How to show if a given array of numbers is a vector?

[shinobi20]

Homework Statement

I'm reading Zee's book Einstein Gravity, I'm in the section where he said that given an array of two numbers p=(ap¹, bp²), it is not a vector unless a=b. He just stated it without really showing how it must be like that. I know that a vector should satisfy a transformation p'=R(θ)p with R as the rotation matrix.

In the exercises he also asked to prove that (p²q³, p³q¹, p¹q²) is not a vector by checking how it transforms under rotation.

Homework Equations

p'=R(θ)p with R as the rotation matrix

The Attempt at a Solution

For the first part, p'=Rp yields (ap¹cosθ - bp²sinθ, ap¹sinθ + bp²cosθ). I'm not sure what this is implying.

 

[andrewkirk]

The question cannot be answered without a great deal more context which, from the sound of it, may need to be the whole containing chapter.

A vector is an element of a vector space, and is defined in relation to that space. To say something is a vector without the context of knowing what vector space we are talking about is like saying a person is a 'member'. A member of what?

Any object can be turned into a vector by constructing a vector space around it, using it as a basis element.

An ordered pair of real numbers is trivially an element of the vector space $\mathbb{R}^2$, but I doubt that's what the author is talking about.

 

[shinobi20]

The author wants to show that it is not a vector if it doesn't preserve the length after rotation. This is tensors actually.