I hope this makes sense, I had a test yesterday and I couldn't answer this question.
The question was laid out something like the following:
I was provided a basis for 5th degree polynomials, $p_1,...,p_5$, and given two polynomials in $\mathbb{P}^5$, $f(x),\,g(x)$.
Where (or something similar):
$f(x) = 5x^4 -3x^3 - 4x^2 + 3x +1\\ g(x) = 4x^4 - 3x^2 - 2x$
The inner product was defined as $\int_{-1}^{1}\big(f(x)g(x)\big)dx$. The polynomials were too long to actually compute inner product in the time allowed and the professor said that all questions related to this can be answered without doing any integrals at all.
The simpler questions that I remember are:
a) Find the magnitudes of $f(x)$ and $g(x)$ and find $<f(x),g(x)>$ (the inner product).
b) This question I can't remember well but it had to do with finding projections of a polynomial onto the basis provided (again without actually calculating any integrals).
I haven't been able to find how to work with these without actually computing any integrals, any help in how this works would be greatly appreciated. I'm sure it has something to do with the fact that $p_1,...,p_5$ actually forms a basis. And in case it matters, the basis elements were not simple, for example, $p_5 = \frac{\sqrt{2}}{\sqrt{5}}(x^4 - 3x)$ (or something similar).
I understand the abstractness of this problem but I hope someone is willing to take a little time to help me understand how all this works without calculating any integrals.
Thank you very much.
Here's what I've thought for part (a). We know that $\mathcal{P}^5$ is isomorphic to $\mathbb{R}^6$, so we consider $f$ and $g$ as vectors in $\mathbb{R}^6$. Thus the magnitudes of $f$ and $g$ are
$$\begin{aligned} |f| & = \sqrt{5^2+3^2+4^2+3^2+1} = 2 \sqrt{15}, \\ |g| & = \sqrt{4^2+3^2+2^2} = \sqrt{29}. \end{aligned} $$
To compute $\langle f, g \rangle$ we use a trick. Note that
$$\langle f+g, f+g \rangle = |f+g|^2 = |f|^2 + 2 \langle f,g \rangle + |g|^2.$$
Compute $|f+g|^2$ as we did before and you'll find $|f+g| = \sqrt{131}$. Therefore
$$\langle f,g \rangle = \frac{\sqrt{131} - 2 \sqrt{15} - \sqrt{29}}{2}.$$