First I'll define what I talk about:
A bilinear form on a vector space V is a mapping:
$F: V \times V \rightarrow \mathbb{R}, (a,b) \mapsto F(a,b)$
which is linear in every argument:
$a, b, c \in V$ and $\lambda, \mu \in \mathbb{R}$:
- $F(\lambda a + \mu b, c) = \lambda F(a, c) + \mu F(b, c)$
- $F(a, \lambda b + \mu c) = \lambda F(a, b) + \mu F(a, c)$
If I get an expression which could be a bilinear form, I check those two. This can be quite long.
A bilinear form F is symmetric, if:
$\forall a, b \in V: F(a, b) = F(b, a)$
Now my question: If I know that a mapping is symmetric, can I make the checks for bilinearity shorter? Something like that:
$F(\lambda a + b, c) = \lambda F(a, c) + F(b, c)$?
If it is not possible, do you have counterexamples where it doesn't work?