Finite Dimensional Quotient Space

Date: 11/01/2008 at 10:41:20
From: Debdeep
Subject: quotient space

If V is a finite dimensional vector space and w is a subspace of V 
then prove that quotient space v/w is also finite dimensional.

Date: 11/02/2008 at 15:52:24
From: Doctor Jordan
Subject: Re: quotient space

Hi Debdeep,

Let V be a vector space and W be a subspace of V.  For v in V, define
[v] by

  [v] = {v + w : w in W}.

In other words, this is the set of co-sets of W in V.  The quotient
space V/W is defined to be the set

  { [v] : v in V }.

Here we define [v_1] + [v_2] = [v_1 + v_2] and a[v]=[av], where a is a
scalar and v_1,v_2 are in V.  You have to verify that the set V/W
defined this way is actually a vector space--in other words that it
satisfies the axioms of vector spaces; you may have already done this
in your class.

Now we will show that V/W is finite dimensional.  When I was thinking
how to do this, I remembered that we know a formula that involves the
dimension of a vector space and the dimensions of the image and kernel
of a linear transformation: dim(V) = dim(Im(T)) + dim(Ker(T)), if
these are all finite.  My idea is to define some linear transformation
T so that one of the terms in this equation is V/W.

Define T:V->V/W by

  T(v) = [v].

You have to verify that this is a linear transformation.  Now, it
should be clear if you look at it for a bit that Im(T) = V/W.  Also,
show that Ker(T) = W.  We know that

  dim(V) = dim(Im(T)) + dim(Ker(T)),

hence

  dim(V) = dim(V/W) + dim(W),

hence

  dim(V/W) = dim(V) - dim(W).

Since dim(V) and dim(W) are finite, this means that dim(V/W) is
finite.  Therefore the quotient space V/W is finite dimensional.

Probably the most important idea to remember from this proof is that
if you want to find the dimension of some vector space, it is often
helpful to use the formula

  dim(V) = dim(Im(T)) + dim(Ker(T)),

and then figure out how we need to define T.

- Doctor Jordan, The Math Forum
  http://mathforum.org/dr.math/