Welcome back! I'm glad to hear that you find the video helpful for your new course as well. It’s always good to revisit and reinforce these concepts. Best of luck with your studies, and if you have any questions, feel free to ask & thanks for watching again 😇
I think most of the books for linear algebra are good and understandable. These books are great: 1. Introductory Linear Algebra: An Applied First Course (International Edition) 2. linear algebra Schaum's outline sixth edition I have studied and taught them as well. Personally, I recommend the international book because it is easy to understand and search about specific things. Good luck & let me know if I can help you further 👍😇
For the infinite solution type, do we have to show the other rows are also consistent? Is there a case where one row is all 0 but some other rows are inconsistent? If we do have to show the consistency, how?
Thanks for asking. First of all, I really recommend for you to see the video in which I have illustrated the consistent & inconsistent. Here is the link: th-cam.com/video/K4SKNWDbuXc/w-d-xo.html If you have any row, which has no solution, then you have to say the system has no solution. For example, if you have one row that has all zeros & one row has no solution then you have to say the system has no solution. Please, let me know if you have any questions & good luck 😇
Each row is an equation, as we have mentioned in the video. So, the last row for the no solution part is 0 x1 + 0 x2 + 0 x3 = c Let say as in the example: 0 x1 + 0 x2 + 0 x3 = 2, this equation has no solution since there is no way to substitute for any x1, x2, and x3 to make the left-hand side and the right-hand side are equal for this equation. Do not hesitate me know if you want me to explain more👍🏻
If the second and third rows are all 0s, yes, it usually means there are infinitely many solutions. This happens because there aren’t enough equations to find all the variables (x_2 and x_3).
Thanks a lot Sir for the explanation. Even after understanding and studying the theory I was unable to solve this question I've an exam tomorrow. It is only because of you that i'm now feeling confident.
You're very welcome! I'm so glad to hear that the explanation helped boost your confidence. I hope your exam went well! If you ever have more questions or need further clarification, don't hesitate to reach out. Thanks for watching & your kind words 😇
@@AgnivoDey Congratulations on your successful exam! I'm delighted to hear that you confidently handled the question about the solutions for the linear system. Your hard work is clearly showing. If you have any more questions or need help in the future, just let me know. Keep up the fantastic work & Good luck 😊
Thanks for watching & I'm glad you found the video helpful!😇 Regarding your question about inf^1 and inf^2 solutions, I'm not quite sure I fully understand what you're asking. Could you please provide a bit more detail or context? I'd be happy to help once I have a clearer understanding of your question.
@@Mulkek Hi thank you for the responds, I'll try to best explain it. If det A=0 --> if R(1) = n-1 R(A/B) = n then imposs. if R(1) = n-1 R(A/B) = n -1 then inf^1 sols. if R(1) = n-2 R(A/B) = n then imposs. if R(1) = n-2 R(A/B) = n -1 then imposs. if R(1) = n-2 R(A/B) = n -2 then inf^2 sols.
Thanks for your question 😊 I'm not completely clear on the notation inf^1 or inf^2, as it's not a standard notation I've used. Also, I haven't covered the topic of rank in this video, but I'm planning to make a video on rank, nullity, and basis in the future. Stay tuned for that & I hope it will help clarify your doubts.@@PortalColin
To solve a 2x2 linear system by RREF (Reduced Row Echelon form) as the following: Firstly, you have to put it as an Augmented Matrix, then you have to apply RREF on the Augmented Matrix. Then, in the last Augmented matrix which you will reach, you get infinitely many solutions if you have a row of zeros. At some time in the future, I will solve it for a 2x2 linear system, and for now, I really recommend to you to see these videos which I have explained as you mentioned but for 3x3: Gauss Jordan elimination (RREF) for Ax=b, infinitely many solutions th-cam.com/video/FN_Lsqexlsk/w-d-xo.html Confirm the infinitely many solutions for Ax=b th-cam.com/video/Xd67U--5dgA/w-d-xo.html Gauss Jordan elimination (RREF) for Ax=0, Infinitely many solutions th-cam.com/video/IaN1NyNitCk/w-d-xo.html Let me know if you need any more explanations 😇
Glad to hear that & you are so welcome 😇 It makes me feel better when I say I'm from this planet since we're all human and we have no difference, and I like helping & supporting all kinds of life equally.
