Let L be a linear transformation defined by its standard matrix AL $\begin{bmatrix}1&-1&1&2\\2&-2&3&4\end{bmatrix}$
As I understand domain and codomain of L are $L:\mathbb{R}^4 \rightarrow \mathbb{R}^2$ ?
How can I write formula definition of transformation L? I know it should look like $L(x,y,z,t) = (\ldots, \ldots,)$.
I'm not sure how to approach this problem. Could anyone explain how to solve this problem? Thanks.

You have to show that in this matrix the second and the fourth column are linear dependent by the first, but not the third.
Indeed 
-1*(1,2)=(-1,-2)   
and   
2*(1,2)=(2,4), 
but 
h*(1,2)+k*(1,3)=0 ===> h+k=0 & 2h+3k=0 ===> h=k=0.
