In engineering, linearity refers to having two properties:
a) f(x+y) = f(x) + f(y) where x, y, are two values, and f is a linear transform/function
b) af(x) = f(ax) where a is constant
In this case, the transform is the conversion between C and F. Since there is +/32 in there depending on direction, neither of these properties hold.
For example, yards to meters is a linear transform. 3 yards converted to meters + 2 yards converted to meters = 5 yards converted to meters. This is not true of temperature. Adding 32F to 32F gives 64F, which is not the same as adding 0C to 0C.