We're defining Matrix addition as $M(k,n) \times M(k,n) \rightarrow M(k,n)$. Can we add two matrices, A = $M(k,n)$ and $B = M(k,n)$ of the same size but over different fields? For example, A is a vector space over $\mathbb{R}$ and B over $\mathbb{C}$? Does this make sense?

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    $\begingroup$ Can you make sense of addition in two different fields? Because that's what you want to do, just in $kn$ independent slots. $\endgroup$ – Randall Nov 8 '19 at 17:06
  • $\begingroup$ Also, your example question isn't the best since $\mathbb{R}$ is naturally a subfield of $\mathbb{C}$, so really you're just adding in $\mathbb{C}$. You should ponder your question in something like $\mathbb{Z}_2$ and $\mathbb{R}$. $\endgroup$ – Randall Nov 8 '19 at 17:07
  • $\begingroup$ @Randall So how can I change my example where one is not a sub-field of the other? I noticed that you can add elements in $\mathbb{R}$ to elements in $\mathbb{C}$ because you are just adding real numbers and not changing the complex part. $\endgroup$ – NinetyNines Nov 8 '19 at 17:10
  • $\begingroup$ Well in my view you cannot do this. This violates all sorts of operational philosophies in algebra. It just doesn't make sense. $\endgroup$ – Randall Nov 8 '19 at 17:12
  • $\begingroup$ The part where a field is not a sub-field of another field, or addition in $\mathbb{C}$? $\endgroup$ – NinetyNines Nov 8 '19 at 17:16

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