Composition of Linear Maps

A matrix of order $m\times n$; where $m$ is the number of rows and $n$ the number of columns ‘represents’ a linear map that takes input from an $n$ dimensional vector space and outputs in a $m$ dimensional vector space.
The rules of multiplying matrices comes from the fact that we want matrix multiplication to reflect the composition of two maps. Since, for composition of maps to work, the (space) for the output of the first map should coincide with the (space) of the input of the next map. Hence the rule that the column of one matrix should coincide with the row of the other.