What are...modules?

[@armchairLibrarian]

Just found your channel and wanted to thank you for creating clear and engaging videos! Please keep posting!

[@VisualMath]

Thank you so much: I hope some of the videos will turn out to be useful and enjoyable ☺

[@shashvatshukla]

Is this notion of module the same as modules over rings? a generalisation of vector space? Why do you choose module and not vector space? Since you keep promoting linear algebra which is the study of vector spaces.

[@VisualMath]

Ah, sorry, the terminology is bad: a module is a vector space, but not necessarily over a field (the ground thingy could be a ring, algebra,...). So still linear algebra 😀, but your matrix entries get more sophisticated.

[@Jaylooker]

Nice way to involve geometry

[@VisualMath]

Isn't it? In practice the geometric actions are not very useful (in higher dimensions it is just easier to write vectors instead of a 100d street sign), but this picture always worked really well as a motivation for me. In the end this is what is really going on, just writing vectors is easier.

[@Jaylooker]

@@VisualMath Using vectors is easier. To elaborate I think representation theory combining something algebraic like a group and something geometric like a module is useful. Note the bijection at 6:58

[@VisualMath]

@@Jaylooker Yes, I totally agree! That bijection is "quite basic" but indeed super useful. I usually abuse the bijection completely and simply identify reps and modules ;-)