My textbook, Introduction to Probability, first edition, by Blitzstein and Hwang, says the following:
In a location-scale transformation, starting with $X \sim \text{Unif}(a, b)$ and transforming it to $Y = cX + d$ where $c$ and $d$ are constants with $c > 0$, $Y$ is a linear function of $X$ and Uniformity is preserved: $Y \sim \text{Unif}(ca + d, cb + d)$. But if $Y$ is defined as a nonlinear transformation of $X$, then $Y$ will not be linear in general. For example, for $X \sim \text{Unif}(a, b)$ with $0 \le a < b$, the transformed r.v. $Y = X^2$ has support $(a^2, b^2)$ but is not Uniform on that interval.
Firstly, isn't what the authors describe here as a linear function actually an affine function - not linear (that is, isn't $Y = cX + d$ an affine function -- not a linear function)?
And lastly, how is $Y = X^2$ not Uniform on the interval $0 \le a < b$? I'm having difficulty understanding how this is the case.
I would greatly appreciate it if people could please take the time to clarify this.
Yes, the authors are saying 'linear' for 'affine'.
If $X$ has uniform distribution on $(0,1)$ then $Y=X^{2}$ does not have uniform distribution on $(0,1)$: $P(Y \leq y)= P(X \leq \sqrt y)=\sqrt y$ for $0<y<1$. Note that $X^{2} <X$. $X^{2}$ assigns higher probabilites than $X$ for values near $0$.