Why Is Matrix Multiplication Defined the Way It Is?

[jacobrhcp]

Something that has bothered me in my linear algebra class was that I learned a lot of techniques but didn't learn why they worked, or what they were useful for.

One of the things is this: why is matrix multiplication so useful in the way it's defined, and not in any other way? Of all the ways we could define the components of a matrix after multiplication, why does the commonly used way turn out to be so great?

I feel this is such a basic property that I should've learned it a long time by now, but I haven't, and I feel sorry about that.

 

[ircdan]

every linear transformation can be represented by a matrix

so take f, g linear and let A, B be their matrix representations respectively, then the matrix representation of fog is the matrix product AB, so it's defined the way it is(however strange it seems at first) so that this works