Consider a set of vectors, $\{{\bf e}_i\}$ in $\mathbb{R}^n$. I am thinking specifically of the standard orthonormal basis. I am having a very difficult time understanding what it means for vectors to be orthogonal. Apparently, this is an inner product-dependent concept. However, I am very used to this being a physical concept. Can you help me separate the two?
Consider two different Euclidean structures for the sake of comparison. Put $\langle {\bf x}, {\bf x} \rangle$ = $\sum_{i=1}^n 2x_i^2 - 2\sum_{i=1}^{n-1}x_ix_{i+1}$. Call this $B_1({\bf x},{\bf y})$. One can show it is a symmetric, positive definite, bilinear form and therefore a Euclidean structure. Now put $\langle {\bf x}, {\bf x} \rangle$ = $\sum_{i=1}^n x_i^2$. Call this $B_2({\bf x},{\bf y})$, yet another Euclidean structure.
We can speak of real Euclidean vector spaces equipped with either $B_1$ or $B_2$.
I am to understand that the components in the formulas for $B_1$ and $B_2$ come off of the same standard orthonormal basis, correct? That is, for each ${\bf x} \in \mathbb{R}^n$, we may express the vector on $\{{\bf e}_i\}$ as ${\bf x} = \sum_{i=1}^n x_i {\bf e}_i$. These $x_i$'s are the same ones as in the formulas for $B_1$ and $B_2$. ${\bf e}_1$ is the vector defined as ${\bf e}_1 = (1,0,0,\cdots,0,0)$.
I can use the bilinear form to induce a norm and hence speak of lengths and angles between vectors. If I use $B_2$, I find that $b_{2ij} = \langle {\bf e}_i, {\bf e}_j \rangle$ yields a diagonal matrix of all 1's. Therefore, $\{{\bf e}_i\}$ is the standard orthonormal basis, and is in fact orthonormal in physical space.
Now if I use $B_1$, I find that all ${\bf e}_i$ have length $\sqrt2$, and that the angle between adjacent ${\bf e}_i$ must be $120^{\circ}$ and is $90^{\circ}$ between all others. This is where I am utterly confused. Why should $\{{\bf e}_i\}$ depend on the defined inner product? I thought these were physically orthogonal and unit vectors in $\mathbb{R}^n$. Are they still given such that ${\bf e}_1 = (1,0,0,\cdots,0,0)$?
What's invariant here? Shouldn't the value of $\langle {\bf x}, {\bf x} \rangle$ be invariant no matter how I define it since it gives me the lengths and angles between vectors? These should be physically invariant quantities no matter what basis I use.
As you can see, I am having trouble reconciling physical orthonormality with mathematical orthonormality. Apparently, mathematical orthonormality is with respect to some arbitrarily defined inner product (a symmetric, positive definite, bilinear form). Could you please help clarify?