Dual Bases and Dual Maps

Linear functionals, dual spaces, dual bases, and dual maps.

[Karim-nqbe]

Amazing book and videos, clear and concise. Thank you very much.

[williamgu]

Hi! This is William from Carnegie Mellon University's course "Matrix Theory" and our professor assigned us this book which proved to be too difficult and abstract😫

[masteranza]

For anyone who gets confused by all the brackets, there's an easier way using the dot composition all the time. Example: T' ∘φ= φ∘T and similarly: D'∘φ∘p=φ∘D∘p=φ∘p'=p'∘3 which is just... p'(3). Note that in this special case the functional can be defined as φ∘p=p∘3, but one should remember what each object should be "fed". So for instance D'∘φ∘p=D'∘p∘3=p∘3∘D would be incorrect, because we've exchanged the functional φ for its value for some p, i.e., we've evaluated it too quickly (using the CS slang). To see that this is the case consider usual vector space in reals with ordinary vectors, map D∘v=-v and functional φ(v)=φ∘v=|v| then what we've just done would correspond to D'∘φ∘v=D'∘|v|=|v|∘D which clearly makes no sense, because D' wants a functional not a scalar.

[omkark]

Prof, Really great slides and explanation ... you are god of Abstract - Linear Algebra

[michellejingdong]

Thank you so much for the videos and the book!

[Eldooodarino]

Very nice. One thing I'm wondering about. In general is not the vector space of polynomials, P, infinite dimensional so that for the examples involving polynomials you need an upper bound on the order of the polynomials in P? Otherwise the proof that the dual space of P has the same dimension as P fails because that proof is valid only for finite dimensional space? Or do those proofs work for countably infinite bases as well?

[miro.s]

It is an audio, so difficult to follow. At least some small effort to illustrate longer monologue parts when screen doesn't change would make it more interesting and understandable. It would become an educative video then. And that would make it easier to follow.