Answer
See solution
Work Step by Step
We need to disprove one of the elements of the definition of a linear transformation That is, we need to prove
\[
\left(\exists c \in \mathbb{R}, \exists x \in \mathbb{R}^{2}, T(c x) \neq c T(x)\right) \vee\left(\exists x, y \in \mathbb{R}^{2}, T(x+y) \neq T(x)+T(y)\right)
\]
Let
\[
c=-1
\]
Let
\[
x=(x 1, x 2) \in \mathbb{R}^{2}
\]
Then
\[
\begin{array}{c}
T(c x)=T(c(x 1, x 2)) \\
=T((c x 1, c x 2)) \\
=\left(-4 x_{1}+2 x_{2}, 3\left|-x_{2}\right|\right) \\
=\left(-4 x_{1}+2 x_{2}, 3\left|x_{2}\right|\right)
\end{array}
\]
And
\[
\begin{aligned}
& c T(x)=c T((x 1, x 2)) \\
&=c\left(4 x_{1}-2 x_{2}, 3\left|x_{2}\right|\right) \\
=&\left(-4 x_{1}+2 x_{2},-3\left|x_{2}\right|\right)
\end{aligned}
\]
$\mathrm{So}$
\[
T(c x) \neq c T(x)
\]
Then
\[
\exists c \in \mathbb{R}, \exists x \in \mathbb{R}^{2}, T(c x) \neq c T(x)
\]
So $T$ is not a linear transformation.
