Proving that a transformation is linear I understand the concept of proving Linear Transformation where you prove that a linear map is closed under addition and multiplication with
$L(U)+L(V)=L(U+V)$ and $XL(U)=L(XU)$
Out of this example I know that the mapping is $(2X,Y)$

But what are the appropriate steps to take when you are only given vectors of the mapping.
 A: (I'm writing the vectors as row vectors instead of column vectors for simplicity's sake)
In general, you assume that $T$ is linear and try to obtain an explicit closed form for $T$. If what you get is indeed a linear mapping, you are set. Else, there doesn't exist one.
For this particular problem, you suppose that $T$ is linear and use the axioms that define a linear transformation, namely the ones you mentioned in your post. You get,
$$T(2,0)=T(1,-1)+T(1,1)=(4,0)=2(2,0)\\ \implies 2T(1,0)=2(2,0)\implies T(1,0)=(2,0)\implies T(x,0)=(2x,0)~\forall~x\in\Bbb R$$
Similarly you get,
$$T(0,2)=T(1,1)-T(1,-1)=(0,2)=2(0,1)\\ \implies 2T(0,1)=2(0,1)\implies T(0,1)=(0,1)\implies T(0,y)=(0,y)~\forall~y\in\Bbb R$$
Add the two results to get $T(x,y)=(2x,y)$. This is what you get assuming that $T$ is linear. It is easy to verify that it is indeed a linear map on $\Bbb R^2$ and you're done.
In general, for a map $T$ on $F^n$ where $F$ is some field, you can assume that $T$ is linear and consider the explicit form $T(x_1,x_2,\ldots,x_n)=\sum_{i=1}^n \alpha_i x_i$ where $\alpha_i\in F~\forall~i\in\{1,2,\ldots,n\}$ and then use the given values to solve for the $\alpha_i$'s. If solutions exist, you're done. Otherwise, no linear map exists.
