Math Help - Invertible Linear Maps

Let L: R3 to R3.

Show that L(x,y,z) = (x-y,x+z,x+y+2z) is invertible. We have been using the theorem that if the KerL is {0} and L is surjective then the map is invertible. But in this case the KerL is not {0}, for example take L(1,1,-1) = (0,0,0). So I don't get it.

This map is not invertible. For e.g. (0,0,1) has no inverse.