Calculating the Kernel of a Matrix - An Example
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- เผยแพร่เมื่อ 14 ต.ค. 2024
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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)
Really he showed me the Bright side of Mathematics.
Well shiiiieeeet! That’s all I had to do? THAAANK YOUUU!
Thx man , after knowing the concept before then start watching this to see the details, everything is finally connected for me. Really appreciate it 🙂.
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.
@@1mol831 The kernel is the vector subspace of all vectors x that solve Ax = 0.
@@brightsideofmaths Hello! Do you have a Discord channel? I am currently a 1st year student at Mathematics and Computer Science University. Thank you!
I have a community server. See description.@@Asdfgadv33423
Cool, calm and good explanation. Well done.
Once again he saved another life
Simple and straight to the point! Well explained
Thank you, You just saved my ass more than you know.
I'm just curious; how have your studies been going?
isn't it x2 = alpha - 0.5 beta? As you should divide by -2 and not by 2?
The best youtube channel in math!
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
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
@@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?
We have exactly as many free variables as vectors in the basis :)@@darcash1738
@@brightsideofmaths gotcha. Thanks :)
Very helpful and clear, thank you
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.
Thank you.
What do you think about Axler's Linear Algebra Done Right?
I like it!
Can you write 'x2' as a new free parameter or does it have to be in terms of alpha and beta?
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.
This video helped me so, so much. Thank you!
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 😭😭
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.
@@PunmasterSTP Thanks a lot 😊😊
@@tinyasira6132 No problem; I just try to help where I can, and I hope I don't confuse more than I help!
@@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?
Thank you for starting uploading in English it will helpful for many others
Thank for saving me sir..highly appreciated 🙏🙌
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?
Ran and kernel are defined for matrices and linear maps in the same way!
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!
Thanks a lot.
Thx bro you have saved me 😂
Thank you I was stuck in the Matrix
it is equal to null space of A can u do the polynomial one?
I wish you were my linear algebra professor!
Yeah I'm sure he'd make a great one. I was just curious; how have your studies been going?
Bilden diese zwei Vektoren, die wir am Ende gekriegt haben und die eine Basis bilden, auch Span?
Der Span von den beiden Vektoren ist ja genau das was dort schon steht, d.h. alle möglichen Linearkombinationen der zwei Vektoren.
Is there a possibility to a Im(A) of this matrix bcs im not really getting Im
Excelente ejemplo .!
How do you decide what are free variables?
Use the variables that correspond to the columns without pivot element.
@@brightsideofmaths ah ok
can the kernel be all 0’s? like the solution to the system would be x=0, y=0, z=0, is that possible?
Yes, the kernel can be trivial and just contain the zero vector!
@@brightsideofmaths ❤️❤️❤️❤️
@@brightsideofmaths and then the dimension of the kernel is considered 1, right?
Nice explanation sir
Thanks and welcome :)
In the first step, you're just converting the matrix to reduced row-echelon form correct?
Yes!
@@brightsideofmaths Thank you! Very helpful video
Ker(A) is the vector space, however the equation Ax=0 seems to be defined as real value ^^
x is a vector :)
Thank you!
Thank you a lot!!!! I LOVE GERMANY!!
here, the dimension of the kernel is 2 right?
Yes :) It's 2.
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
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.
Super Video 👍
Thank you very much!
Danke dir, hat mir sehr geholfen!!
bitte,
What’s the name of the author?
My name? :)
The Bright Side Of Mathematics yes
i dont understand why dont you just complete the gauss jordan
What do you mean with "complete"?
❤
👏👏👏👏
Great tutorial video though Permit my saying that your accent seems german.
Thanks! Yes, it is German :D
terrible way of writing 1's haha thank you though.
Why is it terrible? :)
@@brightsideofmaths looks like an upside down v and not a 1 :') Your video helped with my exam today, thank you.
@@davidmac451 And do not forget my 3's: The look like a rotated omega ;)
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.
@@davidmac451 How did your exam and the rest of your class go?
wo kommt das so vor im echt?
Those ones are weird.
I like them :)
Does anyone feels a little giggling when you hear the word biguss........ Diguss.....
Thx you support me
thwee
that's wight!
I LOVE GERMAN PEOPLE
Nice :D
Aight sir, thanks for the video but ur accent is terrible,ngl
Thanks :D
thank you so much. it was clear and very helpful
Thank you
thank you so much