A set S of vectors is called
linearly independent if for every
vector s
j in S, there do not exist values of a
1, a
2, ... ,
a
n, such that (a
1)(s
1) + (a
2)(s
2) + ... + (a
n)(s
n) =
s
j and a
j=0, where a
1, a
2, ... , a
n are
scalar and s
1,
s
2, ... ,s
n are
elements of S.
You can visualize a vector as a funny kind of arrow. It can be stretched out or shortened as much as you want, but it's direction can never be changed. You can combine vectors by laying them out, the tip of the first one connected to the base of the next.
To say a bunch of vectors are linearly independent means that no matter how you combine them, you can never find a combination of them whose endpoint is the same as one of them. For any point you can get to, there's only one way to get there. If the vector A ends at a certain point, and there is also a combination of B, C, & D that ends there too, then A, B, C & D are not linearly independent.
In the definition, stretching is the same as multiplying.
Linear independence is good because it ensures that there's only one combination of vectors that gets you to each point. So if you ask "how can I get to point X" there will be only one answer. If you are using a non linearly independent set of vectors to give directions to X, then there could be an infinite number of answers to that question.
in the real world you could take a set of vectors A = walk down a road and B = climb a ladder. They are linearly independent, since everywhere you get to there's only one way. But if you add C = ride a bike down the road, then they are not linearly independent - you could get to a point 1 mile down the road by a infinite number ways.