Let $H$ be the set of all points in the first and third quadrants in the plane $V = \mathbb{R}^2$. That is, $H = \lbrace (x,y) \mid xy \geq 0 \rbrace$. Is $H$ a subspace of the vector space $V$? Does $H$ contain the zero vector of $V$? choose H contains the zero vector of V H does not contain the zero vector of V Is $H$ closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in $H$ whose sum is not in $H$, using a comma separated list and syntax such as $\verb+<1,2>, <3,4>+$. Is $H$ closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in $\mathbb{R}$ and a vector in $H$ whose product is not in $H$, using a comma separated list and syntax such as $\verb+2, <3,4>+$. Is $H$ a subspace of the vector space $V$? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. choose H is a subspace of V H is not a subspace of V

In order to get credit for this problem all answers must be correct.