I'm having trouble understanding the conditions of the set
$$H = \{(x,y) \in \Bbb R^{2} \mid\text{ either } x,y \ge0 \text{ or }x,y \le0 \}.$$
To determine if it's closed under addition, let $u$ be $x_1, y_2$ and $v$ be $y_1, y_2$. It is closed if $u + v$ exists. I take $x_1 + x_2$ and $y_1 + y_2$, but I am unsure how to explain because of the conditions.
To determine if it's closed under scalar multiplication, I let $C$ be a negative scalar and $u$ be vector $x,y$. I was thinking if I choose a negative scalar, and choose the option that $x_1, y_2$ would have to be larger than $0$, then it would not be closed under scalar multiplication and thus not a subspace.
Best Answer
$(1,3)+(-3,-2)=(-2,1)$ so $H$ is not closed under addition and therefore is not a subspace of $\mathbb R^2$. (To help you visualise the set, it consists of all elements in the first and third quadrants on the Cartesian plane, including the axes.)