How to prove if something is a linear transformation?

This is almost right, but that third condition shouldn't be there (in fact, I don't know what it means in general--what is "$A$"?). That is, a map $L$ between two vector spaces is a linear transformation if and only if it satisfies $L(ax)=aL(x)$ and $L(x+y)=L(x)+L(y)$ (for any scalar $a$ and any elements $x,y$ in the domain of $L$).

You can check this directly for your example $L(x)=(x_1,x_2,x_1+2x_2)$ (for $x=(x_1,x_2)$). For any scalar $a$, we have $ax=(ax_1,ax_2)$, so $$L(ax)=L(ax_1,ax_2)=(ax_1,ax_2,ax_1+2ax_2)=a(x_1,x_2,x_1+2x_2)=aL(x).$$

If $x=(x_1,x_2)$ and $y=(y_1,y_2)$, then $x+y=(x_1+y_1,x_2+y_2)$, so $$\begin{align} L(x+y)=L(x_1+y_1,x_2+y_2)={} &(x_1+y_1,x_2+y_2,x_1+y_1+2x_2+2y_2)\\={}&(x_1,x_2,x_1+2x_2)+(y_1,y_2,y_1+2y_2)\\={}&L(x)+L(y). \end{align}$$

As you can see, checking something like this can be a kind of lengthy computation sometimes! But it's not hard--you just plug in the definitions and rearrange the terms appropriately.