TITLE: Linear span of the empty set
QUESTION [17 upvotes]: Why is the span of the empty set defined to be $\{0\}$? It is known that the span of any nonempty set of vectors in a vector space $V$, gives a subspace of $V$, and it is stated in “Linear Algebra Done Right” by Axler, that to be consistent with this, the span of the empty set is defined to be $\{0\}$? Is this the only reason or does this definition prove useful in other ways later on?

REPLY [10 votes]: The linear span of a subset of a vector space is the smallest linear subspace which contains it or, equivalently, the intersection of all the linear subspaces which contain it.
If you carry out this for the empty set, you'll get the correct result.