How is a quadratic form and symmetric bi-linear form determine each other?

Question

Let $k$ be a field of characteristic $(\neq 2)$. Let $V$ be a $k$-vector space.

A quadratic form on the vector space $V$ is a map $q:V\rightarrow k$ such that $q(c\cdot v)=c^2q(v)$.

A symmetric bilinear form on $V$ is a map $b:V\times V \rightarrow k$ such that it is linear with respect to each variable.

Now corresponding to a bilinear symmetric form $b$ one can define a quadratic form $q_b(v):=b(v,v)$.

Now, most of the places they say corresponding to a quadratic form $q$, one can define a symmetric bilinear form $b_q(v,w):=\frac{1}{2}(q(v+w)-q(v)-q(w))$.

*I don't see why $b_q$ should be bilinear. * Am I missing something?

Answer 1

As said in a comment there's more than what you think in the definition of a quadratic form. The general definition is the following:

Let $R$ be a ring, $V$ an $R$-module. a map $q\colon V\longrightarrow R$ is quadratic if

1. for all $v\in V$, $c\in R$, $q(cv)=c^2q(v)$;

2. the map $B\colon V\times V\longrightarrow R$ defined by $$B(u,v)=q(u+v)-q(u)-q(v)$$ is a bilinear form (necessarily symmetric).