Seeing that a lot of linear algebra is based on matrices and operations on that representation, I wonder at which point came the notion of transposition of a linear map (LM) (or matrix as well). Both definitions seem very different and the linear map one seems a lot more convoluted with all those details on isomorphisms of dual spaces <-> vector spaces and etc... almost like someone tried very hard for both matrix and LM transposes to type the same.
If the answer to that question is that LM transpose notion came first then it seems more natural to have a notion on matrix transposition as we do now however, I only knew about matrix transposition and I'm not aware of any uses of LM transposition that justify the ubiquity of matrix transposition. It seems like an interesting coincidence following this line of reasoning.
If the answer to that question is that matrix transpose notion came first then it suggests that the "Dual Space" notion was developed/created/discovered as a way to enrich the theory with a way for transposition to work/make sense in terms of linear maps. I say this because, from what I know, only vector spaces have this notion and it seems to have some sort of a more general context waiting to be explored!
Does anyone have any scientific evidence that can answer the question? I'd also appreciate if you could educate me and correct my understanding about this topic or my line of reasoning.