I'm trying to understand what was the motivation behind "creating"
vector spaces. I know what the definition is but I find it difficult
to understand why the definition is constructed like that, what is
their "purpose" and why are they such a important part of linear
algebra.

Part of the answer is that linear maps are very nice, and show up a
*lot*. They show up all over the place. And there are many problems
that are *really* hard (especially in physics, trying to model
physical phenomena), but for which one can find a reasonably good
approximate answer by "linearizing", that is, by pretending that the
answer is a linear function. So, linear functions are pretty much the
bee's knees of functions: they show up a lot, they are cool, and they
are very, very useful.

So what we kind of want is to study linear maps. Vector spaces turn
out to have *just* enough 'structure' to them so that you can talk
about linear maps, and say useful things about them. That is, they
provide you with the scaffolding necessary to be able to talk about
linear maps. Now, when you build scaffolding, you are really trying to
find just the right balance between not building too much and not
building too little. If your scaffolding is too flimsy (too vague, too
general), then it won't support you and you won't be able to build
(you won't be able to say terribly much by way of useful things). On
the other hand, if it is too strong, then it takes too much effort to
build (you don't want a scaffolding that is just as hard to build as
the building you are trying to build). The definition of vector spaces
evolved through some years until it found just that perfect balance:
enough structure so you can say useful stuff, but not so much that it
is too specific; by keeping it as general as possible, it becomes
applicable to many things. This because a desirable thing when people
began to notice that many of the same arguments that were used for
specific instances (linear functions of real numbers) were
"essentially the same" as those used in others (linear functions on
the plane; linear functions between polynomials, etc). They are the
result of a process of 'abstraction', whereby people boiled away all
the stuff that was extra and not needed, and kept just the essence
needed to make sure all the important and interesting arguments could
still be made, and the desirable conclusions reached.

The formalization of vector spaces was done to build an abstract and
thus more powerful theory than the theory of systems of equations and
determinants which appear now as a particular case of vector spaces
(namely finite dimension and real or complex coefficients).

Hope it helps.

With respect to linear algebra, normally it focuses on finite vector
spaces, and in particular on R^n spaces (since all finite vector
spaces are isomorphic to R^n). Consider it as a stepping stone to
functional analysis (which deals with infinite-dimensional vector
spaces), if you wish.

Of course, as someone has already pointed out, finite-dimensional
vector spaces are *still* important, in multi-variable calculus
setting. I recall reading W. Kaplan, Advanced Calculus, he had some
good thoughts why linear spaces are so important in multivariable
calculus. Roughly, when you deal with multivariable setting, often the
only way to deal with the problem in a systematic way is linear
approximation. And once we don't have to deal with non-linearities, we
can use the same techniques for any number of dimensions (as long as
it's finite). Techniques, concepts, the structure of linear spaces
(aka vector spaces) which is the subject of linear algebra. Kaplan's
is another book I recommend to get and just browse through it a bit,
to see how linear spaces are essential for the whole development of
multivariable calculus.

Unfortunately, most of the linear algebra textbooks do not discuss
applications of linear spaces (in calculus or analysis).

      Vectors began life in applied mathematics as quantities with
magnitude and direction, represented by arrows of suitable lengths.  
You'll still find plenty of those in physics.  In the 19th century
(perhaps beginning with Grassmann) their algebraic properties were
increasingly emphasized, until Hilbert and others were prepared to talk
about infinite-dimensional vectors.  Dirac's wave mechanics and
Heisenberg's matrix mechanics were different special cases of the same
theory, which John von Neumann expressed in the more general
Hilbert-space form now used in quantum mechanics.  That well-developed
infinite-dimensional theory was brought down to earth (i.e. down to
finite dimensions and down to undergraduate level) by Halmos's 1942
text-book "Finite-Dimensional Vector Spaces" which is still IMHO better
than most of the variants which have followed it ever since.

      HTH

David C. Ullrich