Calculating the kernel of a matrix - An example

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  • เผยแพร่เมื่อ 24 ธ.ค. 2024

ความคิดเห็น • 106

  • @brightsideofmaths
    @brightsideofmaths  หลายเดือนก่อน +1

    Download the PDF version with the link in the description :)

  • @harshitsrivastava5559
    @harshitsrivastava5559 6 ปีที่แล้ว +83

    Really he showed me the Bright side of Mathematics.

  • @norrishanza
    @norrishanza 3 ปีที่แล้ว +39

    Well shiiiieeeet! That’s all I had to do? THAAANK YOUUU!

  • @環華溫
    @環華溫 5 ปีที่แล้ว +14

    Once again he saved another life

  • @jacobvandijk6525
    @jacobvandijk6525 3 ปีที่แล้ว +9

    Cool, calm and good explanation. Well done.

  • @benhardsim8629
    @benhardsim8629 3 ปีที่แล้ว +32

    Thx man , after knowing the concept before then start watching this to see the details, everything is finally connected for me. Really appreciate it 🙂.

    • @1mol831
      @1mol831 2 ปีที่แล้ว +1

      I'm still wondering where is the difference between Kernel and the vector subspace of matrix solutions. Seems like the two are the same thing.

    • @brightsideofmaths
      @brightsideofmaths  2 ปีที่แล้ว +3

      @@1mol831 The kernel is the vector subspace of all vectors x that solve Ax = 0.

    • @Asdfgadv33423
      @Asdfgadv33423 11 หลายเดือนก่อน +1

      @@brightsideofmaths Hello! Do you have a Discord channel? I am currently a 1st year student at Mathematics and Computer Science University. Thank you!

    • @brightsideofmaths
      @brightsideofmaths  11 หลายเดือนก่อน

      I have a community server. See description.@@Asdfgadv33423

  • @aymanfaghih3604
    @aymanfaghih3604 หลายเดือนก่อน +1

    I watched quite a few videos to understand this, this was the video I needed. Great video :)

  • @darcash1738
    @darcash1738 10 หลายเดือนก่อน +1

    is basis of the kernel different from the kernel? I did reduced row echelon and got x1 = -1/2 * x2+1/10 * x4, x3 = -3/5 * x4. So I thought it was going to be. So lets just call s1 and s2 the arbitrary vars x2 and x4. We would get:
    x1 = -1/2 * s1+1/10 * s2, x2 = s1, x3 = -3/5 * s2, x4 = s2
    s1
    + s2

    • @brightsideofmaths
      @brightsideofmaths  10 หลายเดือนก่อน +4

      The kernel is the subspace and the basis is the minimal spanning set for the subspace. You can see the details in my Linear Algebra series: tbsom.de/s/la

    • @darcash1738
      @darcash1738 10 หลายเดือนก่อน

      ⁠​⁠@@brightsideofmaths i checked my calculator once and for all and I did mess up the calculation 😂
      Btw if the vectors from this were dependent, then we’d choose just one of em for the kernel’s basis, I’d expect?
      Or would we recognize this and before writing the kernel, and redefine one of the free variables in terms of the other?

    • @brightsideofmaths
      @brightsideofmaths  10 หลายเดือนก่อน +1

      We have exactly as many free variables as vectors in the basis :)@@darcash1738

    • @darcash1738
      @darcash1738 10 หลายเดือนก่อน

      @@brightsideofmaths gotcha. Thanks :)

  • @dawafflesupreme
    @dawafflesupreme 5 ปีที่แล้ว +14

    Thank you, You just saved my ass more than you know.

    • @PunmasterSTP
      @PunmasterSTP 2 ปีที่แล้ว +2

      I'm just curious; how have your studies been going?

  • @dark6.63E-34
    @dark6.63E-34 ปีที่แล้ว +2

    Simple and straight to the point! Well explained

  • @evionlast
    @evionlast 3 ปีที่แล้ว +1

    The best youtube channel in math!

