TITLE: Is the empty set linearly independent or linearly dependent?
QUESTION [24 upvotes]: Is empty set linearly independent or dependent?

REPLY [11 votes]: Since the correct definition of "linearly dependent" has not been spelled out in detail in any of the answers so far, let me add a new answer.  A subset $S$ of a vector space $V$ is defined to be linearly dependent if there exist finitely many distinct elements $s_1,s_2,\dots,s_n\in S$ and scalars $c_1,c_2,\dots,c_n$ which are not all $0$ such that $$\sum_{i=1}^n c_is_i=0.$$
(If $S$ itself is finite, you can just state this condition with $s_1,\dots,s_n$ being all the elements of $S$, since given any relation of this form you can just make the coefficients be $0$ for all the elements of $S$ you have not used.)
When $S=\emptyset$, the only possible collection of finitely many distinct elements of $S$ is the empty collection, with $n=0$.  But there does not exist any collection of $0$ scalars, not all of which are $0$.  After all, if you have a collection of no scalars, then vacuously all the scalars in your collection are $0$.
Thus $\emptyset$ is linearly independent (as a subset of any vector space).