Yes/No :Is $f$ a bilnear form on $\mathbb{C}^2$?

Given $$a= (x_1,x_2), b= (y_1,y_2) \in \mathbb{C}^2$$ and $$f(a,b) = x_1\bar y_2 - \bar x_2 y_1$$

Now my question: Is $$f$$ a bilnear form on $$\mathbb{C}^2$$ ?

My attempt: i know that for a bilinear form, we have $$f(ka_1 +b_1 , c_1) = k f(a_1,c_1) + f(b_1, c_1)$$.

But here I don't know how to check whether $$f$$ is bilnear form or not ?

Any hints/solution will be appreciated

• A bilinear form should also have $f(a_1,kb_1+c_1)=kf(a_1,b_1)+f(a_1,c_1)$. I can't wait to see those $\bullet_1$ subscripts come back to bite. – Gae. S. Nov 9 '19 at 9:07
• @Gae.S. Isn't a complex "bilinear" form conjigate linear in one of it's arguments, i.e. sesquilinear? – Botond Nov 9 '19 at 9:19
• @Botond You're answering yourself: "sesquilinear", as opposed to "bilinear". – Gae. S. Nov 9 '19 at 9:20

Let $$a=b=(1,1)$$ and $$\lambda \in \mathbb{C} - \mathbb{R}$$, then

$$f(a,\lambda b)= \bar{\lambda} - \lambda \neq 0=f(a,b)$$.

Thus if the scalar fields of $$\mathbb{C}^2$$ is $$\mathbb{C}$$, then the answer is NO.

Take $$\alpha$$ complex. We check linearity in the first argument.

In general $$f(\alpha a,b)=\alpha x_1 \overline{y_2}-\overline{\alpha}\overline{x_2}y_1$$.

If we take $$x_1=x_2=y_1=y_2=1$$ than $$f(a,b)=0$$ but $$f(\alpha a,b)=2i Im(\alpha)$$. This shows that the operation is not bilinear, not being even linear in the first argument.

UPDATE: This shows that the function cannot be a scalar product over $$\mathbf{C}^2_{\mathbf C}$$. That is if we see $$\mathbf{C^2}$$ as a vector space over $$\mathbf{C}$$. If we see it as a vector space over $$\mathbf{R}$$, that is $$\mathbf{C}^2_{\mathbf R}$$, than we just allow multiplication by real numbers the operation $$f$$ ( which is now a map $$f: \mathbf{C}^2_{\mathbf R} \times \mathbf{C}^2_{\mathbf R} \rightarrow \mathbf{C}_{\mathbf R}$$ ) in that case should be bilinear (in this sense: https://en.wikipedia.org/wiki/Multilinear_map ). To see this quickly I suggest to use that the scalar product is equal to the determinant of a particular 2x2 matrix and use the multilinearity property of the determinant with respect to linear operations on the columns ...