Let $\iota\colon \mathbb{R}^4\longrightarrow\mathbb{C}^2$ be the function defined by $\iota(x,y,u,v):=(x+iy,u+iv)$. Let $A\in \mathrm{GL}(2,\mathbb R)$ be an invertible $2\times 2$-matrix with real coefficients. I want to do two things:
- Prove that $V=\{(x,y,(x,y)A^T)\in \mathbb R^4\mid x,y\in \mathbb R\}$ is a two-dimensional subspace of $\mathbb{R}^4$,
and
- Characterise all matrices $A\in \mathrm{GL}(2,\mathbb{R})$ such that $\iota(V)\subseteq \mathbb C^2$ is a complex one-dimensional subspace of $\mathbb{C}^2$,
where $A^T$ is the transpose matrix of $A$ and $(x,y)A^T$ denotes matrix multiplication. I have already tried a few things but somehow I don’t seem to get the right approach. Can someone help me with this problem? Thanks for your help and stay healthy!