TITLE: Complexification of a real Algebra
QUESTION [5 upvotes]: 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  $M_n(\mathbb C)?$

REPLY [2 votes]: 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