Question

Prove that 0v = 0 for all v in V, and use this to prove that (-1)v = -v for all v in V.

Answer 1

Question 1.

Prove that $0v=0$ for all $v\in V$, and use this to prove that $(-1)v=-v$ for all $v\in V$.

Solution. By distributivity of scalar multiplication with respect to field addition

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

Subtracting $0v$ from both sides of this equality we get $0v=0$.

Furthermore, it follows from the axiom $1v=v$ that

$v+(-1)v=1v+(-1)v=(1-1)v=0v=0.$

This means that $(-1)v=-v$. $\Box$