- #1

torquerotates

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It is stated that I'm only allowed to use the following axioms.

let a,b,c be vectors and V is a vector space, then

1)a&b is in V then a+b is in V

2)a+b=b+a

3)a+(b+c)=(a+b)+c

4)0+a=a+0=a

5)a+(-a)=(-a)+a=0

6)a is in V implies ka is in V

7)k(a+b)=ka+kb

8)(k+m)a=ka+ma

9)k(ma)=(km)a

10) 1a=a

The book does it like this, and i think its wrong

0v=(0+0)v=0v +0v { axioms 4&8}

now subtract 0v from both sides { axioms ?}

we get 0=0v

you see the problem here? there's no justification for the subtraction step because there is no axiom allowing the step. Logically I assume that I'm only allowed to use the 10 axioms to prove this theorem.