So..i agree with you about that we are all humans and there is no difference, but I asked couse i found your English is understandable and clear to me as a non English speaker...and I watched others but I could not understand what they were saying couse of there accent That's it I wish you got it 😌
@@نورالهدىاحمد-ش5ت Occasionally, I find native English speakers to be fast and hard to follow, causing students to be unclear about the lesson, especially international students. It makes me happy to hear that and I'll do my best to make it as simple, understandable, and interesting as possible. Additionally, sometimes if it's hard, I feel like I really want to know, like if I'm playing a game and set it to hard 😆 Good luck👍
Could you please illustrate what do you mean by A+X? or for example, at what time I mentioned A+X? I can explain more about m×n matrices. Each row in an augmented matrix represents an equation, and if there is only one answer for this augmented matrix that satisfied these rows it is called a unique solution.
Thanks for your question! This video covers topics from Linear Algebra that are typically taught at the College or University Level, so it's more Advanced than GCSE. However, in some countries, the introduction to matrices, including operations like Addition, Subtraction, Multiplication, and Solving Systems like Two Equations with Two Variables, is part of the High School Mathematics Curriculum. If you're looking to understand the basics of Matrix Operations, you might find my other videos in the *Linear Algebra playlist* helpful. These include *Scalar Multiplication of Matrices* th-cam.com/video/SK1jBwFh4t4/w-d-xo.htmlsi=Cf2mOTFBGfjMCldD
*Adding and Subtracting Matrices* th-cam.com/video/4WS9xCU07C0/w-d-xo.htmlsi=HTVnlWn15xefi66X *Multiplying Matrices* th-cam.com/video/-_1QKxx_sus/w-d-xo.htmlsi=CrVenkYrVb1aob0i *Solving Two Linear Systems Using the Inverse* th-cam.com/video/ZOFOx3g8Oq0/w-d-xo.htmlsi=DsoG0HgBsqfgELaA *Linear Algebra Playlist* th-cam.com/play/PLPm9fyDNbwj9zUIHKYfNwkfeWWBamk-q_.html These topics might provide a good foundation and are definitely worth checking out if you're interested in Foundational Mathematics. Let me know if you have any other questions 😇
a, b, and c are the values of the result of Ax=b after applying Reduced Row Echelon Form (RREF). The following video link shows the results after applying RREF, which are [-3 6 -4] (a=-3, b=6, c=-4) th-cam.com/video/YbDqFr04EOg/w-d-xo.html at time 12,08 which has a unique solution. In following video link shows the results after applying RREF, which are [-7 6 1] (a=-7, b=6, c=1) th-cam.com/video/jr_OFD3xjkE/w-d-xo.html at time 12,25 which has no solution. In following video link shows the results after applying RREF, which are [1 1 -2] (a=1, b=1, c=-2) th-cam.com/video/FN_Lsqexlsk/w-d-xo.html at time 10,35 which has infinitely many solutions. You can call them a1, a2, a3 instead of a, b, c because b here is different from Ax=b. I wish you the best of luck, and if you have any more questions, feel free to ask👍
I assume that the result vector is equal to (1,2,2) because I wanted to illustrate what is unique, no solution, or infinitely many solutions with an example. You can hear that I have said let's say a=1, b=2, and c=2 which means that I have assumed the result vector is equal to (1,2,2) Please, let me know if you want me to explain this more, and good luck 😇
Hello bro, Here's how I'm gonna illustrate it: Let's write number 1) [1 0 0|1 0 1 0|2 0 0 1|2] This can be represented as three equations: (1*x1) + (0*x2) + (0*x3) = 1 (0*x1) + (1*x2) + (0*x3) = 2 (0*x1) + (0*x2) + (1*x3) = 2 Since anything multiplied by zero is equal to zero, so we can write these equations as 1*x1 = 1 1*x2 = 2 1*x3 = 2 Since anything multiplied by one is equal to the same thing, so we can write these equations as x1 = 1 x2 = 2 x3 = 2 As you can see, we have only one solution, which is called a unique solution. Like us, each of us is unique (we have unique fingerprints & I think everything about us is unique) 😊 Good luck & feel free to ask anything!
