Is this a bilinear function?

Question

I am trying to understand what kind of problem is a bilinear form. Does the following example (zz function) qualify as bilinear?

Matlab code:
[x1,x2]=meshgrid(-1:0.1:+1,-1:0.1:+1);
zz=x1.*x2;
figure(123);surf(x1,x2,zz);xlabel('x1');ylabel('x2');zlabel('z')

(can not post the resulting picture due to low reputation)

Answer 1

You are asking if the function $f(x,y)=x y$ is a bilinear form.

Does it map $\mathbb{R}^1\times \mathbb{R}^1$ onto $\mathbb{R}$ ? Yes.

Is it linear in its first argument? $f(x_1+x_2,y)=x_1 y + x_2 y = f(x_1,y) + f(x_2,y)$ and $f(\alpha x,y) = \alpha f(x,y)$, so yes.

Is it linear in its second argument? $f(x,y_1+y_2)=x y_1 + x y_2 = f(x,y_1) + f(x,y_2)$ and $f(x,\alpha y) = \alpha f(x,y)$, so yes.

In conclusion, yes it is a bilinear form.