The Vector Space of Polynomials: Span, Linear Independence, and Basis

We normally think of vectors as little arrows in space. We add them, we multiply them by scalars, and we have built up an entire theory of linear algebra around these objects. However, it turns out that polynomials of degree less than or equal to n ALSO form a so called vector space; that is, they also have a vector addition and a scalar multiplication that obeys the same long list of algebraic rules as vector. This means we can do linear algebra on these polynomial vector spaces in much the same way. In this video we'll cover the standard ideas of span, linear independence, and basis, and see how these ideas from the vector space R^n extend over to the vector space of polynomials of degree less than or equal to n with real coefficients. We'll do a full example showing how to demonstrate these concepts algebraically.
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Пікірлер: 73

  • @torchickpokemon1165
    @torchickpokemon11652 жыл бұрын

    I took Linear Algebra in my 2nd year Electrical Engineering undergrad 34 years ago. Now I am watching your videos and doing problems with ease. Too bad KZread wasn't around in 1987. Thank you Dr. Bazette.

  • @brandonnyamadzawo1452

    @brandonnyamadzawo1452

    Жыл бұрын

    Kkkkk

  • @aezazi

    @aezazi

    Жыл бұрын

    Industrial and systems engineering in 1986. Feel exactly as you. Learning/relearning this material is so much easier now. So many great resources at your fingertips. Back then we had essentially one instructor and one or two books.

  • @manasandmohit

    @manasandmohit

    Жыл бұрын

    Well well I am currently in EE sem 2

  • @lewiduressa
    @lewiduressa Жыл бұрын

    I used the videos from this channel to fully understand concepts from my calc 3 class and I passed my calc 3 class with a 100%. And now I am using this for linear algebra. Dr. Trefor, thank you so much for explaining things in a visual and simple way for us to understand.

  • @DrTrefor

    @DrTrefor

    Жыл бұрын

    Amazing!

  • @kelvinthehuman
    @kelvinthehuman3 жыл бұрын

    This is the kind of work that KZread should also sponsor for ever! Mr. Bazett is making a significant contribution to humanity.

  • @willthomas6339
    @willthomas63394 жыл бұрын

    Hi, I just wanted to say your Linear Algebra videos are a fantastic resource to learn, especially while we are all remote. Your visuals help me understand the Vector space topics so much better. Thanks

  • @ngouchuy4016
    @ngouchuy40162 жыл бұрын

    Why your videos haven't got million views yet? It's just so good, so clear and straight to the point. Thank you Doc!

  • @utkarshujwal3286
    @utkarshujwal3286 Жыл бұрын

    The best part about your teaching is the fact that we can think this intuitively and also visualize the application of linear algebra. I had matrix operations in my high school but never really understood the motive behind studying it. Its through you that I have come to understand the importance of the same.

  • @kelvinthehuman
    @kelvinthehuman3 жыл бұрын

    Mi hermano, this is a real job, eh? A real wonderful job. You really rock, Mr. Bazett!!

  • @crimfan
    @crimfan4 жыл бұрын

    Super example. This was where a light turned on for me back when I was taking math.

  • @muhdhaziq2422
    @muhdhaziq24222 жыл бұрын

    Thank you Dr, your explanation is so clear

  • @kktech04
    @kktech04 Жыл бұрын

    Great material, and awesomely presented , thank you

  • @salamlababidi3042
    @salamlababidi3042 Жыл бұрын

    thank you thank you thank you!!!!!!!!!!! Absolute life saver I finally understand this chapter

  • @1951split
    @1951split4 жыл бұрын

    Could you do a video about Clifford Algebra (Geometric Algebra) ? There are a few math videos out there explaining the algebra, but I'd love to know how to appy it to real life (engineering) problems. I love all your other courses btw. I was never good at math. I had terrible teachers, I didn't give as much effort as I should, and never managed to get all the rules into my head (well not for the "long term" at least... maily because of a lack of practice) But now watching all your calculus and LA classes and trying to apply your learning strategy and I really feel like I am getting more confident in my problem solving. I set up an anki deck to engrave the rules into my brain once and for all. I am really aiming for a more thourough understanding now. Your videos help a lot. Thank you!

