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### Understanding Vector Spaces

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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.

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/
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Associated Topics:
College Linear Algebra
High School Linear Algebra

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