Homework Help: Vector Spaces, Subsets, and Subspaces

1. Nov 8, 2007

mrroboto

1. The problem statement, all variables and given/known data

What is an example of a subset of R^2 which is closed under vector addition and taking additive inverses which is not a subspace of R^2?

R, in this question, is the real numbers.

2. Relevant equations

I know that, for example, V={(0,0)} is a subset for R^2 that is also a subspace, but I can't figure out how something can be a subset and not a subspace.

3. The attempt at a solution

Does this have anything to do with scalar multiplication being closed on the vector space?

2. Nov 9, 2007

Dick

Think about the set of all (x,y) where x and y are both integers.

3. Nov 9, 2007

mrroboto

So for example, if we let the subset = (a,b) s.t. a,b are elements of Z. Then it is closed under addition but not under scalar multiplication. i.e. Let (a,b) = (1,3) and multiply by 1/2 for example (which is the example we used to figure it out). Then you get (1/2, 3/2), neither of which are in Z.

4. Nov 9, 2007

Dick

Sure, but it does have additive inverses.

5. Nov 9, 2007

andytoh

or both rational numbers

6. Nov 9, 2007

HallsofIvy

It's very easy to be a subset without being a subspace! Just look at any subset that does not satisfy the requirements for a subpace- what about { (1, 0)}?

In order for a subset to be a subspace, it must be closed under addition, have additive inverses, and be closed under scalar multiplication. Since you are asked about a subset that IS closed under addition and has additive inverses, looks like there is only one thing left!