Question

prove that the set of all integers is not a vector space

Answer 1

do you know what conditions must be satisfied for a set to be considered a vector space ?

Answer 2

that was the whole question

Answer 3

the question most likely is relevant to what you've learned in class. what have you learned in class with respect to a vector space ?

Answer 4

i.e., closure rules

Answer 5

one sec

Answer 6

its just general vector spaces and subspaces and linear combinations of vectors

Answer 7

the closure axiom is part of that section

Answer 8

i'd say the set of integers isn't a vector space because the set of integers isn't closed under scalar multiplication

Answer 9

well how do i prove that

Answer 10

it's hard to say. it depends on how you define scalar. if we take all real numbers to be scalars, then any fraction added to an integer results in a fraction (and not an integer). but your question is confusing since the set of integers is closed under + and *

Answer 11

because vectors are closed under division, and integers are not, then the integers do not form a vector space. for proof, take an odd integer and divide it by 2. your result is a number that is not an integer