To show that something is a linear transformation, you have to work in generality to show that it satisfies the conditions in the definition of linear. That is, you would have to check that for all $(x_1,y_1)$ and $(x_2,y_2)$ in $\mathbb R^2$ and for all $\lambda\in R$, $T((x_1,y_1)+(x_2,y_2))=T(x_1,y_1)+T(x_2,y_2)\text{, and }$
$T(\lambda(x_1,x_2))=\lambda T(x_1,x_2).$
On the other hand, to show that $T$ is not a linear transformation, you only need to show that there is one instance of a failure of a condition in the definition of linear, called a counterexample. It is a good idea to try to make counterexamples as easy as possible, and with a numerical problem like this, that means you might want to try using lots of zeros, or maybe some ones.