Understanding Vector Spaces

Date: 03/14/2001 at 09:31:47
From: Matea
Subject: Understanding vector spaces

I have been trying to read about vector spaces everywhere, and I just 
can't seem to grasp the concept. Can you help me? I don't know how to
think about them. What does a vector space in R^n mean? How can I
prove that a list of numbers is a vector space?

Thanks!
Matea

Date: 03/14/2001 at 10:30:26
From: Doctor Mitteldorf
Subject: Re: Understanding vector spaces

Dear Matea,

I can remember having the same question as a college student. For me 
the problem turned out to be that I had all kinds of associations with 
the words "vector" and "space," and I couldn't attach them to the 
examples that were being given. The key was just to think of "vector 
space" as just some nonsense syllables that described a collection of 
objects obeying a few simple postulates:

1. There must be a rule for "addition" of two objects: 
   The space must be closed under addition: if X and Y are in the 
   space, then X+Y should be in the space as well. Addition must 
   commute: X+Y = Y+X.

2. There must be a rule for "multiplication" of any object by a 
   number:
   The space must be closed under scalar multiplication: if X is in 
   the space, and a is a number, then aX should be in the space as 
   well.

3. One of the objects in the space must serve as a "zero."  The scalar
   number 0 multiplied by any member of the space gives this zero 
   vector, and the zero vector can be added to any other vector X 
   without changing it.

4. Addition in the space must parallel addition of scalars. In 
   algebraic language, this can be stated: aX + bX = (a+b)X

If, like me, you're accustomed to thinking of vectors as arrows in our 
3D space that represent force or displacement, then you can see that 
these postulates are true of force vectors. But they're also true for 
collections of objects that might seem less "physical":  Colors made 
up from three primary colors form a 3-D vector space. Profiles of test 
scores on a battery of n tests can be seen as an n-dimensional vector 
space. There is a vector space of all infinite sequences, and a vector 
space of all real-valued functions on a given domain. These spaces 
obey the postulates, even if they don't seem to have much to do with 
arrows or forces.
   
- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/