Motivation for linear transformations

Sometimes I hate classes for mathematicians. It is not their precision and formality in building the concepts, but they never give a motivation.

So in the course my professor started right with the definition of vector space, assuming I suppose that everybody knows what he is talking about. Well, reading some books for beginners like me I've realized that vector spaces are actually a generalization of working with the properties of Euclidean spaces.

Now the topic is about linear transformations. I understand the definition, they are special cases of mappings. Well, the thing is that I don't know why the definition has to be so, what is the real motivation for such a definition?. what is behind the meaning of 'linear'? My first impression is that maybe it has something to do with just preserving the operations of vectors, though I don't know why. What makes linear transformations to be special compared with those that are not linear?

Sorry for this question, maybe it is too naive but it's really important to me.

Vector spaces, as you correctly say, are generalizations of Euclidean spaces, in particular the structure of lines, planes, and hyper planes in Euclidean space.

Now, the definition of vector space is actually an axiomatization of some of the properties of Euclidean spaces. Now, once the abstract notion of vector space is in place one starts noticing that many things that previously did not appear to have much to do with Euclidean spaces are actually vector spaces (e.g., spaces of functions, spaces of polynomials, spaces of solutions to linear equations, spaces of solutions to certain differential equations, etc.).

With so many examples of vector spaces their study is well motivated. Now, once you have two vector spaces $V,W$ it is a very natural question to understand how they relate to each other. Since vector spaces are sets with extra structure it is most natural to consider how the sets relate when the extra structure is preserved. One way to relate sets is by function $f:V\to W$. But we don't want any old function but only those that respect the extra structure. So we demand that $f(u+v)=f(u)+f(v)$ and that $f(\alpha v)=\alpha f(v)$. And voila, there you have linear transformations motivated - you simply want to study not just a particular vector space in isolation but how different vector spaces relate.

As for calling them linear transformations, this is a bit for historical reasons. But basically, linear transformations can be shown to be precisely those that preserve all linear entities in the domain. By linear entities think of lines, planes, and hyperplanes. So such linear entities in the domain of linear transformation will be mapped to such linear entities in the codomain.

I can list two important reasons linear transformations are important.

1. They show up everywhere. For example, the operation of taking the derivative is a linear operation the vector space of polynomials. Projections, shearings, scalings, and so on, are also examples of linear transformations.
2. They are extremely easy to describe. If you have a linear transformation $T:V \to W$, then if you know what happens to the basis vectors $\{ e_i \}$ in $V$, then you know what happens to every vector in $V$! This makes linear transformations very easy to describe: you can describe it entirely by its matrix (once you have chosen a basis for $V$ and $W$).

Agree with most answers here. However, none of those attempt to really make it explicit the importance of linear transformations, and why we really need them . Here is my attempt.

In most systems, we want to study the effect the system has on its inputs. For example, an acoustic engineer may want to study the effect of attenuation by air on sound waves. In this case, sound waves are the input and air within the propagation volume acts as a system which impacts the waves. Now, such systems are usually complex to describe. Linearization is a first approximation to describing these systems. In many cases, such approximations are justified as the nonlinear effects are not very pronounced, or may be very small for most of the input range.

Inputs can also be approximated as a superposition of linear signals (for more info, look at Fourier transform). In particular, we attempt to approximate an input signal as a superposition of certain basis input signals (preferably an orthogonal basis). You might have seen such a basis set while working with 2D vectors in physics. The i and j unit axes used to describe any 2D vector is an example of an orthogonal basis.

This helps a great deal because now we can decompose any signal into a linear combination of such basis signals. And by the property of superposition, the effect of system (linear transformation) on the input signal can be computed as a linear combination of the effect of system on basis signals. Thus, using linear system theory, we can describe the effect of any system on any signal so long as we know the effect of the system on the basis signals.