Instructor's Solutions Manual, Section 1B Exercise 4 **4** The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in the definition of a vector space (1.20). Which one? SOLUTION The additive identity requirement in 1.20 begins "there exists an element 0 $\in$ V ...". This condition is not satisfied by the empty set. Edward Frenkel --- Commentary: $\textbf{Exercise 4.}$ The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in the definition of a vector space (S.20). Which one? $\textbf{Solution 4.}$ The additive identity requirement in S.20 begins "there exists an element $0\in V$ …". This condition is not satisfied by the empty set. --- $\textbf{Exercise 4.}$ The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in the definition of a vector space (S.20). Which one? $\textbf{Solution 4.}$ The additive identity requirement in S.20 begins "there exists an element $0\in V$ …". This condition is not satisfied by the empty set. $\textit{Commentary:}$ This exercise highlights the importance of the existence of the zero vector in the definition of a vector space. The empty set fails to be a vector space because it does not contain a zero vector. This requirement is crucial because the zero vector serves as the identity element for vector addition, which is a fundamental operation in a vector space. Without a zero vector, many of the other vector space axioms wouldn't make sense. This exercise also illustrates the point that a vector space must be non-empty. While the other vector space axioms don't explicitly require the set to be non-empty, the existence of a zero vector implicitly does. $\textit{Example:}$ The set ${0}$ containing only the zero vector is a vector space over any field $\mathbb{F}$, with the usual addition and scalar multiplication. This is the smallest possible vector space. The empty set ${}$ is not a vector space over any field, because it does not contain a zero vector. The set $\mathbb{R} \setminus {0}$ of non-zero real numbers is not a vector space over $\mathbb{R}$, because it does not contain the zero vector $0$, even though it satisfies all other vector space axioms.