<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] It is, for the addition part consider
$$
T\begin{pmatrix}
     x + x'\\
   y + y'\\
     \end{pmatrix} 
= \begin{pmatrix}
    (x- y) + (x'-y')\\
    (x+y) + (x' + y') \\
2x + 2x'
     \end{pmatrix}
= \begin{pmatrix}
    x-y\\
    x+y \\
2x
     \end{pmatrix}
+ \begin{pmatrix}
    x'-y'\\
    x'+y' \\
2x'
     \end{pmatrix}
 = T\begin{pmatrix}
    x\\
    y \\
     \end{pmatrix}+T\begin{pmatrix}
    x'\\
    y' \\
     \end{pmatrix}
$$
</s>