- #1

dcramps

- 43

- 0

## Homework Statement

## Homework Equations

The 10 axioms:

1. If u and v are objects in V, then u + v is in V

2. u + v = v + u

3. u + (v+w) = (u+v)+w

4. There is an object 0 in V, called a zero factor for V, such that 0+u = u+0 = u for all u in V

5. For each u in V, there is an object -u in V, called a negative of u, such that u+(-u)=(-u)+u = 0

6. If k is any scalar and u is any object in V, then ku is in V

7. k(u + v) = ku + kv

8. (k + m)u = ku + mu

9. k(mu) = (km)(u)

10. 1u = u

## The Attempt at a Solution

First of all, are u, v, and w just (x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3}) and so on? If so, am I going about this the right way?

Axiom 1. Since x,y[tex]\in[/tex]R, then any (x,y)[tex]\in[/tex] thus the addition would hold. (Not really sure how to explain this one well)

Axiom 2. [tex](x_{1},y_{1})+(x_{2},y_{2}) = (x_{1}+x_{2}+_{1},y_{1}+y_{2}+_{1}) = (x_{2},y_{2})+(x_{1},y_{1}) = (x_{2}+x_{1}+_{1},y_{2}+y_{1}+_{1})[/tex]

Axiom 3.

[tex]u=(x_{1},y_{1})[/tex]

[tex]v=(x_{2},y_{2})[/tex]

[tex]w=(x_{3},y_{3})[/tex]

[tex]u+(v+w) = (x_{1},y_{1})+(x_{2}+x_{3}+1,y_{2}+y_{3}+1) = (x_{1}+(x_{2}+x_{3}+1)+1,y_{1}+(y_{2}+y_{3}+1)+1) = (x_{1}+x_{2}+x_{3}+_{2},y_{1}+y_{2}+y_{3}+2)[/tex]

[tex](u+v)+w = (x_{1}+x_{2}+1,y_{1}+y_{2}+1) + (x_{3},y_{3}) = ((x_{1}+x_{2}+_{1})+x_{3}+1,(y_{1}+y_{2}+1)+y_{3}+1) = (x_{1}+x_{2}+x_{3}+_{2},y_{1}+y_{2}+y_{3}+2)[/tex]

and so on for the rest...

Is this the correct approach? It seems that it is to me, but I may be way off here.

Last edited: