Confused about the representation of a basis in $V = M_{2×2}(\mathbb{R})$

Question

If the bases of $V = M_{2×2}(\mathbb{R})$ are 2 x 2 matrices, how is it possible that each basis can also be represented as a column matrix?

Answer 1

This only holds for $K^n$, since you have an isomorphism $(K^n)^n \cong M_n(K)$, where $(a_i)_{i \in \{1,\ldots,n \}} \mapsto (a_1,\ldots, a_n)$ and $a_i$ is a vector in $K^n$. For the dimensions we have $\dim((K^n)^n) =n^2 = \dim(M_n(K))$.