Span of an infinite set. Exam Question.

[heshbon]

Homework Statement

Suppose V=<Z> where Z is an infinite set, So Z spans V.
Suppose that every pair of distinct elements of Z is linearly independent.
Is it possible that V is finite dimensional? Justify your answer.

Homework Equations

The Attempt at a Solution

All the vectors must be linearly independent if any two are so...
I don't think you can as a basis for the space would have infinite linaerly independent vectors.

 

[Dick]

All of the vectors don't have to be linearly independent just because every pair is. Let Z be the set of all unit vectors in the first quadrant in the two dimensional plane. Since they all point in different directions, any pair is linearly independent.