The point of introducing vector spaces axiomatically (the 10 rules are called "axioms") is that you can recognize them in many situations, and then a lot of the knowledge you have will apply. The short version of the formal definition is that a vector space is a set of things that can be added,
multiplied [EDIT: subtracted] and scaled (i.e., multiplied by number).
Besides R^n as a vector space over the reals, there are many other things that are vector spaces:
* Polynomials of degree 3
* Sequences of real numbers
* Functions from reals to reals
* Continuous functions from reals to reals
* Complex numbers (they are a 2-dimensional vector space over the reals)
* Quaternions (they are a 4-dimensional vector space over the reals)
* unsigned integers, considering XOR as the addition (you can think of them as a 32-dimensional vector space over the field of two elements)
There will be situations where you will find a vector space while solving a problem, and I give you an example below. But before I lose you in the details of a complicated problem, let me say that the most important benefit of learning about vector spaces, like most of math, is that it will train you to look at problems in new ways.
And now the example. Let's try to find a closed formula for the Fibonacci sequence. That is, we want to find a formula F[n] such that
F = 0, F =1, F[n+2] = F[n+1]+F[n]
Ignore for a second the first two conditions and concentrate on the last one. The set of sequences that satisfy that equation is a vector space, because the sum of two sequences that satisfy that equation satisfies that equation and scaling a sequence that satisfies the equation also satisfies the equation. As an exercise, you should check that all the axioms are satisfied.
Once you learn a bit more from you book, you'll be able to determine that this vector space has dimension two (because the sequence that starts A=1, A=0 and the sequence with B=0, B=1 are linearly independent and can generate any member of the vector space), so all you have to do is find two linearly-independent formulas that satisfy F[n+2] = F[n+1]+F[n] and you know that you can find the formula for Fibonacci as a linear combination of the two. These two formulas are geometric series of the form
F[n] = x^n
In order to find appropriate values for x, write F[n+2] = F[n+1]+F[n], which is x^(n+2) = x^(n+1)+x^n, and after dividing by x^n you get
x^2 = x+1
Solve this second degree equation, find the two values of x that make the formula work and then figure out the coefficients that you have to use to combine the two geometric sequences (Impose F=0 and F=1 and you'll get a system of 2 equations with 2 unknowns which is easy to solve).
Edited by Álvaro, 27 September 2013 - 11:26 AM.