  • @Shaamiix
    @Shaamiix 4 ปีที่แล้ว +5

    isn't it x2 = alpha - 0.5 beta? As you should divide by -2 and not by 2?

  • @navierstokes2356
    @navierstokes2356 2 ปีที่แล้ว +1

    What do you think about Axler's Linear Algebra Done Right?

  • @PunmasterSTP
    @PunmasterSTP 2 ปีที่แล้ว +1

    Calculating the kernel? More like "Can do it now that I know!" Thanks for making all of these videos which help countless students understand the concepts and solve problems.

  • @alevelmathematics1
    @alevelmathematics1 6 ปีที่แล้ว +2

    Can you write 'x2' as a new free parameter or does it have to be in terms of alpha and beta?

    • @brightsideofmaths
      @brightsideofmaths  6 ปีที่แล้ว +3

      Keep in mind: x2 is a leading variable and not a free one! Therefore, it has to be a fixed value or given in terms of alpha and beta.

  • @yaroslava6910
    @yaroslava6910 7 ปีที่แล้ว +2

    Bilden diese zwei Vektoren, die wir am Ende gekriegt haben und die eine Basis bilden, auch Span?

    • @brightsideofmaths
      @brightsideofmaths  7 ปีที่แล้ว +2

      Der Span von den beiden Vektoren ist ja genau das was dort schon steht, d.h. alle möglichen Linearkombinationen der zwei Vektoren.

  • @bikespike4150
    @bikespike4150 ปีที่แล้ว

    The math does make sense here but what exactly would the difference between the kernel and the span then be? Could it be understood that the Span is then just referring to things like vectors and planes etc whilst kernels are more specific to matrixes?

    • @brightsideofmaths
      @brightsideofmaths  ปีที่แล้ว +1

      Ran and kernel are defined for matrices and linear maps in the same way!

  • @gustavohernandez57
    @gustavohernandez57 6 ปีที่แล้ว +2

    How do you decide what are free variables?

    • @brightsideofmaths
      @brightsideofmaths  6 ปีที่แล้ว +2

      Use the variables that correspond to the columns without pivot element.

    • @gustavohernandez57
      @gustavohernandez57 6 ปีที่แล้ว +1

      @@brightsideofmaths ah ok

  • @tinyasira6132
    @tinyasira6132 2 ปีที่แล้ว +2

    why did u take alfa n bita only for x3, x4 and not for x1/x2?
    Also how can we take alfa,bita when we do determinant method? please reply it's so confusing 😭😭

    • @PunmasterSTP
      @PunmasterSTP 2 ปีที่แล้ว

      I think he mentioned this in the video and replies to other comments, x1 and x2 have pivots in their columns, while x3 and x4 do not. So it is best to think of x1 and x2 in terms of x3 and x4.

    • @tinyasira6132
      @tinyasira6132 2 ปีที่แล้ว +1

      @@PunmasterSTP Thanks a lot 😊😊

    • @PunmasterSTP
      @PunmasterSTP 2 ปีที่แล้ว +1

      @@tinyasira6132 No problem; I just try to help where I can, and I hope I don't confuse more than I help!

    • @tinyasira6132
      @tinyasira6132 2 ปีที่แล้ว +2

      @@PunmasterSTP no I get it! Thanks! But if we do matrix determinant method instead of row reduction then can u please tell me how to choose what is free variable and what is not?

  • @hamzasabri5610
    @hamzasabri5610 6 ปีที่แล้ว +9

    Thx bro you have saved me 😂

  • @Anas-im4gt
    @Anas-im4gt 3 หลายเดือนก่อน

    Thank you I was stuck in the Matrix

  • @adambcc44
    @adambcc44 ปีที่แล้ว

    can the kernel be all 0’s? like the solution to the system would be x=0, y=0, z=0, is that possible?

