Post by quasi
In my opinion, you have succeeded in capturing the main motivation,
ignoring the side issues.
It was my intent, as stated, to understand the initial motivation
for the theory in terms of ideas and not the terminonolgy or ideas
Too often educators, at least many that have taught me, focus on
definitions and terminology or even far too abstract ideas before
initially getting across the actual motivation and specific ideas of
what is being taught. They forget that all of the mathematical
definitions are not created out of thin air, but rather are created in
human brains which are driven by intent and motivation. And before we
students can properly understand the mathmatics we must understand the
specific and fundamental ideas in human terms and not definitions
created after the fact or be assumed to have it all fit together in our
minds once we've memorized a set of definitions.
At least, for me, learning the ideas in human terms before anything
else is preferable. Since it's my choice when learning on my own, that
is what I do. And after a while of contemplation and searching the
fundamental ideas are found and after that I can understand why
definitions are given as such and theorems are true. It makes actual
understanding possible for me. I can't be alone in this manner.
What else is interesting is then determining just what it means
that matrix algebra can be applied to some physical problem based on
ideas on not just algebra. That is, where the multiplications,
additions, compositions have physical meaning and then to find that
For instance, if a physical problem obeys the linear function
requirement, f(x + y) = f(x) + f(y), then that may hint at a more
fundamental concept of what constitutes x and y. I mean, if f
represents a law, then things, if broken down to x and y, may be able
to be more simply understood than x + y. Well, it is hard to explain.
Post by quasi
Note that by viewing matrices as representing linear maps, matrix
multiplication automatically satisfies a number of laws since the
underlying linear maps do, such as the associative law (since
composition is associative), and the distributive law.
Exactly! That is why I sought the motivation for matrix algebra. It
is discovering that motivation that leads to further ideas that extend
and propel the construction of the theory.
In a text I have it lists the axioms that must be satisfied. They
are introduced in a manner that makes one wonder just why they are
defined as such. The axioms are then used to construct the theory and
show that linear compositions, etc, can be represented. But in
actuality it was the other way around! The compositions motivated the
axioms. To put the theory on firm logical and mathematical ground the
theory is then expressed beginning with such axioms, and of course, it
works out in way such that the mathematics has wide applicability.
I'm off to further my understanding of matrix algebra now. Knowing
what is behind many of the definitions that I've previously learned.