Is this linear mapping positive definite, symmetric, bilinear?

Question

$V=\mathbb{R}^2$ and let be$s:V \times V \rightarrow \mathbb{R^2} $ and $ s(x,y)=2x_1y_2+2x_2y_2$

Well I found out that the matrix is: $$\begin{pmatrix} 0 & 2\\ 0 & 2 \end{pmatrix}$$

i) It's clear that the matrix is not positive definite

ii) It's is also easy to show that the matrix is not symmetric

iii) But how can I see that $s$ is a bilinear form? Is there probably an easy way to see it? I'm struggling to use the definition. I would appreciate your help.

Answer 1

The word bilinear has to to with the form: $s(x,y) = xAy^T$, where $x = (x_1,y_1)$, $A = \begin{pmatrix} 0 & 2 \\ 0 & 2 \end{pmatrix}, y = (x_2,y_2)$.