# Thread: Sum of sets of vectors

1. ## Sum of sets of vectors

S and T are subsets of space V

If S and T are not subspaces S+T is not necessarily a subspace.

However, if S and T are subspaces S+T is a subspace.

My question is: Is it possible that S and T are not subspaces but their sum S+T is? If so, can I please see an example. If not, can you explain why not.

Thanks

2. One example comes to mind:

In $\mathbb{R}$, the subsets $S=\{1\}$ and $T=\{-1\}$ are not subspaces. But their sum $S+T=\{0\}$ is a subspace.

I know this is a "trivial" counterexample. I'll let you know, if I come up with a better one (if a better one exists anyway).

3. A slightly more complicated example: Let U be the subset of all vectors in $R^2$, <x, y> such that y= 2x and the single vector <1, 3>. The set of all <x, y> such that y= 2x is, of course, a subspace but adding <1, 3> to it, which is not in that subspace, gives a set that is not a subspace.

Let V be the subset of all vectors in $R^2$ such that y= 3x [b]and[b/] the single vector <1, 2>. The set of all <x, y> such that y= 3x is a subspace but adding <1, 2>, which is not in that subspace, gives a set is not a subspace.

Their sum, now, is the set of all vectors of the form <a+ b, 2a+ 3b>= a<1, 2>+ 3<1, b>, which is a subspace.