A set $S$ in a vector space $V$ is linearly dependent if it contains finitely many linearly dependent vectors.
What is the significance of this definition? Obviously for finite sets, this is no different than the original definition. So what significance does it hold for infinite sets? I know it's not saying that infinite sets are always linearly independent, because that's not true (for example, the set of all real numbers, an infinite set, is linearly dependent).
Can you give me an example of a way in which this definition can be useful? Any help is appreciated!