I was wondering if there is a non-geometric interpretation of the dot product (or the inner product more generally). That is, an interpretation that has no concept of length and angle.
My motivation for asking this lies in how, in quantum mechanics, the inner product between the bra and ket vectors are often taken. However, these vectors do not represent spatial concepts, but are rather, a vector containing probabilities (which isn't a spatial concept).
Thanks for any help!
On
Well, the simplest interpretation is that they take a vector and recover its coordinates. So, for instance, if you have a vector (2, 3, 1), and you want to express the fact that the second coordinate is 3, the most effective way of doing this is by writing something like
$$(2, 3, 1) \cdot (0, 1, 0) = 3$$
The geometric interpretation is really secondary much of the time.
Well, it's a symmetric bilinear function $G:\>V\times V\to{\mathbb R}$ such that the associated quadratic form $q(x):=G(x,x)$ is positive definite. This means that $$G(x,y)=G(y,x),\quad G(x+x',y)=G(x,y)+G(x',y),\quad G(\alpha x,y)=\alpha G(x,y), $$ and $G(x,x)>0$ for all $x\ne0$. If such a $G$ has been selected once and for all one writes, e.g., $\langle x,y\rangle$ instead of $G(x,y)$, and calls $\langle\cdot,\cdot\rangle$ a scalar product on $V$.
Given a basis $(e_i)_{1\leq i\leq n}$ of $V$, such a $G$ is determined by the data $g_{ik}:=\langle e_i,e_k\rangle$. If the matrix $[g_{ik}]$ is the identity matrix the basis $(e_i)_{1\leq i\leq n}$ is called orthonormal.
Similar things hold in a complex environment.