# Vector spaces

how do you prove the set of vectors "all ordered quadruples of positive real numbers" make a vector space?

## Answers and Replies

you show that the axioms for a vector space are satisfied. i'm sure they're in your book. it shouldn't be very hard, just direct verification. you put ""s around the phrase ordered quadruples.... that just means 4-dimensional vectors whose entries are real numbers, that's all.

The problem is that it isn't a vector space since there is no additive inverse vector, nor is scalar multiplication valid (unless you redefine vector addition, the zero vector and th field over which the vector space is defined).

Hurkyl
Staff Emeritus
Gold Member
how do you prove the set of vectors "all ordered quadruples of positive real numbers" make a vector space?
You don't because they don't.

To make a vector space, you need 3 things:
(1) A set of "vectors".
(2) A field of "scalars".
(3) An "addition" operation that takes two vectors and produces a vector.
(4) A "multiplication" operation that takes a scalar and a vector and produces a vector.

You've only specified (1), so that's certainly not enough to make a vector space. Once you specify the other three things, then we can start discussing the question of whether it's a vector space or not.