As i am sitting here reading some lecture notes on lie algebras I found myself getting stock because of the word "rank". As I understand rank, it's just the dimension of the image of a linear map and therefore rank is just a fixed number associated to a particular linear map. I have tagged a part of the notes, and as you can see there are several possible rank's for a certain exterior derivative say $d: \wedge^1 g^* \rightarrow \wedge^2 g^* $. The notes goes on to rank 2 and 3 as well. I'm sure this is just me missing something obvious, but I can't figure out how rank is being defined in this context.
Can someone help me understand this please?
