In this problem we have the polynomial vector space$P_{2}$ is all polynomials with degree smaller or equal to 2 and the scalar product is defined as: $$\langle p,q \rangle = \int_{-1}^1 p(t)q(t) dt$$. \
I need to first find an orthonormal Basis Q for the Vectorspace $P_{2}$. I did that using the Gram-Schmidth process and got the following for Q. $$\{ {1\over \sqrt 2}, {3\over 2}x, {45\over8}(x^2-1/3)\}$$. \ noe I need to find the transformation Matrix to transform the coordinate vectors from $P_{2}$ to Q. I'm not sure how to do that. I got the following for T. is this correct? please help. thanks.
More general, how can I find the transformation matrix from a basis $P_{2}$ to Q? this confuses me everytime!
$$T = \begin{pmatrix} 1/\sqrt2 & 0 & 0 \\ 0 & 3/2 & 0 \\ -15/8 & 0 & 45/8 \\ \end{pmatrix} $$
Your method is correct, but it seems that you have forgot that the norm is the square root of the inner product: $||f||=\sqrt{\langle f,f\rangle}$. The correct answer is that the orthonormal basis, with your inner product, is: $$ \left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}}x,\sqrt{\frac{5}{8}}(3x^2-1)\right\} $$
Note that these are Legendre Polynomials normalized in a different way than usual.
You can correct the matrix with these corrected polynomials.