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. 

I can list two important reasons linear transformations are important.


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*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. 

*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 $W$). 

