9.1. Definition of vector spaces331We conclude this section by deriving some initial consequences of the vector space axioms.Proposition 9.9: Elementary consequences of the vector space axiomsIn any vector space, the following are true:(a) The additive unit is unique. In other words, wheneveru+v=u, thenv=0.(b) Additive inverses are unique. In other words, wheneveru+v=0, thenv=−u.(c)0u=0for all vectorsu.(d) The followingcancellation lawholds: ifu+w=v+w, thenu=v.Proof.We prove the first three properties, and leave the last one as an exercise. AssumeVis any vectorspace over a fieldK.(a) Consider arbitrary vectorsu,v∈Vand assumeu+v=u.Applying the law (A1) (commutative law) to the left-hand side, we havev+u=u.Adding−uto both sides of the equation, we have(v+u)+(−u)=u+(−u).Applying the law (A2) (associative law) to the left-hand side, we havev+(u+(−u))=u+(−u).Applying the law (A4) (additive inverse law) to both sides of the equation, we havev+0=0.Applying the law (A3) (additive unit law) to the left-hand side, we havev=0.This proves that wheneveru+v=u, thenv=0, or in other words,v=0is the only element actingas an additive unit.(b) Consider arbitrary vectorsu,v∈Vand assumeu+v=0.Applying the law (A1) (commutative law) to the left-hand side, we havev+u=0.