To sketch: (i) The properties, (ii) Linear transformation. Let {T}:mathbb{R}^{2}tomathbb{R}^{2} be the linear transformation that reflects each point through the x_{1} axis. Let {A}={left[begin{matrix}{1}&{0}{0}&-{1}end{matrix}right]}

Assume the value of $u=\left[\begin{array}{c}2\\ 4\end{array}\right],v=\left[\begin{array}{c}6\\ 3\end{array}\right].$
Consider some random points for u and v.
The value of $u+v.$
$u+v=\left[\begin{array}{c}2\\ 4\end{array}\right]+\left[\begin{array}{c}6\\ 3\end{array}\right]$
$=\left[\begin{array}{c}8\\ 7\end{array}\right]$
The transformation $x\mapsto Ax\text{reflects points through the}{x}_1$ -axis.
Determine the value of $T(u):$
$T(u)=Au$
$=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}2\\ 4\end{array}\right]$
$=\left[\begin{array}{c}2\\ -4\end{array}\right]$
Determine the value of $T(v):$
$T(v)=Av$
$=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}6\\ 3\end{array}\right]$
$=\left[\begin{array}{c}6\\ -3\end{array}\right]$
Determine the value of $T(u+v):$
$T(u+v)=A(u+v)$
$=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}8\\ 7\end{array}\right]$
$=\left[\begin{array}{c}8\\ -7\end{array}\right]$
Show the values of the vectors as in Figure 1.

Linear transformation:
Consider the value of c as 2.
Thus the value of cu is:
$cu=2\times \left[\begin{array}{c}2\\ 4\end{array}\right]=\left[\begin{array}{c}4\\ 8\end{array}\right]$
Determine the transformation of cu.
$T\left(cu\right)=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}4\\ 8\end{array}\right]$
$=\left[\begin{array}{c}4\\ -8\end{array}\right]$
Show the linear transformation of T as in Figure 2.