Answer to Question #187896 in Linear Algebra for sabelo Zwelakhe Xu

Question

Show that in the definition of a vector space v the condition about existence of additive inverse can be replaced with the condition:0v=v for all v€V

Answer 1

Given that V(F) is a vector space

$0.v=(0+0).v=0.v+0.v$

$\Rightarrow 0.v+0=0.v+0.v$ Where 0 is the identity element in v

$\Rightarrow 0.v=0$ (by left cancellation law)

hence $0.v=0 \text{ for all } v\in V$