This comes from a homework question: For $\bf x, \bf y$ $\in \mathbb{R}^n$, put $\langle {\bf x}, {\bf x} \rangle$ = $\sum_{i=1}^n 2x_i^2 - 2\sum_{i=1}^{n-1}x_ix_{i+1}$. Show that the corresponding symmetric bilinear form defines on $\mathbb{R}^n$ a Euclidean structure.
A few questions about this, mostly definitions.
First, I see that, on any vector space $\mathcal{V}$, with ${\bf x} \in \mathcal{V}$, a given formula for an inner product $\langle {\bf x},{\bf x} \rangle = Q({\bf x})$ defines a quadratic form, which corresponds to a symmetric bilinear form $B({\bf x}, {\bf y})$. My first question is: Is the matrix of the mapping $Q: \mathcal{V} \to \mathbb{R}$ ALWAYS the same as the matrix of the mapping $B: \mathcal{V} \times \mathcal{V} \to \mathbb{R}$?
Now, there is a fundamental theorem of linear algebra that any real finite dimensional vector space $\mathcal{V}$ is isomorphic to the standard coordinate space $\mathbb{R}^n$, where $n = \dim(\mathcal{V})$. My second question is: Are all inner products for $\mathcal{V}$ typically given in $\mathbb{R}^n$? i.e do mathematicians always think of any space $\mathcal{V}$ as its isomorphic equivalent $\mathbb{R}^n$? It seems that way.
In the inner product given in my homework problem, a quadratic form is given that we are to check and see if we can use it as a Euclidean structure (Euclidean inner product) on $\mathbb{R}^n$. In this formula we are given $x_i$'s, components of a vector $\bf x$ on SOME BASIS. My third and main question is: Am I to assume that these $x_i$'s are components with respect to the standard orthonormal basis of $\mathbb{R}^n$, $\{{\bf e}_i\}$? Or are they in fact relative to some unknown basis $\{{\bf f}_i\}$, not necessarily orthonormal? If it is with respect to $\{{\bf f}_i\}$, can we find a formula for these vectors in terms of $\{{\bf e}_i\}$?
For example, when we are given $\langle {\bf x}, {\bf x} \rangle$ = $\sum_{i=1}^n x_i^2$, it is clear to me that we are in the standard orthonormal basis for $\mathbb{R}^n$. Fourth question: What is this inner product called? It must be given a special name since we use it so often.
Thanks for helping to clear up all the subtleties!
Edit: I should add that the second part of the question is: Find the angles between the standard coordinate axes in $\mathbb{R}^n$. The answer in the back of the book is: $\theta({\bf e}_i , {\bf e}_{i+1}) = \frac{2 \pi}{3}$, while all other pairs are perpendicular. Huh? I thought the standard coordinate axes for $\mathbb{R}^n$ were ALWAYS orthogonal??