# Linear Algebra (Subspaces)

1. Jul 11, 2009

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

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

2. Relevant equations
None

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. Jul 11, 2009

### VeeEight

Try to find a counterexample

3. Jul 11, 2009

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

4. Jul 11, 2009

5. Jul 11, 2009

### 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.

6. Jul 11, 2009