## Vector Spaces, Subsets, and Subspaces

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?

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 Recognitions: Homework Help Science Advisor Think about the set of all (x,y) where x and y are both integers.
 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.

Recognitions:
Homework Help

## Vector Spaces, Subsets, and Subspaces

Sure, but it does have additive inverses.

 Quote by Dick Think about the set of all (x,y) where x and y are both integers.
or both rational numbers

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