Multiplication can have a number of meanings, but there are two important ones: - multiplication as function composition, eg the product of two matrices is the

Multiplication can have a number of meanings, but there are two important ones: - multiplication as function composition, eg the product of two matrices is the linear transform obtained by applying the right matrix and then the left (onto a column vector) - multiplication as “the operation that distributes over addition” — in the theory of rings and algebras, if you have one operator “+” which forms an abelian group, then any operator “•” such that (a+b)•(c+d) = ac+ad+bc+bd for all a,b,c,d then you have a ring (if • is associative) or an algebra (otherwise) and you can apply all kinds of structure theorems. This is very oversimplified but it’s important to remember that you can multiply lots of things which don’t look like integers at all.