle}{\langle x,x\rangle}x$$
Then normalize the vectors. If they're not orthonormal, then you probably made a mistake onyour calculations or wriing the inner product (unless I wrote something wrong).