Dimension of Subspaces Involving Linear Inequalities

Question

In my introductory linear algebra course, one of our assignment questions is to determine whether the set of all vectors in $\mathbb{R}^2$ such that for every$\begin{bmatrix}x \cr y\end{bmatrix}$, $x\le2y$, is a subspace or not. By my thinking, this should not be a subspace because it is neither a point, a line, or all of $\mathbb{R}^2$. I've tried to prove this using the definitions of a subspace (for every $\bar{v}\in S$, $\bar{w}\in S$, and scalar $c$, $\bar{v}+\bar{w}\in S$, $c\bar{v}\in S$, and $\bar{0}\in S$), but I don't really know how to do that with inequalities. Can I represent S as the span of a set of vectors? I really just don't know what to do.

Answer 1

It is not a subspace because for instance, $\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]$ belongs to it, but $-\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]\left(=\left[\begin{smallmatrix}0\\-1\end{smallmatrix}\right]\right)$ doesn't.