Question

Let hh be the set of all points in the second quadrant in the plane v=ℝ2v=r2. that is, h={(x,y)∣x≤0,y≥0}h={(x,y)∣x≤0,y≥0}. is hh a subspace of the vector space vv?

Answer 1

No, $H$ is not a subspace of $V$. A simple counter-example to the contrary: let $h\in H$ with $h=(-1,0)$. However, scaling by -1 gives the vector $-h=(1,0)$ and $-h\not\in H$.