Let $W$ be a subspace of a vector space; is the set $(V\setminus W)\cup\{0\}$ a vector space?

Question

I know the union of two subspace is only a subspace if one is contained in the other. I have this question,

Let $W$ be a subspace of vector space $V$ over field $\mathbb{F}$. Is the set $(V\setminus W)\cup \{0\}$ necessarily a subspace of $V$?

Answer 1

No -- and in fact it will always never be a subspace. It is a subspace if and only if $W=V$ or $W=\{0\}$.

Let's look, for example, at $V=\mathbb R^2$ and $W$ being the $x$ axis. Your set then contains both $(0,1)$ and $(1,-1)$, but doesn't contain their sum $(1,0)$. So it is not a subspace.

This argument can be repeated whenever $V$ contains both a vector $v\notin W$ and a nonzero vector $w\in W$. Then $v$ and $w-v$ are both in your set, but their sum is $w$ which isn't.

Answer 2

Suppose $W\ne V$ and $W\ne\{0\}$, otherwise the answer is trivially yes.

Let $x\in V\setminus W$ and $y\in W$, $y\ne 0$. Then $x+y\in V\setminus W$ and $-x\in V\setminus W$.

What about $(x+y)+(-x)$?