<s>[INST] Determine if $T$ is a Linear Transformation or Not Suppose that
$$
T\begin{pmatrix}
x \\
y \\
\end{pmatrix} = 
\begin{pmatrix}
x-y\\
x+y \\
2x
\end{pmatrix}
$$
I tried multiplying with a scalar as follows
$$
T \left\{\alpha\begin{pmatrix}
    x\\
    y \\
     \end{pmatrix}\right\} 
 =T\begin{pmatrix}
    \alpha x\\
   \alpha y \\
     \end{pmatrix} 
= \begin{pmatrix}
   \alpha x- \alpha y\\
    \alpha x+\alpha y \\
2\alpha x
     \end{pmatrix}
= \alpha \begin{pmatrix}
    x-y\\
    x+y \\
2x
     \end{pmatrix}
 =\alpha T\begin{pmatrix}
    x\\
    y \\
     \end{pmatrix}
$$
Is the above correct? 
And I don't know how to show the addition part.
 [/INST] The first part for the multiplication is correct.
For the addition, consider two different vectors 
$V_1=\begin{pmatrix}
x\\
y \\
\end{pmatrix}$
and 
$V_2=\begin{pmatrix}
u\\
v \\
\end{pmatrix}$
and show that 
$$T(V_1+V_2)=T(V_1)+T(V_2)$$
</s>