$B: \mathbb R\times \mathbb R \to \mathbb R$ be the function $B(a,b) = ab $. Which of the following is true?

Question

$B: \mathbb R\times \mathbb R \to \mathbb R$ be the function $B(a,b) = ab $. Which of the following is true?

1) $B$ is a linear transformation

2) $B$ is a positive definite bi-linear form

3)$B$ is symmetric but not positive definite.

4) $B$ is neither linear nor bi-linear.

I can not understand how to proceed. This is certainly not linear. But how can I check whether it is bi-linear or not. Can anyone help me to understand what is the concept of bilinear and how to check whether it is bi-linear or not?

Answer 1

1) $B$ is not linear because $B(\lambda a, \lambda b) = \lambda^2 ab \neq \lambda B(a,b)$ in general.

2) $B$ is bilinear : indeed, you can check that $a \mapsto B(a,b)$ (when $b$ is fixed), and $b \mapsto B(a,b)$ (when $a$ is fixed) are both linear. Moreover $B$ is positive definite because for all $a$, $B(a,a) = a^2 \geq 0$, and this vanishes only when $a=0$.

3) $B$ is symmetric because $B(a,b)=B(b,a)$.