Duality in Linear Algebra: Dual Spaces, Dual Maps, and All That

An exploration of duality in linear algebra, including dual spaces, dual maps, and dual bases, with connections to linear and bilinear forms, adjoints in real and complex inner product spaces, covariance and contravariance, and matrix rank.
More videos on linear algebra: • Linear and Multilinear...

[@tomasvaldivieso126]

Your expository style is incredibly clear. It maintains the right amount of abstraction while still providing "concrete" examples to help the listener make sense of the material. Thank you for your hard work in making these videos :)

[@aziz0x00]

great work

[@M3GAprincess]

THANK YOU!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@timpani112]

This is great content for sure. Over the years the importance of duality has slowly started to become more evident to me, but I feel like my journey there could've been much more efficient if more focus had been placed on the concept from an early stage in my mathematical education (at university level). At what stage in a student's education do you think it is appropriate to start focusing on duality as a concept a bit more earnestly? I'm starting to suspect that it can be done already at the undergrad level (maybe even lightly introduced as early as one's initial linear algebra course if done the right way), but I'd be interested in getting the perspective of others.

[@blargoner]

Thanks for your comment, and good question! I agree duality can be introduced to undergrads if done right. I initially saw it as an undergrad but it was presented poorly, as is commonly the case. It only clicked for me later when I saw the symmetric definition, with the scalar product as a generalization of the inner product allowing for similar geometric reasoning, and how that can be used in proofs like the cyclic decomposition theorem, etc. I think all of that can be conveyed to undergrads. I'd be curious what others think.

[@jalepezo]

BIUTIFUL