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$\matrix M$ and $\matrix M_n$ are matrices of equivalent size

Multiplication by $\matrix M=\matrix M_n$ maps the...

What does the above statement mean? Since you can't multiply by an equation what is the above statement trying to say? Is it simply stating multiply by $\matrix M$ , which is equal to $\matrix M_n$? Is it simply a combination of statements?

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I'm not sure without context, but technically you can multiply equations. If I have $a=b$, and $c=d$, I can multiply both sides by $c$: $$ac=bc$$

and then note that since $c=d$, I can replace the $c$ on the right side with $d$, giving $$ab=cd$$ as if I multiplied the two equations directly. You can add equations similarly. If doing the same thing to both sides preserves equality, and the second equation asserts that those seemingly different things are in fact equal, doing one to the first side and the other to the second is valid.

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I'm almost sure your interpretation is correct --- it just means, "multiplication by $M$, which happens to be equal to $M_n$, ...."

Of course, it would be a lot easier to know for sure if you'd give a little more context.

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