system
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human
What exactly is a linear transformation?

This is a basic question I know but I cannot wrap my head around it. I understand the conditions under which a transformation is linear (although I could not tell you exactly what the significance of it is) given here. What I don't get it this. The transformation y = ax + b is non-linear but it is the equation of a line. It seems to me that if you took all of the x points and scaled them then shifted them that should be a linear transformation. Please help me.
I stumbled upon this question when looking into the relation between Celsius and Fahrenheit (F = (9/5)C+32). Both Fahrenheit and Celsius scales are linear correct? If they are, how can the transformation between them be non-linear?
gpt
Linear transformations preserve addition and scalar multiplication (as you've said you already know). 

Let's see what happens when we try the transformation f(x) = ax + b. 

*Addition*

We need f(x1 + x2) = f(x1) + f(x2)

f(x1 + x2) = a(x1 + x2) + b 

f(x1) + f(x2) = ax1 + b + ax2 + b = a(x1 + x2) + 2b

*Scalar multiplication*

We need f(kx) = k f(x)

f(kx) = a(kx) + b = kax + b

k f(x) = k(ax + b) = kax + kb

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So f(x) = ax + b doesn't meet either of the criteria for a linear transformation. You are right, though, in saying that it describes a line. As you can see from the calculations above, the problem in both cases is the constant term (b). Without the constant, the transformation f(x) = ax would be a linear transformation. 

f(x) = ax + b describes an *affine transformation*, which is not linear, but does still map lines to lines.