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Question: Consider the inner product space $C_{[-1,1]}$ with $$\langle f,g\rangle = \int_{-1}^{1} f(x)g(x)~dx$$

Let $$W = span(1, x+1, x^2)$$

Find an orthonormal basis for the subspace of $W$.

The way I approached this question is by using the Gram-Schmidt process to find an orthogonal basis and then normalize the vectors by dividing out their sizes:

Here's my approach:

$$W = sp(1, x+1, x^2) = \{(1,0,0), (1,1,0), (0,0,1)\}$$

$$x_1 = (1,0,0), x_2 = (1,1,0), x_3 = (0,0,1)$$

$$\mathbf v_1 = x_1 = (1,0,0)$$ $$\mathbf v_2 = x_2 - proj_{v_1}x_2= (1,1,0)-\frac{(1,1,0)(1,0,0)}{(1,0,0)(1,0,0)}(1,0,0) = (0,1,0)$$ $$\mathbf v_3 = x_3 - proj_{v_1}x_3 - proj_{v_2}x_3= (0,0,1)-\frac{(0,0,1)(1,0,0)}{(1,0,0)(1,0,0)}(1,0,0) - \frac{(0,0,1)(0,1,0)}{(0,0,1)(0,1,0)}(0,1,0) = (0,0,1)$$

Therefore, the orthogonal basis is: $\{(1,0,0), (0,1,0), (0,0,1)\}$

and in order to find an orthonormal basis I normalized the vectors by dividing out their sizes which ended up being the same thing: $\{(1,0,0), (0,1,0), (0,0,1)\}$

However, I got a zero on this question and I don't know where I went wrong. Was I supposed to somehow use this fact: $\langle f,g\rangle = \int_{-1}^{1} f(x)g(x)~dx$ in this question? I'm not too sure how to. If someone could help me out with this, that would be really appreciated.

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here is what i got doing gram-schmidt:

i will first find three orthogonal functions $q_1, q_2, q_3$ and will make length one later.

set $q_1 = 1.$ now $1 + x = k q_1 + q_2$ where $k$ is determined by $<q_2, q_1> = 0$ so $$\int_{-1}^1 (1 + x)dx = k \int_{-1}^1 dx \mbox { gives } k = 1, q_2 = x$$

now to find $q_3$ set $x^2 = k q_1 + l q_2 + q_3$ and require $<q_1, q_3> = 0 = <q_2,q_3>$ which is $$\int_{-1}^1 x^2 dx = k \int_{-1}^1 dx, \int_{-1}^1 x x^2 = l \int_{-1}^1 x^2 dx \mbox{ giving } k = \frac{1}{3}, l = 0 \mbox{ so } q_3 = x^2 - 1/3$$

putting all these together $\{1, x, x^2 - 1/3\}$ is an orthogonal basis. making it length one this basis becomes $$\{1/2, \sqrt{3/2} x^2, 3/2\sqrt 2(x^2-1/3)\}. $$

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Hint: The projection operator is defined as: $$proj_{\textbf{u}}(\textbf{v}) = \frac{\left<\textbf{u},\textbf{v}\right>}{\left<\textbf{u},\textbf{u}\right>}\textbf{u}$$

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