I'm kind of bored at the moment, so I'm going to explain the basics. An ordered pair of real numbers is written as (x,y). It's "ordered" in the sense that (x,y) is usually not the same as (y,x). For example, (3,2) isn't equal to (2,3). Two ordered pairs (a,b) and (c,d) are equal if and only if a=c and b=d.
The sum of two ordered pairs is defined by this formula: (a,b)+(c,d)=(a+c,b+d)
The product of a real number and an ordered pair of real numbers is defined by this formula: a(b,c)=(ab,ac).
In this context, the ordered pairs are called "vectors" and the numbers are called "scalars". The operations defined above are called "addition" and "scalar multiplication"
It's common to write vectors in bold and real numbers not in bold. For example, ##\mathbf x=(x_1,x_2)##. For a person who has some experience with proofs, it's not hard to show that these ordered pairs, and the operations of addition and scalar multiplication defined above, satisfy a number of conditions.
1. For all ##\mathbf x, \mathbf y, \mathbf z##, we have ##\mathbf x+(\mathbf y+\mathbf z)=(\mathbf x+\mathbf y)+\mathbf z##.
2. For all ##\mathbf x, \mathbf y##, we have ##\mathbf x+\mathbf y=\mathbf y+\mathbf x##.
...
There are 8 of these conditions. The full list can be found here:
http://en.wikipedia.org/wiki/Vector_space#Definition
Ordered pairs are not the only thing that satisfies these conditions. Consider e.g. functions that take real numbers to real numbers. We can define the sum of two functions by saying that f+g is the function such that (f+g)(x)=f(x)+g(x) for all real numbers x. We can define the product of a number k and a function f by saying that kf is the function defined by (kf)(x)=k(f(x)) for all x. Then we can prove that these two operations satisfy the same list of conditions as the two operations we defined earlier:
1. For all f,g,h, we have f+(g+h)=(f+g)+h.
2. For all f,g, we have f+g=g+h.
...
The "vector space" concept generalizes this idea by saying that if V is any set (like the set of ordered pairs of real numbers, or the set of functions that take real numbers to real numbers) with two operations that satisfy these conditions, then we call that set a "vector space". The elements of that set are then called "vectors". So the two examples above are actually two examples of vector spaces.
The
dimension of a vector space can be defined in several different ways. I like to define "linearly independent" first, and then define the dimension like this: A vector space V is said to be n-dimensional if it contains a linearly independent set with n vectors, but none with n+1 vectors. If V is n-dimensional, we call n the dimension of V.
Unfortunately the concept of linear independence is a bit tricky and takes a while to understand. I won't try to explain it here. You will have to look it up.
If a vector space contains a linearly independent set with n vectors, for all positive integers n, then that vector space is said to be infinite-dimensional. A vector space that isn't infinite-dimensional is said to be finite-dimensional. Linear algebra is (roughly) the study of finite-dimensional vector spaces. It doesn't have a lot to do with lines.
I have mentioned two different vectors spaces in this post. The first one is 2-dimensional. The second is infinite-dimensional.
