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.

Then $\mathbb{R}^{N \times M}$ and $\mathbb{R}^{N M}$ are D1496: Isomorphic vector spaces.