Vector spaces and groups
Name: harvey s reall
Date: Around 1995
As a first year math undergraduate I have done a lot of work on vector
spaces, groups and the like. Why do I need to study them? Is it just
because so many mathematical sets have theses structures so studying the
abstract structure, gives information on many different subjects? If so,
why do the lecturers not point this out?
It seems to me that your suggested reason is a major one. When an efficient
set of properties describes a lot of different ideas in a lot of different
contexts then there is real value in studying these structures abstractly.
A general language is then valuable for communication and for focusing
thought. It is true that lecturers do not try hard enough to communicate
this to the students. They are probably discouraged by a feeling that the
attempt would not be productive since it is so much easier to understand
after "having been there."
Your question is a very good one and illustrates one of the deficiencies of
math education in this country. The math teachers (as opposed to the
mathematicians) have isolated themselves from the other disciplines and
impart gibberish to their students without providing any context.
The language of vector spaces is extremely useful in quantum mechanics
and classical mechanics and heaven only knows where else. There are many
beautiful mathematical structures, and many are only dimly understood. Go
and talk to your nearest friendly physics or electrical engineering profes-
sor for more information.
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Update: June 2012