Which of the folllowing transformations are linear? how can u tell?

[mr_coffee]

http://img499.imageshack.us/img499/9875/untitled1copy0oi.jpg

Hello everyone I'm not looking for someoene to tell me the answer, but I'm really confused on how you can tell if somthing is a linear stransformation or not? I'm not understand what operations I'm suppose to go through to find this out, any help would be great! thanks!
The image is above which explains the problem!

 

[matt grime]

If you cannot verify the axioms of linearity, try to find a counter example. Ie try to show L(x+y)=L(x)+L(y), and L(ky)=kL(y), and if that's impossible, show why (simply find a single counter example)

 

[HallsofIvy]

Checking the DEFINITION of "Linear Transformation" would seem like a good idea. As I recall, the definition requires that
L(u+ v)= L(u)+ L(v) and L(av)= aL(v) where u and v are vectors (your examples all involve 2 or 3 dimensional vectors) and a is a number.