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What happens if we define $$ \begin{pmatrix} 1 & 2 \\ 1 & 2 \\ 1 & 2 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 2 & 4 & 3 \\ 2 & 4 & 3 \\ 2 & 4 & 3 \end{pmatrix} $$ I think computers do this to refresh a part of the screen. But why can't we do it by matrices by filling the remaining rows/columns with zero? Is it related to the definition of vector spaces?

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You can do it if you find any use for it... –  Karolis Juodelė Apr 12 at 4:16

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Then you are just implying zeros in a third invisible column. If you really want to call that a 3x2 matrix (that there really is no last invisible column of zeros) then I would argue that your idea of subtraction is not a closed operation, which is not desirable. You would also have non-unique additive inverses, which is also not desirable.

There is nothing wrong with "doing it" but that is why it is not defined that way I suppose. It is rather like adding a 2d vector (a,b) to a 3d vector (c, d, e). You can say it is (a+c, b+d, e) if you want, but you didn't really add a 2d vector to a 3d vector, you added a 3d embedding of the 2d vector in the 3d space to a 3d vector. I think (a,b) in R² can be identified naturally with (a,b,0) in R³, but they are not the same -object.-

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