Are $\mathbb{C}^2$ and $\mathbb{R}^4$ isomorphic to one another? Two vector spaces are isomorphic if and only if there exists a bijection between the two. We can define the linear map $T: \mathbb{C}^2 \mapsto \mathbb{R}^4$ as
$$
T\left(\left[\begin{array}{cc} a + bi \\ c + di \end{array}\right]\right) = \left[\begin{array}{cccc} a \\ b \\ c \\ d \end{array}\right]
$$
Does this bijection suffice to show that they're isomorphic?
Best Answer
As T. Bongers points out, we also need the map $T$ to be linear if we would like it to be an isomorphism. It isn't too hard to see that $T(v + w) = T(v) + T(w)$, and if $\alpha \in \mathbb{R}$, then $T(\alpha v) = \alpha T(v)$. This shows that as real vector spaces, $\mathbb{C}^2$ and $\mathbb{R}^4$ are isomorphic. However, for $\alpha \in \mathbb{C}$, we do not have $T(\alpha v) = \alpha T(v)$ because the right hand side isn't defined, this is because $\mathbb{R}^4$ is not a complex vector space.
In summary, $\mathbb{C}^2$ and $\mathbb{R}^4$ are isomorphic as real vector spaces, but $\mathbb{C}^2$ is also a complex vector space, while $\mathbb{R}^4$ is not.