    • @brightsideofmaths
      @brightsideofmaths  ปีที่แล้ว +1

      Yes, the kernel can be trivial and just contain the zero vector!

    • @adambcc44
      @adambcc44 ปีที่แล้ว

      @@brightsideofmaths ❤️❤️❤️❤️

    • @adambcc44
      @adambcc44 ปีที่แล้ว

      @@brightsideofmaths and then the dimension of the kernel is considered 1, right?

  • @buqet
    @buqet 3 ปีที่แล้ว

    Is there a possibility to a Im(A) of this matrix bcs im not really getting Im

  • @marllymbungu1721
    @marllymbungu1721 6 ปีที่แล้ว +5

    thank you so much. it was clear and very helpful

  • @ccuuttww
    @ccuuttww 6 ปีที่แล้ว +2

    it is equal to null space of A can u do the polynomial one?

    • @brightsideofmaths
      @brightsideofmaths  หลายเดือนก่อน

      Null space and kernel are different names for the same thing :)

  • @sohailshah1574
    @sohailshah1574 ปีที่แล้ว

    Nice explanation sir

  • @christopherandrewmartin494
    @christopherandrewmartin494 3 ปีที่แล้ว +1

    This video helped me so, so much. Thank you!

  • @mindovermath2042
    @mindovermath2042 4 ปีที่แล้ว +2

    I found you through the suggested videos. Your videos are vey informative! I just made a TH-cam Channel due to your inspiration. Keep up the Videos! Just Subbed!

  • @banquo4223
    @banquo4223 2 ปีที่แล้ว

    In the first step, you're just converting the matrix to reduced row-echelon form correct?

  • @Fanaindel3
    @Fanaindel3 ปีที่แล้ว

    here, the dimension of the kernel is 2 right?

  • @ZadronXion
    @ZadronXion 5 ปีที่แล้ว +2

    I wish you were my linear algebra professor!

    • @PunmasterSTP
      @PunmasterSTP 2 ปีที่แล้ว

      Yeah I'm sure he'd make a great one. I was just curious; how have your studies been going?

    • @ZadronXion
      @ZadronXion หลายเดือนก่อน +1

      @@PunmasterSTP I have 14 credit hours left, no way to pay for it, and no valid cosigner available for loans

    • @PunmasterSTP
      @PunmasterSTP หลายเดือนก่อน

      @@ZadronXion Oh man, I'm really sorry to hear that! Is there any way you can find a job/another job now to help pay for stuff?

  • @suryabharathi7924
    @suryabharathi7924 5 ปีที่แล้ว +4

    Thank you for starting uploading in English it will helpful for many others

  • @vincentgh9292
    @vincentgh9292 ปีที่แล้ว

    Ker(A) is the vector space, however the equation Ax=0 seems to be defined as real value ^^

  • @nikurou9025
    @nikurou9025 6 ปีที่แล้ว +4

    Thank you

  • @meerak.3390
    @meerak.3390 3 ปีที่แล้ว +1

    Thank for saving me sir..highly appreciated 🙏🙌

  • @NG-we8uu
    @NG-we8uu 5 ปีที่แล้ว +1

    What’s the name of the author?

    • @brightsideofmaths
      @brightsideofmaths  5 ปีที่แล้ว +1

      My name? :)

    • @NG-we8uu
      @NG-we8uu 5 ปีที่แล้ว +1

      The Bright Side Of Mathematics yes

  • @REHOCHELENH
    @REHOCHELENH 6 ปีที่แล้ว +2

    thank you so much

  • @jesusbastian5201
    @jesusbastian5201 3 ปีที่แล้ว +1

    Excelente ejemplo .!

  • @LightChu2.7183
    @LightChu2.7183 4 ปีที่แล้ว +1

    Thank you a lot!!!! I LOVE GERMANY!!

  • @alexeul9843
    @alexeul9843 5 ปีที่แล้ว +1

    Thank you!

