I am reading "Linear Algebra Done Right" by Sheldon Axler.
This book contains the following exercise:
The empty set is not a vector space. The empty set fails to satisfy only
one of the requirements listed in 1.19. Which one?
This exercise says that the empty set fails to satisfy only one of the axioms of vector spaces.
I think that the empty set fails to satisfy the following two axioms:
There exists an element $0\in V$ such that $v+0=v$ for all $v\in V$.
For every $v\in V$, there exists $w\in V$ such that $v+w=0$.
Does the empty set really fail to satisfy only one of the axioms of vector spaces?
There doesn't exist an element $0\in \emptyset$ such that $v+0=v$ for all $v\in \emptyset$.
But the following axiom uses $0$.
No problem?
For every $v\in V$, there exists $w\in V$ such that $v+w=0$.
Best Answer
The first axiom you cite is the one you are looking for.
The second one is satisfied by the empty set. It states that if you take some $v\in\emptyset$, then there is some $w\in\emptyset$ for which $v+w=0$. This is an implication whose antecedent (the part including the "if") never holds, hence the implication itself is true (since "false implies true" and "true implies true" are implications which are true).