## fleauworld 3 years ago Let Z denote the set of all integers with addition defined in the usual way, and define scalar multiplication, denoted o, by: alpha o k = [[alpha]].k for all k in Z where [[alpha]] denotes the greatest integer less than or equal to alpha, for example, 2.25 o 4 = [[2.25]].4 =2..4 = 8 show that Z, together with these operations, is not a vector space. Which axioms fail to hold?

1. myko

Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v

2. fleauworld

I agree with the second answer, but why the Distributivity of scalar multiplication with respect to field addition fail to hold ?

3. experimentX

straight out of wikipedia Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv let a=2.5 and b=3.5

4. fleauworld

oh I see never mind, thx :)

5. myko

let them better be a=2,5 and b=3,6

6. fleauworld

what was ur approach to finding the solutions please ? did you test for all axioms one by one ?

7. myko

not really, becouse all the rest are not afected by any weird definition. Just this two

8. experimentX

the other one also seem to work Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v a=4 and b=2.5

9. fleauworld

hum ok I see, thank you mate

10. myko

you have to choose two numbers where decimal part sum is bigger than 1