I have been working on this problem, we are given the below system of vectors
$f_{1} = x, f_{2} = \cos(x), f_{3}= \sin(x)$ from the inner product of $C_{\mathbb{R}}[-1,1]$
and we have to orthogonalise the system
I know that the Gram-Schmidt algorithm works like below
$w_{1} = v_{1}$
$w_{2} = v_{2} - \operatorname{proj}_{w_{1}}(v_{2})$
and so on, and I am just struggling how to use all of this to orthgonalise the vectors. Do I have to integrate over the inner product space? Or am I just simply substituting into the Gram- Schmidt Algorithm?
Any help would be much appreciated
The vector space of real functions whose domain is an closed interval [a,b] with inner product $\langle f_1, f_2 \rangle $ is $\int_{a}^{b} f_1 f_2\,dx$. So yes, you need to integrate over the inner product space.
You need to start by normalizing $f_1$. Let's say $w_1$ is $f_1$ normalized.
$$ w_1 = \frac{f_1}{||f_1||}\, ||f_1||^2=\int_{a}^{b}f_1^2(x)dx \qquad(1) $$
After normalizing $f_1$, you need to find a vector orthogonal to $w_1$. We can use $f_2$ to find this vector $W_2$ as
$$ W_2 = f_2-\bigl(\;\langle\, w_1,f_2\,\rangle\;\bigr)w_1\; $$ where $\langle\, w_1,f_2\,\rangle$ is the inner product of $w_1$ and $f_2$ $$ W_2 = \sin x - \Bigl(\int_a^bw_1\sin x\,dx\Bigr)w_1 \qquad(2) $$
$W_2$ is not normalized and it can be normalized into $w_2$ in the same way as in eq(1).
Next $W_3$ and $w_3$ is found in the same way as in eq(2), which will give you 3 orthonormal vectors