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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?
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?

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myko
 3 years ago
Best ResponseYou've already chosen the best response.1Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v

fleauworld
 3 years ago
Best ResponseYou've already chosen the best response.0I agree with the second answer, but why the Distributivity of scalar multiplication with respect to field addition fail to hold ?

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.2straight 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

fleauworld
 3 years ago
Best ResponseYou've already chosen the best response.0oh I see never mind, thx :)

myko
 3 years ago
Best ResponseYou've already chosen the best response.1let them better be a=2,5 and b=3,6

fleauworld
 3 years ago
Best ResponseYou've already chosen the best response.0what was ur approach to finding the solutions please ? did you test for all axioms one by one ?

myko
 3 years ago
Best ResponseYou've already chosen the best response.1not really, becouse all the rest are not afected by any weird definition. Just this two

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.2the other one also seem to work Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v a=4 and b=2.5

fleauworld
 3 years ago
Best ResponseYou've already chosen the best response.0hum ok I see, thank you mate

myko
 3 years ago
Best ResponseYou've already chosen the best response.1you have to choose two numbers where decimal part sum is bigger than 1
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