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Linear Algebra (Subspaces)

  • #1

Homework Statement


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.
[/tex]

Homework Equations


None


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.
 

Answers and Replies

  • #2
614
0
Try to find a counterexample
 
  • #3
I guess there is the possibility of the trivial solution. Is this enough to say it is false?
 
  • #4
Is there a better way to think about this?
 
  • #5
614
0
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.
 
  • #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?
 
  • #7
66
0
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.
 

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