Contravariant/covariant basis question

This isn’t a question that will be of interest to most science buffs here so I apologize if you wasted your time clicking on it. I’m hoping a mathematician or maybe Chronos will know the answer.

The above site says that given an Euclidean orthonormally represented vector

V = < 5 , 12 > you can define two new basis sets:

A[sub]1[/sub] = < .5 , 0 >
A[sub]2[/sub] = < .75 , .5 >


A[sup]1[/sup] = < .2 , -3 >
A[sup]2[/sup] = < .0 , 2 >

He then says that the first set is contravariant and the second set is covariant.

How can you have a covariant or contravariant basis set for fixed vector? And if you can, then what is that that makes them covariant and contravariant?

Isn’t it the using of subscripts versus superscripts all that makes the two basis sets covariant vs contravariant here? He could as easily have written
A[sub]1[/sub] = <2, -3>
A[sub]2[/sub] = <0, 2>
and called that the contravariant basis. (Well, that’s kind of backwards; subscript vs superscript is determined by whether it’s contravariant or covariant, not the other way around). I think maybe you’re looking for something too deep for this simple example.

I’m not exactly sure what your question is, but it seems that the author is just saying that you can represent the vector <5,12> with any number of different coordinate systems depending on the basis you choose. if you choose your basis to be the standard basis which will yeild the 2x2 (in R^2) identity matirx. but if you choose any 2 independent vectors you will be able to refer to the same vector with a vector that looks different than <5,12> but is really just a differnet way of writing the same thing. I doubt I answered your question, but maybe I’ll get the discussion started