SubspaceDate: 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/ |
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