Subspace

Date: 01/25/2001 at 23:33:22
From: anthony
Subject: Linear algebra

Question: Determine whether the following is a subspace of R^3:

W3 = {(a1,a2,a3) exists in R^3:  a1-4a2-a3=0}

I don't understand how to do the problem.

Date: 01/26/2001 at 13:52:33
From: Doctor Schwa
Subject: Re: Linear algebra

Hi Anthony,

A subspace is a set of vectors, part of another vector space (in
this case R^3).

To be a subspace, it has to:

1) include the identity vector (zero vector),
2) be CLOSED under addition (that is, if two vectors a and b are
   in it, a+b has to be in it too),
3) be CLOSED under scalar multiplication (so if a vector a is in
   it, ka has to be in it too).

So in your case, you'll need to show that if (a1,a2,a3) and (b1,b2,b3) 
satisfy the restriction, then their sum (a1+b1, a2+b2, a3+b3) 
satisfies it too, and so does the product (ka1, ka2, ka3).

Also prove that (0,0,0) satisfies the restriction.

If all of that works, then it's a subspace; if it doesn't work, then 
it's not a subspace. 

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