always it amuses me the incredulity of people when they receive an explanation of the covariant and contravariant behavior of linear algebra matters.
If one has a change of basis , you gotta to specify the basis that is assigned.
If you begin by chosing the canonical basis , where
and another basis, say
then we have and . From here one can resolve: and .
So if a vector is an arbitrary (think… ) element in then its expression in the new basis is:
So, if are the component in the old basis, then are the components in the new one.
Matricially what we see is this:
This corresponds to the relation
which -contrasted against in the change between those basis- causes some mind twist :D… :p
One says then that the base vectors co-vary, but components of vectors contra-vary.
For more: Covariance and contravariance of vectors.