Calculating the Kernel of a Matrix - An Example

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This is my video series about Linear Algebra. I hope that it will help everyone who wants to learn about it.
Here I present some short calculation for the kernel of a matrix. I apologise for my pronunciation. The focus is on the mathematics and not my English skills :)
#LinearAlgebra
(This exercise fits to lectures for students in their first year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers, Linear algebra and so on)

Опубликовано:

3 дек 2017

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Комментарии : 96

@harshitsrivastava5559 6 лет назад

Really he showed me the Bright side of Mathematics.

@norrishanza 2 года назад

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

@jacobvandijk6525 2 года назад

Cool, calm and good explanation. Well done.

@dark6.63E-34 Год назад

Simple and straight to the point! Well explained

@user-nl7mx3uc1s 5 лет назад

Once again he saved another life

@marllymbungu1721 5 лет назад

thank you so much. it was clear and very helpful

@benhardsim8629 3 года назад

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 2 года назад

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 2 года назад

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

@Asdfgadv33423 5 месяцев назад

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

@brightsideofmaths 5 месяцев назад

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

@benwinstanleymusic 3 года назад

Very helpful and clear, thank you

@dawafflesupreme 5 лет назад

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

@PunmasterSTP 2 года назад

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

@christopherandrewmartin494 3 года назад

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

@mindovermath2042 4 года назад

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

@evionlast 3 года назад

The best youtube channel in math!

@meerak.3390 3 года назад

Thank for saving me sir..highly appreciated 🙏🙌

@nikurou9025 6 лет назад

Thank you

@alexeul9843 5 лет назад

Thank you!

@hamzasabri5610 5 лет назад

Thx bro you have saved me 😂

@Christopher-ew7jw 5 лет назад

Thank you.

@REHOCHELENH 5 лет назад

thank you so much

@PunmasterSTP 2 года назад

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.

@Sal9862 4 года назад

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

@jesusbastian5201 3 года назад

Excelente ejemplo .!

@ZadronXion 4 года назад

I wish you were my linear algebra professor!

@PunmasterSTP 2 года назад

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

@suryabharathi7924 4 года назад

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

@ninakoch1799 5 лет назад

Danke dir, hat mir sehr geholfen!!

@b3n0sgaming88 Год назад

bitte,

@alevelmathematics1 5 лет назад

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

@brightsideofmaths 5 лет назад

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.

@olivioarmandocordeirojunio8280 2 года назад

Thanks a lot.

@ccuuttww 5 лет назад

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

@sohailshah1574 7 месяцев назад

Nice explanation sir

@brightsideofmaths 7 месяцев назад

Thanks and welcome :)

@LightChu2.7183 3 года назад

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

@yaroslava6910 6 лет назад

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

@brightsideofmaths 6 лет назад

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

@buqet 2 года назад

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

@navierstokes2356 Год назад

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

@brightsideofmaths Год назад

I like it!

@bikespike4150 Год назад

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 Год назад

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

@vincentgh9292 Год назад

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

@brightsideofmaths Год назад

x is a vector :)

@gustavohernandez57 5 лет назад

How do you decide what are free variables?

@brightsideofmaths 5 лет назад

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

@gustavohernandez57 5 лет назад

@@brightsideofmaths ah ok

@vanshika3971 6 лет назад

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 6 лет назад

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.

@amarhadzipasic6637 9 месяцев назад

Super Video 👍

@brightsideofmaths 9 месяцев назад

Thank you very much!

@darcash1738 4 месяца назад

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 4 месяца назад

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 4 месяца назад

⁠⁠@@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 4 месяца назад

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

@darcash1738 4 месяца назад

@@brightsideofmaths gotcha. Thanks :)

@Fanaindel3 7 месяцев назад

here, the dimension of the kernel is 2 right?

@brightsideofmaths 7 месяцев назад

Yes :) It's 2.

@tinyasira6132 2 года назад

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 2 года назад

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 2 года назад

@@PunmasterSTP Thanks a lot 😊😊

@PunmasterSTP 2 года назад

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

@tinyasira6132 2 года назад

@@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?

@adambcc44 Год назад

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 Год назад

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

@adambcc44 Год назад

@@brightsideofmaths ❤️❤️❤️❤️

@adambcc44 Год назад

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

@banquo4223 Год назад

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

@brightsideofmaths Год назад

Yes!

@banquo4223 Год назад

@@brightsideofmaths Thank you! Very helpful video

@linushuber2984 7 месяцев назад

❤

@NG-we8uu 4 года назад

What’s the name of the author?

@brightsideofmaths 4 года назад

My name? :)

@NG-we8uu 4 года назад

The Bright Side Of Mathematics yes

@ericlol1337 3 месяца назад

i dont understand why dont you just complete the gauss jordan

@brightsideofmaths 3 месяца назад

What do you mean with "complete"?

@isaactonui21 2 года назад

👏👏👏👏

@templetonpeck1084 2 года назад

Those ones are weird.

@brightsideofmaths 2 года назад

I like them :)

@georgebaker5038 2 года назад

thwee

@brightsideofmaths 2 года назад

that's wight!

@lekemorfaw4375 3 года назад

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

@brightsideofmaths 3 года назад

Thanks! Yes, it is German :D

@tonikaiser2823 4 года назад

wo kommt das so vor im echt?

@davidmac451 5 лет назад

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

@brightsideofmaths 5 лет назад

Why is it terrible? :)

@davidmac451 5 лет назад

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

@brightsideofmaths 5 лет назад

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

@ninakoch1799 5 лет назад

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 2 года назад

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