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:

* Matrices

* Polynomials of degree 3

* Polynomials

* 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] = 0, F[1] =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[0]=1, A[1]=0 and the sequence with B[0]=0, B[1]=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]=0 and F[1]=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.**