Covariance and contravariance in a coordinate co/basis

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I'm having trouble with the paragraph beginning "There is a convention which refers to "vectors"...

I think it's easily shown to be wrong with a scalar example. Suppose the change of basis is  . Then the dual transformation is  . And in higher dimensions, a change of basis   induces the dual transformation  . (Note: this verifies the invariance of the contraction   under a change of basis) This correction can remediate the third sentence of the paragraph, as well.

The fourth/final sentence is correct, although the terminological confusion might be explained by the fact that, while the pull-back of the cobasis   transforms contravariantly, the components of a covector (with respect to its cobasis) transform covariantly, and vice versa for the basis and components of a vector.--ScriboErgoSum (talk) 08:17, 21 November 2021 (UTC)Reply