Proof for Problem 1.1.1 of Shankars Prin of QM

[Mindstein]

I think I may be over-thinking this as I have had one formal course in QM and an independent study course in QM, but any help is MORE THAN GREATLY APPRECIATED!

Homework Statement

Prove that |-V> = -|V>

Homework Equations

He instructs us to begin with |V> + (-|V>) = 0|V> = |0>

The Attempt at a Solution

Let -|V> = |W>

Then

|V> + |W> = 0|V> = |0>

which implies that

|W> = |-V> ?

I'm really confused.

 

[Cruikshank]

Okay, first I think you mean problem 1.1.2, unless you have a different edition.
This is the kind of proof where you have to doubt everything. Act as if you are in a mathematical minefield. Only do the things you can completely justify.

I find myself wanting to say that 1V = V, but that actually isn't an axiom...

Assuming that for the moment,
V + (-1)V = 1V + (-1)V = (1 + -1)V = 0V = 0 (by part 1 of the same problem)
And thus (-1)V is the unique vector -V such that V + -V = 0

Looking up in other books, 1V = V is listed as an eighth axiom for linear spaces. Shankar described it by interpretation in a paragraph, but didn't actually state it overtly as an axiom.