I am trying to grasp the idea of vector spaces in my linear algebra textbook. The only thing I got out of it was that a vector space is set of elements or vectors (elements and vectors seems to be used interchangably which causes confusion) that must satisfies 10 rules that involves matrix addition and scalar multiplication in order to be called a vector space.

This concept is a whole lot more abstract compared to what was covered before: which was on system of linear equations and echileon forms.

Question: I still do not see the big picture or the whole point of learning vector spaces. I am trying to find the motivation to appreciate vector spaces. It has been 2 days and it is still an abstract concept to understand fully. Am I approaching vector spaces the wrong way?

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).

Matrices and quaternions aren't very good examples of vector spaces since there aren't any rules to multiply elements together in vector space, which is the thing you usually do with matrices and quaternions. They are good examples of (non-commutative) rings though.

Yes, they are non-commutative rings, but they are also vector spaces, and sometimes it's a good idea to think about them that way. By the way, they are also Lie groups. There are many ways to look at the same object. Sometimes you may want to think of the reals as a vector space over the rationals.

The short version of the formal definition is that a vector space is a set of things that can be added, multiplied and scaled (i.e., multiplied by number).

Are you sure you are thinking of a vector space? Vector spaces don't define multiplication between elements, only addition of vectors and multiplication by a scalar. I think you are thinking of a ring.

Oh, damn it. I meant to say "subtracted", not "multiplied". Sorry about that.

beside the axioms a set must prove to be a vector space, there are a few short cuts to explain vectors space(s). A vector can exist in any amount of vector spaces, so to keep its identity, it is expressed in every vector space differently depending on the vector space's basis vectors. Little ToDo for you:

vector (2,1,4) in a vector space H defined by basis vectors (1,0,0)x(0,1,0)x(0,0,1) over R(eal numbers) is expressed as (1,0.5,2) in vector space G over R(eal numbers), Find the basis vectors of vector space G. Define the matrix that transforms from vector space G to vector space H.

if you solve this, you will have a little decent understanding of vector space(s).

JohnnyCode's language is non-standard. I think what he means to say is that a vector can be expressed using coordinates, but the coordinates depend on the basis we are using.