Thread: [SOLVED] Vector space

I am sure this is easy but I have been trying all day and just don't understand the question.

Determine whether the set, together with the indicated operations, is a vector space.

1. The set {(x, -x) : x is a real number} with the standard operations.

2. The set {(x, y) : x, y are a real numbers such that x > y} with the standard operations.

Any help would be appreciated. Thanks

Originally Posted by Neoxl

1. The set {(x, -x) : x is a real number} with the standard operations.

Let . Now if then and so so in . Similarly in where . So it is a vector space.

Originally Posted by ThePerfectHacker

Let . Now if then and so so in . Similarly in where . So it is a vector space.

don't you have to verify a list of like 10 axioms or something. that's what we had to do in my linear algebra class. i guess if the standard operations hold, then verifying that it is closed under addition and scalar multiplication is fine, but i don't know

Originally Posted by Jhevon

don't you have to verify a list of like 10 axioms or something. that's what we had to do in my linear algebra class. i guess if the standard operations hold, then verifying that it is closed under addition and scalar multiplication is fine, but i don't know

You do not need to. For example, since is commutative then certainly any subset! So it just comes down to checking the closure of addition and scalar multiplication.