@Daniel Leo 1*x3 means 1 multiplied by x_3. I have used the same notation as @How lam cherry Chung. So, 3 in x_3 is a subscript for x. Therefore, 1*(x_3)= x_3. I hope this is what you asked for. I haven't seen your comment before now. I wish you the very best, and if you have any more questions, feel free to ask 👍
It will be the same idea since I put Ax=b and b it can be any number including zero. However if b=0 you will just have two possible solutions: 1. The system has a unique solution which means only one solution. 2. The system has infinitely many solutions. So, you will not have a no solution since we mentioned in the video it has to be "the left of the line has a zero row and the right of it a number not equal 0". To be more clear if you have Ax=0 9:25 2. no solution it will be the same as 10:50 3. infinitely many solutions so, since you have a zero row, then, it will be infinitely many solutions. I recommend you to see the following video which I have compared Ax=b Vs Ax=0. th-cam.com/video/K4SKNWDbuXc/w-d-xo.html Let me know if you need more explanation & Good luck 👍
This is exactly what I was missing from my notes. I'm going to ace this exam now!
Glad it helped, and good luck 😇
@@Mulkek For sure just got a 100 and aced the bonus thanks!
@@Vassimau Great to hear that! Everything in math is simple and just need time to make it sense and then apply it 😃
So far the best videos in explaining linear algebra. Just straight, precise and clearly🙌🙌
Glad you liked it & great to hear that 😇
This is what I have been missing!! Thank you
Glad it helped, and good luck 😃
I've been here b4; the video was already liked. Now i'm back with a new course where i need to relearn this XD
Welcome back! I'm glad to hear that you find the video helpful for your new course as well. It’s always good to revisit and reinforce these concepts. Best of luck with your studies, and if you have any questions, feel free to ask & thanks for watching again 😇
You’re so patient
Thanks 😊
The best video I have ever seen about solution types in matrix, thanks a lot
Glad you liked it & thank you for watching 😇
I understand this better than my math teacher.
Thanks, and I'm glad to hear that😇
Can you recommend some good book for linear algebra?
I think most of the books for linear algebra are good and understandable.
These books are great:
1. Introductory Linear Algebra: An Applied First Course (International Edition)
2. linear algebra Schaum's outline sixth edition
I have studied and taught them as well. Personally, I recommend the international book because it is easy to understand and search about specific things.
Good luck & let me know if I can help you further 👍😇
@@Mulkek Thank you! I’ll start it now!
@@livinginncity7870 Great! Good luck👌
For the infinite solution type, do we have to show the other rows are also consistent? Is there a case where one row is all 0 but some other rows are inconsistent? If we do have to show the consistency, how?
Thanks for asking. First of all, I really recommend for you to see the video in which I have illustrated the consistent & inconsistent. Here is the link:
th-cam.com/video/K4SKNWDbuXc/w-d-xo.html
If you have any row, which has no solution, then you have to say the system has no solution.
For example, if you have one row that has all zeros & one row has no solution then you have to say the system has no solution.
Please, let me know if you have any questions & good luck 😇
this is what i was looking for understood it perfectly
thank you
keep up the good work
Glad to hear that & and thanks for watching 😇
Thank you for this ! have an exam soon and this helped me so much
Glad it helped! Good luck and best wishes 😇
Tq...bcs of u now i can understand this😁
I'm glad to hear that😇
No words just thank u!!!🙌
My pleasure, and thank you for watching 😊
i don't understand the no solution part. What do you mean the left line has a zero row and the right of it a number =/ 0?
Each row is an equation, as we have mentioned in the video. So, the last row for the no solution part is
0 x1 + 0 x2 + 0 x3 = c
Let say as in the example:
0 x1 + 0 x2 + 0 x3 = 2,
this equation has no solution since there is no way to substitute for any x1, x2, and x3 to make the left-hand side and the right-hand side are equal for this equation.
Do not hesitate me know if you want me to explain more👍🏻
Thanks..... Very clearly explained
Glad it helped, and thank you for watching 😇
This video is very helpful.. Thank You Sir. 😊
Happy to help and thanks for watching 😇
it makes sense for me, thank you
I'm glad to hear that it made sense to you! You're very welcome. If you have any more questions, feel free to reach out. Thanks for watching 😇
very precise and short,but great,
tysm!!!
