Linear Algebra Linear Independence I was looking through a textbook and came across the following theorem: A finite set of vectors is linearly dependent if and only if one of the vectors is a linear combination of the vectors that precede it, in the ordering established by the listing of the vectors in the set. First of all, I thought sets were unordered lists of objects. Secondly, why does it have to be a linear combination of elements _preceding_? That portion of the theorem doesn't make much sense to me. Any explanations? Thanks.

I agree that this phrasing is not as clear as possible. What the author means is the following. Start selecting vectors from your set and each time you select a new vector check whether it is a linear combination of the ones you have already selected. This will happen for some vector if and only if this set of vectors is linearly dependent.

Note: if the set is linearly dependent, you will find a vector that is linear combination of the ones you selected prior to it, irrespectively of the order with which you are selecting your vectors.