Question

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?

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

oh I see never mind, thx :)

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

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

a=4 and b=2.5

Answer 9

hum ok I see, thank you mate

Answer 10

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