Is the Set of Solutions to a Homogeneous System of Equations a Subspace?

[DanielFaraday]

Homework Statement

Okay, this is the last True/False question I will post.

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

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.

 

[VeeEight]

Try to find a counterexample

 

[DanielFaraday]

I guess there is the possibility of the trivial solution. Is this enough to say it is false?

 

[DanielFaraday]

Is there a better way to think about this?

 

[VeeEight]

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.

 

[DanielFaraday]

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?

 

[blerg]

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.