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.

http://home.pacbell.net/bbowen/covariant.htm

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 >

And

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?