  • @pralay1762
    @pralay17624 жыл бұрын

    Amazing videos Great work Keep it up Simply awesome

  • @awazin4031
    @awazin40313 жыл бұрын

    I'm just a french guy who likes mathematics. I really enjoy watching your videos. Thank you for your excellent work :)

  • @DrTrefor

    @DrTrefor

    3 жыл бұрын

    Glad you like them!

  • @savmelonjr.3592
    @savmelonjr.35923 жыл бұрын

    youre killin it

  • @suhailawm
    @suhailawm4 жыл бұрын

    Wow. Tnx aLot prof. Nice nd simple xplain.

  • @2false637
    @2false6374 жыл бұрын

    Very insightful!

  • @DanijelAleksicMath
    @DanijelAleksicMath4 жыл бұрын

    As a 3rd year math student, who studied systematically most of the topics that you covered in your videos, I can say that i still learnt so many new things, but mostly filled "lack of intuition" gaps in my knowledge. I am glad someone like you exists on KZread. If I knew for this channel 3 years before, I would have gone through much less pain. Greetings from University of Belgrade! P. S. I think you should change your microphone position, you sound kind of shallow. Maybe it's just me.

  • @navjotsingh2251

    @navjotsingh2251

    2 жыл бұрын

    It's all to do with who is teaching you. If you have a bad teacher, they will show you the bare minimum and nothing else.

  • @thebreath6159

    @thebreath6159

    Жыл бұрын

    If you want to know more about this read Linear Algebra done right.

  • @kamrulhassan7157
    @kamrulhassan7157Ай бұрын

    Thanks for the explanation

  • @WildBlue533
    @WildBlue5332 жыл бұрын

    Very helpful!

  • @andreluisal
    @andreluisal3 жыл бұрын

    Really excellent explanation!!!

  • @DrTrefor

    @DrTrefor

    3 жыл бұрын

    Thank you!!

  • @liamohanlon8616
    @liamohanlon8616 Жыл бұрын

    shout out to you my man you clutchin up like kyrie in game 7

  • @simonribas4625
    @simonribas46252 жыл бұрын

    1000th like woo. By the way I am just curious, is there a case where the polynomials are linearly independent but don't span full P2? other than the obvious case that we don't have a polynomial representing x^2 or x or x^0.

  • @Joshua-rk7bl
    @Joshua-rk7bl2 жыл бұрын

    Thank you very much

  • @5STARanmol
    @5STARanmolАй бұрын

    i love this man

  • @kilianschulz5186
    @kilianschulz51862 жыл бұрын

    Does a polynomial vector space have to reduce to the identity matrix in order to span a space?

  • @biaschatterjee9836
    @biaschatterjee98364 жыл бұрын

    Hey can you make videos on solving differential equations for epidemic models?

  • @dangrigsby5915
    @dangrigsby5915 Жыл бұрын

    Thanks!

  • @AaaAaa-rq6ry
    @AaaAaa-rq6ry4 жыл бұрын

    Hi! Could we use wronskian to conclude linear independence in this case?

  • @simplemathandfun4613
    @simplemathandfun46133 жыл бұрын

    Wonderful

  • @user-gj1zg3gb9q
    @user-gj1zg3gb9q Жыл бұрын

    جزيل الشكر دكتور على جهودك في شرح المادة بشكل مبسط ❤ تحية لك من العراق

  • @MrBARA1000
    @MrBARA10003 жыл бұрын

    Thanks alot!

  • @DrTrefor

    @DrTrefor

    3 жыл бұрын

    You're welcome!

  • @gurleensingh4152
    @gurleensingh41522 жыл бұрын

    Really helpful 👏👏

  • @DrTrefor

    @DrTrefor

    2 жыл бұрын

    Glad it was helpful!

  • @frankdearr2772
    @frankdearr2772 Жыл бұрын

    hello, a question: at 2.30 minutes I can see (c+d)*vecU = c*vecU+d*vecV, but the last seems to be d*vecU ?? Thanks for you reply :)

  • @chinthamgopi8705
    @chinthamgopi8705 Жыл бұрын

    Great teaching sir what is your country

  • @simonpeterj
    @simonpeterj3 ай бұрын

    Thanks

  • @DrTrefor

    @DrTrefor

    3 ай бұрын

    Thank you so much!!!

