You're on the right track. You need to calculate L(u+v) and L(u)+L(v) separately. You calculated L(u+v) perfectly, now do the linear transformation of and show that this is the same as the first transformation. You calculated the first part and then just set it equal to the second. You must justify this. That's the whole point.
As for something not being linear, think about a trig function: such that