Show that $T: \mathbb{R}^2\times\mathbb{R}^2 \rightarrow \mathbb{R}$ given by $T(x,y)=x_1y_1+x_2y_2+x_1y_2+x_2y_1+1$ is not a bilinear form. So by this property: $$f(u,0)=f(0,u)=0$$ can i prove that is not a bilinear form? Since:
$$T(x,0)=x_1(0)+x_2(0)+x_1(0)+x_2(0)+1=1$$
Yes, that suffices. If you fix any of the coordinates, say $a,$ then $T(a,-):\Bbb{R}^2\longrightarrow \Bbb{R},$ must be a linear map, and since $T(a,0)\ne 0,$ then $T(a,-)$ is not linear, and certainly not bilinear.