Hello,

My question is,

A bilinear form $w$ on $U \oplus V$ is degenerate if, as a function of one of its two arguments, it vanishes identically for some non-zero value of its other argument; otherwise it is non-degenerate.

(a) Give an example of a degenerate bilinear form (not identically zero) on $\mathbb{C}^2 \oplus \mathbb{C}^2$.

(b) Give an example of a non-degenerate bilinear form on $\mathbb{C}^2 \oplus \mathbb{C}^2$.

The definition we have for a bilinear form is: $w$ on $W=U \oplus V$ is a bilinear form if, $$w(a_1x_1 +a_2 x_2, y)=a_1 w(x_1, y)+a_2w(x_2,y) \: \text{and} \: w(x, a_1 y_1+a_2 y_2)=a_1w(x,y_1)+a_2w(x, y_2).$$ And the only example we have of a bilinear form is the dot product in $\mathbb{R}^n \oplus \mathbb{R}^n \to \mathbb{R}$.

I'm not entirely sure what is so confusing here for me. I want to just take my only example, adapt it for a 2 dimensional case, and shove in complex entries for each array. So, $$\begin{pmatrix}x_1\\x_2\end{pmatrix}^T \begin{pmatrix}y_1 \\ y_2 \end{pmatrix}=x_1y_1+x_2y_2,$$ now just set this as being equal to zero. But that doesn't make any sense to me.

The only other thing I have is, $$w(x_1 + x_2 , y_1 + y_2)=w(x_1, y_1)+w(x_1, y_2)+w(x_1,y_1)+w(x_2,y_2),$$ which is just the full definition expanded out. But this seems to be too much as the question wants to restrict attention to "as a function ofoneof its two arguments".

So I'm not sure what to do.