Determine if the transformation T: is linear if T(x, y)= (x+1, 2y)

1. T(u + v) = T(u) + T(v)

2. T(c*u) = cT(u)

3. T(0) = 0

I believe I have to use the above provided equations to determine whether T(x, y)= (x+1, 2y) is linear or not. If I use the third equation above, I get T(0,0) = T(0 + 1, 0) = (1, 0). Therefore the transformation is not linear. Am I right?