# Is a vector space containing values in $\mathbb{R}^n$ itself a subspace of $\mathbb{R}^n$?

Such that if $$W$$ is a subspace of $$ℝ^n$$ and $$V=$${$$v∈ ℝ^n | v ∉ W$$}, is $$V$$ also a subspace of $$ℝ^n$$?

I think that it is, considering that if all elements of $$V$$ are in the set $$ℝ^n$$ that would mean that $$V$$ has to be as well, but another student disagrees. Why is it/is it not?

• You state your reason for thinking that $V$ is a subspace as “if all elements of $V$ are in $\mathbb{R}^n$ that would mean that $V$ has to be as well”. That sounds to me like you are confusing subspace with subset. You are correct that $V$ is a subset of $\mathbb{R}^n$. A subspace is a subset that satisfies certain properties. To check if a subset is a subspace, you have to check if it satisfies those properties. – Joe Oct 1 '19 at 2:49

$$W$$ is a subspace, so it contains the zero vector $$\mathbf{0}$$. Hence $$V$$ cannot contain $$\mathbf{0}$$, so it cannot be a vector space.
Here's a slightly different take, if this helps. Take $$W$$ to be the $$x$$-axis in $$\mathbb{R}^2$$. Then your $$V$$ consists of all vectors off the $$x$$-axis. Now, $$(1,1)$$ and $$(-1,-1)$$ are both vectors in $$V$$. But their sum is $$\mathbf{0}$$, which isn't in $$V$$ (because it is in $$W$$). So, $$V$$ isn't closed under addition.
• Your very definition of $V$ tells me that $\mathbf{0} \notin V$. If $\mathbf{0} \in V$ then this means $\mathbf{0} \notin W$, which is silly. Have you looked at any concrete examples? Consider $W$ as the $x$-axis inside $\mathbb{R}^2$. – Randall Oct 1 '19 at 0:42