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.
 A: 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.
A: The ordering is really not important. In my opinion, a better definition would be:
A set of vector is linearly dependent if there is at least one vector that can be expressed as a linear combination of the rest.
Another one:
A set of vectors $\lbrace v_1,v_2,...,v_n \rbrace$ is linearly independent if and only if the only solution to:
$$a_1v_1+a_2v_2+\cdots+a_nv_n=0$$
is: $a_i=0$ for all $i$.
Your deffinition is right though, the idea is that if you can go through all vector of the set in any order, and in some point you find one that is a linear combination of the preceding, then the set is linearly dependent.
