Prove/disprove: there exists a surjective linear transformation from skew-symmetric 2x2 complex matrices to symmetric 2x2 complex matrices

Question

Find a surjective linear transformation $T:V\to W$, as $V=\{A \in M_2(\mathbb{C})|A=-A^t \}, W= \{A\in M_2(\mathbb{C})|A=A^t \}$

In words $V$ is skew-symmetric 2x2 complex matrices $W$ is symmetric 2x2 complex matrices

I'm really confused about this question as I believe that as $\dim(V)=\dim(W)$ there must be such linear transformation, but I really can't figure it out. The fact that this linear transformation is above $\mathbb{C}$ probably changes the answer, But I can't find any example.

Answer 1

Such a transformation cannot exist, because we have $\dim(V)=1$ and $\dim(W)=3$ by Dimensions of symmetric and skew-symmetric matrices (you could also construct bases yourself to see this in the $2\times 2$ case). For a surjective transformation $T:V\to W$ to exist, we require that $\dim(V)\geq \dim(W)$.