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?

Question

Douglas Parrish

Mathematics

16 October, 23:13

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?

+3

Answers (1)

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Myla Blevins · 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