Defining the dot product without reference to an arbitrary coordinate system

Question

The dot product is of course independent of the basis or coordinate system. This would suggest that the dot product is a quantity that could be defined without recourse to such a system. However, every definition I can find forces you to express the vectors in some essentially arbitrary coordinate system.

This seems to be missing something. Is there a way to define the dot product without reference to any coordinate system?

Answer 1

A dot product is geometric. Consider three points $O,A,B$ where we assume the line segment $OA$ is unit length. Assuming the concept of orthogonality, the perpendicular projection of $B$ onto line $OA$ gives a fourth point $C$. The signed length of $OC$ is the dot product of $OA$ and $OB$. If $OA$ is not unit length, then scale the dot product by the length of $OA$.

Answer 2

The abstract (axiomatic) definition of an inner product doesn't refer to a basis (or coordinates) :

https://en.wikipedia.org/wiki/Inner_product_space

Answer 3

I would call it, not "dot product", but "inner product". In any vector space, V, an "inner product" is a function from VxV to R, notated as $\langle u, v\rangle$, such that $\langle au+ bv, w\rangle = a\langle u, w\rangle + b\langle v, w\rangle $, $\langle u, v\rangle = \langle v, u\rangle $, and $\langle v, v\rangle \ge 0$ with $\langle v, v \rangle= 0$ if and only if v= 0.

Given an n dimensional vector space there are many different inner products. And one can show that for each such inner product there exist a coordinate system such that that inner product is the dot product in that particular coordinate system.

Answer 4

It's wrong to say without qualification that the dot product is independent of the basis (or coordinate system).

What you might want to say instead is that one can precisely describe which bases the dot product is independent of, namely: the dot product is independent of the choice of an orthonormal basis; but the dot product will change if you instead pick a non-orthonormal basis.

Then you could go on to tailor your actual question: Is there a way to define the dot product without reference to an orthonormal basis?

And the answer to that tailored question is given by @Somos.