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?
$W$ 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.