Let Z denote the set of all integers with addition defined in the usual way and define the scalar multiplication, denoted ◦, by
a ◦ k = [[α]] ∙ k for all k ∊ Z
where [[α]] denotes the greatest integer less than or equal to α. For example,
2.25 ◦ 4 = [[2.25]] ∙ 4 = 2 ∙ 4 = 8
Show that Z, together with these operations, is not a vector space. Which axioms fail to hold?