This is a problem in the first chapter of Linear Algebra Done Right. The example given in the answer is the union of coordinate axes. Using x and y as coordinates, this would be the set of points U such that either x=0 or y=0. It would be closed under scalar multiplication, but not close under addition; e.g. (1,0) (0,1) = (1,1) which isn't in the subset.

I have been thinking about this for a while-- is there another example? The only thing I could think of was some situation where you "rotate" the axes, like the union of y=x and y=-x. It would still be closed under scalar multiplication but open under addition: (1,1) (1, -1) = (2, 0) which isn't in the subset.

Is this the only type of solution possible? Is there a way to prove that?

Thank you very much! I'm sorry if this isn't phrased well -- I'm very new to this.