1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Linear Algebra (Subspaces)

  1. Jul 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Okay, this is the last True/False question I will post.

    True or False:
    [tex]\text{The set of all solutions to the }m\times n\text{ homogeneous system of equations }Ax=0\text{ is a subspace of }\mathbb{R}^m.

    2. Relevant equations

    3. The attempt at a solution
    I thought the answer was true, but it is actually false. I know that in order to qualify as a subspace, all linear combinations of the solutions must present a solution that remains in that subspace, but I wasn't sure how to justify my answer.
  2. jcsd
  3. Jul 11, 2009 #2
    Try to find a counterexample
  4. Jul 11, 2009 #3
    I guess there is the possibility of the trivial solution. Is this enough to say it is false?
  5. Jul 11, 2009 #4
    Is there a better way to think about this?
  6. Jul 11, 2009 #5
    There is the possibility of the trivial solution but the question asks for the set of all solutions, not just one. Regardless, the trivial solution will be a subspace

    Go thorough a few examples and see if you can find such a set of solutions that is not a subspace of R^m.
  7. Jul 11, 2009 #6
    I've been trying to come up with a good example, but everything I try seems to be a subspace of R^m. Does anyone have a counter-example?
  8. Jul 11, 2009 #7
    instead of just making up systems of equations and checking if it is a subspace, try to prove that it is a subspace (I know this isn't true). This will tell you which condition of being a subspace it fails to satisfy. At this point, creating a counterexample is simple.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook