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I have recently replied to a comment posted by Ingeniero concerning the relation between contravariant/covariant vector components and basis vectors transformation matrix.
Hope it makes things clearer.
See the comment

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New! Scan our messenger code to be easily and directly in touch with us via facebook messenger chat!
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Four months after having been lauched, the stats show an increasing interest for einsteinrelatively.com. I am hoping such a growing trend for the next following months also!
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You can now follow us on facebook by liking the following page
https://www.facebook.com/einsteinrelativelyeasy
It will be a place where i will give an update of any new article, any interesing reading or more generally of any interesting news about Relativity.
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We are happy to announce the start of the article series dedicated to General Relativity.
You can already read the Introduction to General Relativity
From now on, we will be writing articles related to both Special and General Relativity theories in parallel.