I was reading different definitions of vector space but still I have some dark places without meaning... I can't understand exactly what type of relation is defined between the vector space and the field over which is defined...
I can't understand what exactly is the meaning of over. A example that I take from Wikipedia:
A vector space over a field $F$ is a set $V$ together with two operations that satisfy the >eight axioms listed below. Elements of $V$ are called vectors. Elements of $F$ are called >scalars. In this article, vectors are distinguished from scalars by boldface.
I can't understand the relation between scalar of field $K$ and the $n$-tuple of any $V$ vector space. Where is the relation on dimensions, tuples and scalars? Of course the tuples are expressed in terms of $K$ but the expression over is a bit confusing.
More obscure is to call to the scalar multiplication as a "external" operation where the space vector is defined over $K$, how it can be "external" if the vector space itself is defined over $K$? External over itself?
These types of definitions seem to me just justified by practical uses (for physics or so) but lack A LOT of real deepness in a mathematical sense. I don't read something like this on other mathematical abstract areas as topology or measure theory ($\rightarrow$ forget this if you want, these are my REAL thoughts, maybe not good to express publicly, and I get angry when I dislike something, sorry, just don't read if you don't want to know my REAL opinions that, of course, are not "correct" in any political sense).
I will appreciate any enlightenment about this.
Here's a neat way to think about it, I think.
A vector space is a mathematical structure that can be built as follows. We start with a set of objects that, first of all, you are able to add/subtract together (or, in the language of algebra, an abelian group).
As long as you are able to add things, it makes sense to define, at the very least, integer multiplication as a shorthand for addition. So, for example, let's say that our vector space consists of the vectors "$x$" and "$y$". We can say things like $$ x + x = 2x\\ 2x + 3x = (2+3)x = 5x\\ x - x = ``0"\\ 2 \cdot (3x) = (2 \cdot 3)x = 6x\\ 2x + 2y = 2(x + y) = (x+y) + (x + y) $$ Now, remember, we haven't said anything about multiplying by scalars, we've just said that multiplying by integers already sort of makes sense anyway.
What we do in a vector space is extend that multiplication to include multiplication by elements in a field, and say that our multiplication should obey the same "common sense" rules (the axioms of a vector space).
So, if we take the field over which the space is defined to be $\Bbb Q$, it now makes sense to talk about, in addition to things like $2x$ and $3y$, things like $\frac 12 x$ and $\frac 23 y$. The advantage of using a field is that scalar multiplication is nice in that (among other properties) it is now "invertible", unless you multiply by $0$.
So, for example, we can now say that $\frac 12 \cdot (2x) = (\frac 12 \cdot 2) x = x$.
In a sense, you can imagine that we've added a whole bunch of elements (such as $\frac 12 x + \frac 23 y$) to our original set by adding in the rule "you can also multiply by scalars".