Complexification of a real Algebra Let $\mathbb R$ be the field of real numbers and $\mathbb C$ be the field of complex numbers. Consider the complexification of the real matrix algebra $M_n(\mathbb R)$ that is $\mathbb C\otimes_{\mathbb R}M_n(\mathbb R)$. It is known that 
$$\mathbb C\otimes_{\mathbb R}M_n(\mathbb R)\cong M_n(\mathbb C).$$
What is an example of an isomorphism from $\mathbb C\otimes_{\mathbb R}M_n(\mathbb R)$ to  $M_n(\mathbb C)?$
 A: Let $e_{jk}$ denote the "unit matrix" which is zero everywhere exept in the $j,k$ entry, where it has a 1.
Map each basis element of the form $x\otimes e_{jk}$ where $ x\in\{1,i\}$ and $1\leq j,k\leq n $ to $xe_{j,k}\in M_n(\mathbb{C})$. This produces an $\mathbb{R}$ algebra isomorphism.
If you want a $\mathbb{C}$ algebra isomorphism, you can check that the same map is a $\mathbb{C}$ algebra isomorphism, but the elements of $\mathbb{C}\otimes M_n(\mathbb{R})$ would no longer be linearly independent over $\mathbb{C}$. Long story short, the map would be:
$$
(\alpha i+\beta)\otimes M\rightarrow (\alpha i+\beta)M
$$
A: A little bit more abstract, though without dealing with bases:
Let $A:=M_n(\mathbb{R})$, $A_\mathbb{C}=A\otimes_\mathbb{R}\mathbb{C}$ its complexification and $B:=M_n(\mathbb{C})$. We have a canonical embedding $\iota:A\hookrightarrow B$ of real algebras, which uniquely lifts via a morphism $\iota_\mathbb{C}:A_\mathbb{C}\to B$ of $\mathbb{C}$-algebras, by the universal property of $A_\mathbb{C}$, i.e. we have $\iota_\mathbb{C}\circ\alpha=\iota$, where $\alpha$ is the canonical map $A\ni a\mapsto a\otimes 1\in A_\mathbb{C}$. Now since $\iota_\mathbb{C}(A_\mathbb{C})$ is a $\mathbb{C}$-subalgebra of $B$ containing $\iota(A)$, $\iota_\mathbb{C}$ must be surjective and hence is the desired isomorphism, by counting dimensions.
Cheers, Robert
