For a function to be a linear transform it needs two properties[1]:

  1. F(c1 + c2) = F(c1) + F(c2)
  2. F(k*c1) = k*F(c1)

Converting from Celsius->Fahrenheit (or vice versa) is does not have these properties.

  F(c) = 9/5 * c + 32

  F(c1 + c2) = 9/5 * (c1 + c2) + 32
  F(c1) + F(c2) = (9/5 * c1 + 32) + (9/5 * c2 + 32) = 9/5 * (c1 + c2) + 64

  F(k*c1) = 9/5 * (k*c1) + 32
  k*F(c1) = k * (9/5 * c1 + 32) = 9/5 * (k*c1) + k*32

Admittedly, I'm not entirely sure why being a non-linear transform gets the OP so riled up.

[1]http://mathworld.wolfram.com/LinearTransformation.html