Do we say that an $m \times n$ matrix A exists in $\mathbb{R}^m$ or $\mathbb{R}^n$?

As per the title,

Do we say that an $$m \times n$$ matrix A exists in $$\mathbb{R}^m$$ or $$\mathbb{R}^n$$ ?

• Neither -- it exists in $\mathbb R^{mn}$. The dimension $mn$ is just the number of entries in the matrix -- literally $m\times n$. – MPW Aug 13 at 19:35
• And an nxn matrix doesn't exist in $\mathbb{R}^n$ either. – It'sNotALie. Aug 13 at 19:35
• Thanks for the clarification! – HonsTh Aug 13 at 19:41

Usually, we use the notation $$\mathbb{R}^{m\times n}$$. Although it is Clearly isomorphic to $$\mathbb{R}^{mn}$$.
• However, $\ \mathbb R^{\{1..5\}\times\{1..9\}}\$ is truly different from $\ \mathbb R^{\{1..3\}\times\{1..15\}}.$ – Wlod AA Aug 13 at 20:05
• @hlcrypto123 Techincally, all $\Bbb R$-vector spaces $V$ such that $1\le \dim_{\Bbb R}V\le 2^{\aleph_0}$ are isomorphic as additive groups. – Gae. S. Aug 13 at 22:43
• @Gae.S. how are $\mathbb{R}$ and $\mathbb{R}^2$ isomorphic then? – hlcrypto123 Aug 14 at 4:57