Sylvester figured all this stuff out in 1852.

Let V be an n-dimensional vector space over the reals. A quadratic form is a function q(v) from vectors to reals—from V to R.
q: V→R

Concretely: (in the following i, j and k each take on n values.)
For any quadratic form q on V and any basis {ei} for V, there are n2 reals gij such that
gij = gji and for any n reals {αi}, q(Σkαkek) = ΣiΣjgijαiαj

For a fixed basis the g’s determine q and vice versa.
f(Σkαkek, Σkβkek) = ΣiΣjgijαiβj is a symmetric bilinear function.
The quadratic form determines the bilinear function and conversely.

If gij is a metric tensor then f is the inner product and conversely.

Abstractly:
a function q from V to R is a quadratic form when there is a bilinear function f(x, y) from V to the reals and q(x) = f(x, x).
f can be recovered from q thus: f(x, y) = (q(x+y) − q(x) − q(y))/2.
Given a quadratic form q there is a basis {ek} for which
for each n reals {αk}, q(Σαkek) = Σσkαk2 and for each k, for some n reals {σk}:
σk ∊ {−1, 0, 1} and σk ≥ σk+1. There may be more than one such basis but the quadratic form uniquely determines the σ’s, collectively the signature of the form. The form is degenerate iff any of the σ’s are zero. For non-degenerate forms we say that the signature is (p, q) when there are are no 0’s, p 1’s and q −1’s

Given two quadratic forms, p and q, on an n-dimensional vector space V, (there is linear transformation A on V such that for all x, y in V, p(x, y) = q(Ax, Ay)) just in case p and q have the same signature.

See Sylvester’s law.

The bilinear forms over some vector space themselves form a vector space. Ditto symmetric bilinear forms.

### Vector Spaces Without Quadratic Forms?

Once you have learned about quadratic forms, or equivalently pseudo metrics, it may seem that any applied vector space has such a form naturally associated with it. Perhaps, depending on your meaning of ‘naturally’. But consider the temperatures at the zones described here. The available tools used to argue about the stability of these equations treat these temperatures as the components of a vector in its own vector space. The linear transformation, A, of that space which controls the progress of the temperatures, is not symmetric after you include varying heat capacities of the zones. The sum of the squares of the temperatures is not interesting because of varying heat capacities. Still A has eigenvalues that tell you about the stability of the calculations. I recall that the temperatures times the square roots of the heat capacities, taken as coordinates results in a symmetric A. This was not known when the stability results were derived. It remains to be shown that you can always find an appropriate quadratic form.
There is little difference between a pseudo-metric with signature x and another with signature −x; the geometry of {− + + +} is the same as the geometry of {+ − − −}. Their respective Clifford algebras are not isomorphic, however. What gives?