1. ## properties

Show the following:

• $\displaystyle |0 \rangle$ is unique
• $\displaystyle 0|V \rangle = |0 \rangle$
• $\displaystyle |-V \rangle = -|V \rangle$
• $\displaystyle |-V \rangle$ is the unique additive inverse of $\displaystyle |V \rangle$

So there exists a null vector $\displaystyle |0 \rangle$ obeying $\displaystyle |V \rangle + |0 \rangle = |V \rangle$. Suppose there is another vector $\displaystyle |0' \rangle$ such that $\displaystyle |V \rangle + |0' \rangle = |V \rangle$. Then $\displaystyle |0 \rangle + |0' \rangle = (|V \rangle + |V \rangle)-(|V \rangle + |V \rangle) = |0 \rangle$. Hence $\displaystyle |0 \rangle$ is unique.

We know that $\displaystyle |0 \rangle = (0+1)|V \rangle + |-V \rangle$. This is equaled to $\displaystyle 0|V \rangle + |V \rangle + |-V \rangle = 0|V \rangle$.

We know that $\displaystyle |V \rangle + (-|V \rangle) = 0|V \rangle = |0 \rangle$. Then by associativity, $\displaystyle |V \rangle + (|-V \rangle) = 0|V \rangle = |0 \rangle$. Hence $\displaystyle |-V \rangle = -|V \rangle$.

Suppose there is some $\displaystyle |W \rangle$ such that $\displaystyle |V \rangle + |W \rangle = |0 \rangle$. Then $\displaystyle |V \rangle + |W \rangle = |V \rangle + |-V \rangle$. So $\displaystyle |W \rangle = |-V \rangle$ since $\displaystyle |0 \rangle$ is unique.

Is this correct?

2. Your ideas are mostly correct, but can be simplified to be more elegant.

If $\displaystyle |0\rangle$ and $\displaystyle |0'\rangle$ are both identity elements, then $\displaystyle |0\rangle=|0\rangle+|0'\rangle=|0'\rangle$, where the left equality comes from $\displaystyle 0'$ being the identity, and the right equality comes from $\displaystyle 0$ being the identity.

$\displaystyle 0|V\rangle=(1-1)|V\rangle=|V\rangle-|V\rangle=|0\rangle$

In this next one I assume $\displaystyle |-V\rangle$ is the inverse element of $\displaystyle |V\rangle$. Then $\displaystyle |-V\rangle=|-V\rangle+ 0|V\rangle=|-V\rangle+ (1-1)|V\rangle=|-V\rangle+ |V\rangle-|V\rangle=|0\rangle-|V\rangle=-|V\rangle$

Suppose $\displaystyle |V\rangle+|W\rangle=|0\rangle$. Then $\displaystyle |W\rangle=|W\rangle+|0\rangle=|W\rangle+|V\rangle+ |-V\rangle=|0\rangle+|-V\rangle=|-V\rangle$