  • @shutupimlearning
    @shutupimlearning Жыл бұрын

    holy damn im week 2 into my linear algebra course and watching this video hurts my brain

  • @arkansh.h.1313
    @arkansh.h.13132 жыл бұрын

    what about the span of these functions 1, x, cosx, sinx?

  • @prabhatsojitra556
    @prabhatsojitra5562 жыл бұрын

    If we take u1=x and u2=-x then u1 + u2= 0 , it is zero polynomial. And degree of zero polynomial is not defined, so, 0 will not be in this space. So how this space becomes vector space?! Please correct me if I am wrong..

  • @muzammilahmed8646
    @muzammilahmed86462 жыл бұрын

    at 11:03 shouldn't we have t_1 to t_(n+1) as the co-efficients

  • @ivyunhinged9521
    @ivyunhinged95213 жыл бұрын

    Th only reason I could keep up with you is because I drowned myself in linear algebra the past month. Very informative!

  • @Christian-mn8dh
    @Christian-mn8dh Жыл бұрын

    2:16 why isn't commutative multiplication a property of vector spaces?

  • @julioquejidobarrera8539
    @julioquejidobarrera8539 Жыл бұрын

    Hello!!! I think , and i say i THINK (i could be wrong), around the minute 11' , wouldn't it be the formula t1.1 + t2.x + ... + tn.x^n-1 ? Or am i making and error. I love your videos by the way ♥ ♥

  • @julioquejidobarrera8539

    @julioquejidobarrera8539

    Жыл бұрын

    Forget about it, i think i understood the meaning

  • @d0fabur5st82
    @d0fabur5st8211 ай бұрын

    6:03 why would the coefficients need to be 0?

  • @adresscenter
    @adresscenter3 жыл бұрын

    👍

  • @achalcharantimath5603
    @achalcharantimath56034 жыл бұрын

    hi, have you learnt machine learning .... can you make a course on math of machine learning??

  • @navjotsingh2251

    @navjotsingh2251

    Жыл бұрын

    Math of machine learning is just calculus, linear algebra and statistics 😀 you may also benefit from learning how to do these topics with a programming language.

  • @thomasolson7447
    @thomasolson74473 жыл бұрын

    There is also scalar's inside the factorization. I seem to be seeing it everywhere now that I know where to look. The scalars are the zeros only. It doesn't scale the regular function. (x-sqrt(2)*r1_He[5])(x-sqrt(2)*r2_He[5])(x-sqrt(2)*r3_He[3])(x-sqrt(2)*r4_He[3])(x-sqrt(2)*r5_He[3])

  • @aidosmaulsharif9570
    @aidosmaulsharif95703 жыл бұрын

    Oooohh it is so sad that the video has such few views

  • @alexhelmers2148
    @alexhelmers21482 жыл бұрын

    thank you, you just literally saved my dumbass

  • @KatzeArtemis
    @KatzeArtemis2 жыл бұрын

    funny how right at the end you say its a n+1 * n+1 matrix, but you count the t_n from 1 to n instead of 0 to n

  • @confused6526
    @confused65265 ай бұрын

    Should "Span {v1,...,v2}" denoted as "Span {v1,...,vn}"? And that "Span {v1,...,v2} = V" denoted as "Span {v1,...,vn} belongs to (not equal) vector space V" ?

  • @confused6526
    @confused65265 ай бұрын

    Another condition beside not all t1...tk are zero is that all vectors in vector space are NOT zero vectors.

  • @MussaKisinza
    @MussaKisinzaАй бұрын

    Sorry sir reduce speed of explanation because am not understand

  • @aku7598
    @aku75983 жыл бұрын

    I thought linear algebra only involves linear equations. But polynomial like x square is not linear.

  • @DrTrefor

    @DrTrefor

    3 жыл бұрын

    The polynomials are like vectors, and then it is linear in those vectors. Change of perspective.

  • @nehalamba2747

    @nehalamba2747

    Жыл бұрын

    It deals with linear combinations of vectors and that vector can be a lot of things

  • @MrChaluliss
    @MrChaluliss Жыл бұрын

    Your microphone game is painful here. Otherwise good stuff.

  • @billygaspar84
    @billygaspar843 ай бұрын

    Thanks!

  • @DrTrefor

    @DrTrefor

    3 ай бұрын

    You're amazing, thank you!