Chapter 5.2, Problem 35E

Let,

$V=\left\{x:x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right],\text{where }{x}_1\text{ and }{x}_2\text{ are in }R\right\}$.

For $u$ and $v$ in $V$ and $c$ in $R$, define the operations of additions and scalar multiplication on $V$ by,

$\begin{array}{l}u+v=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]+\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{c}{u}_1+v_1\\ u_2+v_2\end{array}\right]\text{and}\\ cu=\left[\begin{array}{c}0\\ 0\end{array}\right]\end{array}$ $\left(6\right)$

Show that the operations defined in $\left(6\right)$ satisfy all the properties in definition $1$ except $\left(m4\right)$. (Note that the addition given in $\left(6\right)$ is usual addition of $R^2$. Since $R^2$ is a vector space, all the additive properties of definition $1$ are satisfied.