Examples of linear transformations: Fourier transform
transform, Lorentz transformation
Finite dimensional linear transformations are what linear algebra is all about. In linear algebra, you can represent a finite dimensional linear transformation by a matrix. Then, you can transform a vector by multiplying it with the matrix. Multiplying two matrices by matrix multiplication corresponds to the transformation that you get when you do first one transform, then another, etc.
Linear transformations are used _everywhere_ in science and engineering! For example, in games like doom and quake, the processor probably spends much time doing linear algebra, because, for example, _rotation_ of objects is a linear transformation. So every time the player rotates himself into a new position, the game must linear transform the things that the player sees, to determine where the player will see them in his new position. These linear transforms are actually a type of Lorentz transform :s!
Infinite dimensional linear transformations are also interesting. For example, ordinary differentiation is a linear transformation, which assigns a function to its derivative function instead of a vector into another vector. There are many similarities between finite dimensional and infinite dimensional linear transformations. A branch of mathematics called functional analysis deals with transformations of functions.