Let $\mathbb{R}^{N \times M}$ be a D1294: Vector space of finite real matrices.
Let $\mathbb{R}^{N M}$ be a D1256: Euclidean real vector space.
Let $\mathbb{R}^{N M}$ be a D1256: Euclidean real vector space.
Then
\begin{equation}
\mathbb{R}^{N \times M}
\cong \mathbb{R}^{N M}
\end{equation}