Why does matrix multiplication need to follow the rule of joined corner marks? [closed]
We know the matrix multiplication formula must meet joined corner marks
$ A_{ms} \cdot B_{sn} = C_{mn} $
but why? why we must follow this rule? how to understand it in terms of linear transformations?
Solution 1:
In terms of linear maps, it means that if you have a linear map $f\colon\Bbb R^n\longrightarrow\Bbb R^s$ and another linear map $g\colon\Bbb R^{s'}\longrightarrow\Bbb R^m$, then you can consider $g\circ f$ if and only if $s=s'$. If $s\ne s'$, then that composition makes no sense.