Question

Prove that 0v = 0, for all v in V: I put down: from property 5 and property 8 we know that: v = 1v = (1+0)v = 1v + 0v = v + 0v = v+0 =v. thus, -v+v = -v+(v+0v) = (-v+v) + 0v. He said it was right but not totally right, and to prove it for anything (general case), ,and something about expressing W in terms of v in order to do that. Anyone?

Answer 1

Question 1.

Prove that $0v=0$ for all $v\in V$.

Solution. By distributivity of scalar multiplication with respect to field addition (axiom 6) we have

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

By axiom 4 there exists an additive inverse $-0v$ of $0v$, such that $0v+(-0v)$ is the zero vector $0\in V$. Then adding $-0v$ to both sides of the above equality by associativity of addition (axiom 1) and definition of the zero vector we get:

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

Thus, $0v=0$ as desired. $\Box$