Gram-Schmidt procedure on functions

Question

I have been applying the Gram-Schmidt procedure with great success however i am having difficulty in the next step, applying it to polynomials.

Here i what i understand If i have 2 functions, say $x^2$ and $x^3$, and want this to work over -1 to 1, then

$$ V_1=U_1 = x^2$$ great, now for $V_2$

$$ V_2 = U_2 - \dfrac{<U_2,V_1>}{||V_1||^2}V_1 $$

This is where i dont understand $$ V_2 = x^3 -\dfrac{\int_{-1}^1 B}{\int_{-1}^1x^4}x^2 $$ If i have done everything correctly, then how do i proceed, how do i get B and get $V_2$?

Answer 1

You are just using the integral to define your inner product:$$\langle f,g\rangle := \int_{-1}^1f(t)g(t)dt.$$

In your case you have $U_1 = V_1 = x^2$, $U_2 = x^3$, hence, as you correctly wrote, the formula for $V_2$ is:

$$ V_2 = U_2 - \frac{\langle U_2,V_1\rangle}{||V_1||^2}V_1 = U_2 - \frac{\langle U_2,V_1\rangle}{\langle V_1 ,V_1\rangle}V_1 =x^3 - \frac{\int_{-1}^1t^3 \cdot t^2\,dt}{\int_{-1}^1t^2\cdot t^2\,dt}x^2 = x^3$$ since the integral in the numerator equals $0$.