Symmetric bilinear form is determined by values on same-vector pairs

Statement

Suppose $V$ is a real vector space, $b:V\times V\to \mathbb {R}$ is a symmetric bilinear form. Then, $b$ is completely determined by the values:

$q(x):=b(x,x),x\in V$

In fact, it suffices to specify $q$ for one vector in every line, because knowing $q(x)$, automatically tells us what $q(\lambda x)$ is. In the particular case where we have already put a Euclidean inner product on $V$, we can say that it suffices to describe $q$ on all unit vectors.

Proof

The proof rests on the following polarization identity:

$b(x,y)={\frac {1}{2}}\left(b(x+y,x+y)-b(x,x)-b(y,y)\right)$