Is the set of integers a vector space?

My friend says it's not, because he thinks vectors have to have have at least two elements, otherwise they're scalars. I say that (Z, +, *) satisfies the axioms of a vector space and that's all you need.

What does /sci/ think?

>>11 >>12
Fail, >>10 defined no norm. {0} need not be a normed vector space, so the concept of magnitude is not intrinsically defined over its elements. >>10 is correct; it does not have magnitude.

>>1
As everyone has been saying, your friend is correct for incorrect reasons. First of all "elements" is not the term you're looking for; the number of elements is literally the total number of unique vectors in the space, so it's always either zero, one, or infinity (with some cardinality). You're thinking of the term "dimensions", and your friend is wrong when he says you need at least two.

The set of integers is not a vector space because it's not a field. The set of real numbers, however, is a field, so it is a vector space (even though it has only one dimension).