ThmDex – An index of mathematical definitions, results, and conjectures.
Result R678 on D1294: Vector space of finite real matrices
Vector space of finite real matrices is isomorphic to a euclidean real vector space
Formulation 0
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.
Formulation 1
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 \begin{equation} \mathbb{R}^{N \times M} \cong \mathbb{R}^{N M} \end{equation}
Proofs
Proof 0
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.
Results
(i) R5486: Dimension of a euclidean real vector space
(ii) R5487: Dimension of a vector space of finite real matrices

show that the dimension of both vector spaces is $N M$. Hence, by result R676: Finitely generated vector spaces isomorphic iff dimensions coincide, they are isomorphic. $\square$