I was reading a proof of the fact that $\varphi: \text{Sym}(n) \to \text{GL}_n(\mathbb{R})$ such that $\varphi(\sigma) = A_{\sigma}$ is a group homomorphism. Here $A_{\sigma} = [e_{\sigma(1)} \: e_{\sigma(2)} \: \cdots e_{\sigma(n)}]$ where the $e_i$'s are the standard basis vectors for $\mathbb{R}^n$.
Now the proof showed that $\varphi$ is a homomorphism in the following way:
Let $\sigma, \tau \in \text{Sym}(n)$ then $A_{\sigma} \cdot A_{\tau} = [e_{\sigma(1)} \: e_{\sigma(2)} \: \cdots e_{\sigma(n)}] \cdot [e_{\tau(1)} \: e_{\tau(2)} \: \cdots e_{\tau(n)}] = [e_{\sigma \circ \tau(1)} \: e_{\sigma \circ \tau(2)} \: \cdots e_{\sigma \circ \tau(n)}] = A_{\sigma \tau}$
My question regards the second equality. Though I can see that we need this equality to hold, it is not clear to me that the matrix multiplication will result in the permutation composition. If someone can provide some algebraic reasoning behind this, that would be helpful.