My question is are there subsets $A$ and $B$whereby $A+B$ is a subspace if $A$ and $B$ aren't subspaces?
I know that $A+B$ is a subspace if $A$ and $B$ are and can prove it by showing it's closed under addition and scalar multiplication.
However, if $A$ and $B$ aren't subspaces I think I'd have to prove that $A+B$ isn't a subspace by going through every property of them not being a subspace, e.g if $A$ isn't a subspace then it doesn't contain the zero vector so $A+B$ can't be a subspace, then do if $A$ isn't a subspace it's not closed under addition so $A+B$ can't be etc...
I feel like this is not a very efficient proof and was wondering if there are any other ways of doing it? Sorry for the long description.
Try a line that passes through the origin and cut it in two parts.