A problem in the Algebra by Artin.
Is $GL_n(\Bbb{C})$ isomorphism to a subspace of $GL_{2n}(\Bbb{R})$
I think there is an isomorphism. Because I know that when $n=1$, $$\{A\mid A=\left(\begin{matrix}a&b\\-b&a\end{matrix}\right)\}\simeq\mathbb{C}$$, so I think maybe it can be genaralized into $n\ge1$. But I have no idea how to do it.
You can take the idea of replacing $a+bi$ by the matrix $\pmatrix{a&b\\-b&a}$ and run with it. For instance for $n=2$ map $$\pmatrix{a_{11}+b_{11}i&a_{12}+b_{12}i\\ a_{21}+b_{21}i&a_{22}+b_{22}i}\mapsto\pmatrix{a_{11}&b_{11}&a_{12}&b_{12}\\ -b_{11}&a_{11}&-b_{12}&a_{12}\\a_{21}&b_{21}&a_{22}&b_{22}\\ -b_{21}&a_{21}&-b_{22}&a_{22}}.$$
More theoretically, an $n$ by $n$ matrix over $\Bbb C$ represents a linear map from $V=\Bbb C^n$ to itself. But $V$ is also a vector space over $\Bbb R$, of dimension $2n$, so choosing an $\Bbb R$-basis allows one to express that linear map as a $2n$ by $2n$ real matrix.