Chapter 5.5, Problem 24E

Consider the linear space P of all polynomials, with inner product
$\langle f,g\rangle ={\displaystyle {\int }_0^{1}f\left(t\right)g\left(t\right)dt}$ .
For three polynomials f, g, and h we are given the following inner products:

For example, $\langle f,f\rangle =4$ and $\langle g,h\rangle =\langle h,g\rangle =3$ .
a. Find $\langle f,g+h\rangle$ .
b. Find $\Vert g+h\Vert$ .
c. Find ${\text{proj}}_{\text{E}}h$ , where $E=\text{span}\left(f,g\right)$ . Express yoursolution as linear combinations of f and g.
d. Find an orthonormal basis of $\text{span}\left(f,g,h\right)$ . Expressthe functions in your basis as linear combinations off, g, and h.