What are covariant and contravariant coordinates?

Usually when people first study special relativity or general relativity, they will get introduced to the concept of covariant and contravariant coordinates. They will learn that these two sets of coordinates are connected through the metric tensor $g_{ij}$ and that these coordinates are tensor themselves. As always in physics, this notion and the way to talk about it is obscured and the connection to a proper mathematical treatment is left out. This post is a short introdcution into covariant and contravariant coordinates in terms of linear algebra. Continue reading “What are covariant and contravariant coordinates?”