Chapter 5.2, Problem 36E

Let

$V=\left\{x:x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right],\text{where}{x}_2>0\right\}$.

For $u$ and $v$ in $V$ and $c$ in $R$, define addition and scalar multiplication by

$\begin{array}{rllll}u+v& =& \left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]+\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]& =& \left[\begin{array}{c}{u}_1+v_1\\ u_2{v}_2\end{array}\right]\text{and}\\ cu& =& \left[\begin{array}{c}c{u}_1\\ u_2^c\end{array}\right].\end{array}$ $\left(7\right)$

With the operations defined in $\left(7\right)$, show that $V$ is a vector space.