4. Problems

Problem D-1. Prove these facts, using any laws above:

(a) $rv =$   0$\Rightarrow r = 0$ or $v =$   0.

(b) $rv = rw$ and $r \neq 0$ imply $v = w$ (cancellation of a scalar).

(c) $rv = sv$ and $v \neq$   0 imply $r = s$ (cancellation of a vector).

Note: (a) is on p. 31.

Problem D-2. Equivalent definitions of a vector space are possible. For instance, since $(-1)v = -v$, we could get away with not mentioning $-v$ in the definition of a vector space. In that case, we would omit (d). However, then we can't prove (j), so we need to make that one of our defining properties.

The problem: Starting from (a)-(c), (e)-(h), and (j) as the ``new'' definition of a vector space, define what $-v$ means and then prove (d), quoting the laws you are using.