  • @amarhadzipasic6637
    @amarhadzipasic6637 ปีที่แล้ว

    Super Video 👍

  • @vanshika3971
    @vanshika3971 6 ปีที่แล้ว +1

    Thanku 😆 sir ,ur way of teaching is very good but I have one question if we have to calculate kernel of tha given matrix then is there a need to write the basis of the kernel also can't we just end up by writing only kernel

    • @brightsideofmaths
      @brightsideofmaths  6 ปีที่แล้ว +2

      That depends what you want in the end. It is always better to have a basis of a subspace (here the kernel) since then you have all information in a compact form.

  • @ninakoch1799
    @ninakoch1799 6 ปีที่แล้ว +6

    Danke dir, hat mir sehr geholfen!!

  • @ericlol1337
    @ericlol1337 9 หลายเดือนก่อน

    i dont understand why dont you just complete the gauss jordan

    • @brightsideofmaths
      @brightsideofmaths  9 หลายเดือนก่อน

      What do you mean with "complete"?

  • @templetonpeck1084
    @templetonpeck1084 2 ปีที่แล้ว +1

    Those ones are weird.

  • @tonikaiser2823
    @tonikaiser2823 5 ปีที่แล้ว

    wo kommt das so vor im echt?

  • @davidmac451
    @davidmac451 6 ปีที่แล้ว +4

    terrible way of writing 1's haha thank you though.

    • @brightsideofmaths
      @brightsideofmaths  6 ปีที่แล้ว +5

      Why is it terrible? :)

    • @davidmac451
      @davidmac451 6 ปีที่แล้ว +4

      @@brightsideofmaths looks like an upside down v and not a 1 :') Your video helped with my exam today, thank you.

    • @brightsideofmaths
      @brightsideofmaths  6 ปีที่แล้ว +4

      @@davidmac451 And do not forget my 3's: The look like a rotated omega ;)

    • @ninakoch1799
      @ninakoch1799 6 ปีที่แล้ว +2

      Bob Balooga in Switzerland (and Germany where he probably comes from) we write 1‘s exactly like he does. My cousin from the States told me they write them as just a vertical line. Anyways, he has a nice handwriting, he‘s just not American.

    • @PunmasterSTP
      @PunmasterSTP 2 ปีที่แล้ว

      @@davidmac451 How did your exam and the rest of your class go?

  • @georgebaker5038
    @georgebaker5038 2 ปีที่แล้ว +1

    thwee

  • @lekemorfaw237
    @lekemorfaw237 4 ปีที่แล้ว +1

    Great tutorial video though Permit my saying that your accent seems german.

  • @NiragireJeanclaude-fm9vl
    @NiragireJeanclaude-fm9vl 2 หลายเดือนก่อน

    Thx you support me

  • @MuhammadAkbar-tw7ln
    @MuhammadAkbar-tw7ln 2 ปีที่แล้ว

    Does anyone feels a little giggling when you hear the word biguss........ Diguss.....

  • @linushuber2984
    @linushuber2984 ปีที่แล้ว

  • @isaactonui21
    @isaactonui21 2 ปีที่แล้ว

    👏👏👏👏

  • @aureuse7174
    @aureuse7174 5 หลายเดือนก่อน

    I LOVE GERMAN PEOPLE

  • @zeynepcyrusss
    @zeynepcyrusss 6 หลายเดือนก่อน +1

    Aight sir, thanks for the video but ur accent is terrible,ngl

  • @benwinstanleymusic
    @benwinstanleymusic 3 ปีที่แล้ว +4

    Very helpful and clear, thank you

  • @Christopher-ew7jw
    @Christopher-ew7jw 6 ปีที่แล้ว +3

    Thank you.

  • @olivioarmandocordeirojunio8280
    @olivioarmandocordeirojunio8280 3 ปีที่แล้ว +1

    Thanks a lot.