learning mathematics from a pure perspective is very fun, but It also puts holes in a lot of math that are taught to non-mathematicians. For instance, I hated the cross product because it was useful but it lacked generality (n-dimensional). I took physics twice before I found the wedge product. And let me tell you how much I felt vindicated.
the thing is, I have that same feeling for matrix multiplication. I feel like there should be a general form of a matrix product that can handle two matrices of varying orthogonality.
This is a miss use of orthogonality, but its the best way to communicate that if you have a n by m matrix [A] and a p by q matrix [B] then as [A] moves in and out of the space containing [B] then the perpendicular product between these matrices is also changing.
I feel like there is a more generalizable form of this product, but It could just be that I am a young mathematician. I'm mostly looking into this because one of my projects this semester is trying to describe this idea I had involving systems of differential equations.
