Why does linear algebra make sense?

Matrices are but representations of Linear function, typically over a Vector space. So forget about the matrices themselves and try to develop intuitions on those underlying concepts. read more

That is the way to make sense of linear algebra. But now here is the kicker: It is not just linear algebra! All your mathematical career you have been taught to manipulate symbols on a page, with hardly a word (unless you had an exceptional math teacher) about the spatial concepts that the symbols represent. read more

Our operations matrix is 2×3 and our input matrix is 3×2. Writing them together: [Operation Matrix] [Input Matrix] [operation count x operation size] [input size x input count] [m x n] [p x q] = [m x q] [2 x 3] [3 x 2] = [2 x 2] Notice the matrices touch at the “size of operation” and “size of input” (n = p). read more

All these questions make sense (and carry an awful lot of information on the linear map) only when the domain and codomain coincide. The determinant, characteristic polynomial, etc. are there to answer these questions. read more