Jacques Carette writes:
Henning Thielemann wrote:
I don't see the problem. There are three very different kinds of multiplication, they should also have their own signs: Scalar product, matrix-vector multiplication, matrix-matrix multiplication.
You see 3 concepts, I see one: multiplication. Abstract Algebra is the branch of mathematics where you want to abstract out the *unimportant* details. Much progress was made in the late 1800's when this was discovered by mathematicians ;-).
One of the things I appreciate and I hate simultaneously in your postings is that you are so categorical. This time you will *not* convince me that there is "one concept: multipli- cation", moreover "abstracted over unimportant details". If matrices represent operators, their multiplication is a *group* operation, the op. composition. Acting of a matrix on a vector is not. "Multiplication" of two vectors giving a scalar (their contration) is yet another beast. I believe that some progress has been done in math, when people discovered that mixing-up things is not necessarily a good thing, and different entities should be treated differently. Jerzy Karczmarczuk