TITLE: what does the set containing only the zero vector actually span?
QUESTION [10 upvotes]: I apologize if this sounds stupid but I am struggling to grasp the following concept. I understand that the span of the empty set is the zero vector. However, what does the set only containing the zero vector span? The zero vector as well? Also, are the following two phrases logically equivalent, "The span of the empty set is the zero vector" and "The empty set spans the zero vector". I believe they do but I don't want to make any assumptions since I'm not entirely sure myself. Thank you.

REPLY [5 votes]: In general, adding to a set $S$ of vectors a vector $v$ that is already in the span of $S$ does not change the span; in other words, in this case $S\cup\{v\}$ spans the same subspace as $S$ does.
This applies in particular to $S=\emptyset$ and $v=\vec0$; you already indicated that the span of the empty set contains $\vec0$, and so $\emptyset\cup\{\vec0\}=\{\vec0\}$ spans the same ($0$-dimensional) subspace as the empty set does.