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As per the title,

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

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

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  • $\begingroup$ Thanks for that $\endgroup$ – HonsTh Aug 13 at 19:41
  • $\begingroup$ However, $\ \mathbb R^{\{1..5\}\times\{1..9\}}\ $ is truly different from $\ \mathbb R^{\{1..3\}\times\{1..15\}}.$ $\endgroup$ – Wlod AA Aug 13 at 20:05
  • $\begingroup$ They are isomorphic as additive groups tho $\endgroup$ – hlcrypto123 Aug 13 at 21:16
  • $\begingroup$ @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. $\endgroup$ – Gae. S. Aug 13 at 22:43
  • $\begingroup$ @Gae.S. how are $\mathbb{R}$ and $\mathbb{R}^2$ isomorphic then? $\endgroup$ – hlcrypto123 Aug 14 at 4:57

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