The beautiful geometric view of FOURIER SERIES // The Linear Algebra Perspective

Fourier Series can sometimes seem very computational and full of integrals, but there is actually a very deep and important geometric picture that is analogous to Linear Algebra. So many core ideas from linear algebra like bases, linear independence, orthogonality, inner products, and projections all have their analogs in Fourier Series. Effectively the idea of computing a Fourier Series is just decomposing a vector in an orthonormal basis. Cool!

0:00 Vector Space of Periodic Functions
2:55 Bases
5:57 Inner Product Space
10:47 Projections
13:27 The Big Picture

OTHER COURSE PLAYLISTS:

OTHER PLAYLISTS:
► Learning Math Series
►Cool Math Series:

BECOME A MEMBER:

MATH BOOKS & MERCH I LOVE:

SOCIALS:

[BentHestad]

This series on Fourier is just brilliant, and this episode especially so! Thank you very much, professor Bazett!

[elltwo]

This was actually the only thing I knew about the Fourier transform before watching your series, that is, that the Fourier transform is an isometric isomorphism of l2(Z) and L2(T). Also, showing that the functions f_n(t) = e^(int) actually give an orthonormal basis for L2(T) can be done with an application of the Stone-Weierstrass theorem, one of my favourite theorems of all time!

[anirudhrowjee]

I did everything short of jumping in surprise when I realized what was about to happen.. wow!
This entire series is excellent.
Thank you, Dr, Bazett! Your videos are the reason I've been enjoying math of late.

[punditgi]

Awesome explanation! Truly, a revelation. Many thanks for these amazing insights! 😃

[sergiolucas]

edit: great video, indeed, the best video about math that ive ever seen, thank you :)

[hdheuejhzbsnnaj]

This is such an interesting take. I'm sure I'll come across this later in my studies, but I really appreciate getting exposed to it at this early juncture even if I'm really only here to get a grasp on the basics. The big picture view helps things stick.

[fordtimelord]

Wow…this was absolutely beautiful! Thank you! This connection between linear algebra and Fourier series is so intuitive and easy after you explained it. For some reason, understandings Fourier series in the context of linear algebra makes it easy lists intimidating.

[treksis]

This is the best video in Fourier Series list

[janeh]

this video gave me that absolutely wonderful "AHA!" moment two different times. thanks so much!

[MrFriday]

Really great video. It's great to see the links explicitly pointed out. Thank you.

[salmael_badry]

The more I watch videos made by you, the more I believe you were born as an artist.

[andrewharrison]

That closure moment when you realise that the same structure underlies 2 "obviously" different parts of mathematics.
Made my evening - fabulous.

[michaelzumpano]

Trefor, that was excellent. I think everyone going into quantum mechanics should watch this series before doing the “Math’ section in the beginning of the book. You did it without getting your foot stuck in Hilbert space!

[bernardoramirez]

As soon as I get employed I’ll make sure I make a donation of some kind to this amazing channel

[oraange]

I'm studying QM on my own, it's really needed to be enough good in linear algebra. Your videos are outstanding! Keep it up!!!

[manishrathi]

I had no clue about this part of the beauty, thankyou professor

[mathalysisworld]

This was so awesome, man. This was a super suprise for me. What a beautiful journey I had today!

[mnada]

😯 This video explained a lot ... This kind of connecting the dots and filling the gaps is what makes mathematics enjoyable. I wish you extend your explanation to include fourier transform.

[brothberg]

This has opened my eyes to the formulae I was taught in engineering.
Thank you so much

[aashsyed]

Thanks so! Much!
You explain better than teachers! :DD