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.
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?
He only claimed that the result about the dimension of the dual space holds when the primal space is finite dimensional. He didn't claim that it holds for infinite dimensional spaces such as P.
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😫
it has never meant to be a feasible fast-comprehension medium, whence better as 2nd exposure to lin alg, and also requires some mathematical maturity, akin to that of set theory based proof & proportional logic
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.
I do not agree with the comment above. First, this is a video, not an audio as stated in the comment. Second, the screen does change frequently. Finally, about the comment above that it is difficult to follow: I agree that learning linear algebra just from a video may be difficult. These videos are intended as a supplement for the book Linear Algebra Done Right. Different people have different learning styles. Some people will find the videos useful to supplement the book; other people may be better off by spending more time with the book.
@@sheldonaxler5197 Indeed, I fully agree with you and you shouldn't feel obliged to anything by lazy viewers. The videos are concise and perfect for the review of the topic. If one wants to learn something rather than just passively view, it's easy to stop and write a line or two on paper. It has worked for me perfectly. That being said, I'm happy to contribute something in return and to answer some of the questions posed by the viewers as an exercise to myself.
@@sheldonaxler5197 I agree with the original poster. This video is no more than an audio-book with pictures. It is difficult to maintain interest when the presenter simply reads out definitions, without interesting motivation and development of the ideas behind the definitions, and without digressions to forge interesting links with other ideas.
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.
Thank you so much for the videos and the book!
Amazing book and videos, clear and concise. Thank you very much.
I love your shit sheldon, keep it up
Prof, Really great slides and explanation ... you are god of Abstract - Linear Algebra
Math audio book! Amazing!
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?
He only claimed that the result about the dimension of the dual space holds when the primal space is finite dimensional. He didn't claim that it holds for infinite dimensional spaces such as P.
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😫
it has never meant to be a feasible fast-comprehension medium, whence better as 2nd exposure to lin alg, and also requires some mathematical maturity, akin to that of set theory based proof & proportional logic
Perfect!!! Thank you.
Please continue this vlog
bravo
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.
I do not agree with the comment above. First, this is a video, not an audio as stated in the comment. Second, the screen does change frequently. Finally, about the comment above that it is difficult to follow: I agree that learning linear algebra just from a video may be difficult. These videos are intended as a supplement for the book Linear Algebra Done Right. Different people have different learning styles. Some people will find the videos useful to supplement the book; other people may be better off by spending more time with the book.
@@sheldonaxler5197 Indeed, I fully agree with you and you shouldn't feel obliged to anything by lazy viewers. The videos are concise and perfect for the review of the topic. If one wants to learn something rather than just passively view, it's easy to stop and write a line or two on paper. It has worked for me perfectly. That being said, I'm happy to contribute something in return and to answer some of the questions posed by the viewers as an exercise to myself.
@@sheldonaxler5197 I agree with the original poster. This video is no more than an audio-book with pictures. It is difficult to maintain interest when the presenter simply reads out definitions, without interesting motivation and development of the ideas behind the definitions, and without digressions to forge interesting links with other ideas.
@@last3239 I have coped. I've watched videos from people who can actually teach. But thanks for your concern.