Thank you so much for your kind words & I'm glad you liked it 😇
what if the second and third rows are all 0? does it include having infinite solutions?
If the second and third rows are all 0s, yes, it usually means there are infinitely many solutions. This happens because there aren’t enough equations to find all the variables (x_2 and x_3).
Excellent explanation
Glad you liked it 😇
You saved my life thank you so much !
Glad it helped!
Thank you sir...I was messed up in this topic it was really helpful
Thank you too for watching, and good luck👍🏻
THANK YOU SO MUCH
Glad it helped😇
@@Mulkek THANK YOU SOO SOOO MUCH FOR THIS VIDEO ITS STRAIGHT TO THE POINT
@@za___moss8445 Thank you too for watching the video, and I'm really happy to hear that 😄
Amazing explanation thanks a bunch!!! ❣
Thank you so much! I’m so glad you enjoyed the explanation 😇
Thanks alot this will help me in my exam
You are so welcome! Good luck and best wishes 😇
يعطيك العافيه 🥰🥰
Thank you for watching 😇
Well done... Keep it up
Thanks, and I will try my best! 😇
Each leading entry must be one ?
As you know, the important thing is to solve the linear system and the standard way to do it is to put each of leading entry 1.
You're just amazing💖💖
Thank you so much! I’m really glad you enjoyed the video 😇
you save me！！！thank you so much ！😭😭😭
No worries. We are here for help😇
Nice video sir 😊
Keep watching😊
thank you so much, this is just what I needed
Glad it helped & thank you for watching 😇
Thank you.. For such a helping video 😊
Thanks for watching, and glad it was helpful 😇
Thank you 🙏🏽
You are so welcome
awesome video, thank you!
My pleasure! You are so welcome 😇
Great video!
Glad you enjoyed it 😇
Thank you bro this helped a lot ❤️
Glad it helped & Good luck 😇
Thanks a lot Sir for the explanation. Even after understanding and studying the theory I was unable to solve this question I've an exam tomorrow. It is only because of you that i'm now feeling confident.
You're very welcome! I'm so glad to hear that the explanation helped boost your confidence. I hope your exam went well! If you ever have more questions or need further clarification, don't hesitate to reach out. Thanks for watching & your kind words 😇
@Mulkek thank you sir. Exam went very well and there was a problem on no of solutions in the paper which I was able to solve without any difficulty 😀
@@AgnivoDey Congratulations on your successful exam! I'm delighted to hear that you confidently handled the question about the solutions for the linear system. Your hard work is clearly showing. If you have any more questions or need help in the future, just let me know. Keep up the fantastic work & Good luck 😊
@@Mulkek thanks a lot sir.
Great explanation 👏
Glad you liked it
Wonderful thank uu 💗
Glad to hear that & thank you for watching.
Cool video, understand it now, one question I have is, there are inf^1 solutions and inf^2 solutions, how do I know which one is which?
Thanks for watching & I'm glad you found the video helpful!😇
Regarding your question about inf^1 and inf^2 solutions, I'm not quite sure I fully understand what you're asking. Could you please provide a bit more detail or context? I'd be happy to help once I have a clearer understanding of your question.
@@Mulkek Hi thank you for the responds, I'll try to best explain it.
If det A=0 -->
if R(1) = n-1
R(A/B) = n
then imposs.
if R(1) = n-1
R(A/B) = n -1
then inf^1 sols.
if R(1) = n-2
R(A/B) = n
then imposs.
if R(1) = n-2
R(A/B) = n -1
then imposs.
if R(1) = n-2
R(A/B) = n -2
then inf^2 sols.
Thanks for your question 😊
I'm not completely clear on the notation inf^1 or inf^2, as it's not a standard notation I've used. Also, I haven't covered the topic of rank in this video, but I'm planning to make a video on rank, nullity, and basis in the future. Stay tuned for that & I hope it will help clarify your doubts.@@PortalColin
fantastic video
Thanks😇
awesome thanks! just a suggestion try keeping the voice more human??if that makes sense?
Yeah, sure. Thanks for watching 😇
@@Mulkek bruh thanks for making the video
@@helloworld-hv9oy Happy to help, and you are always welcome 👍🏻
How can i get many infinitely solution if i have 2x2 linear system
To solve a 2x2 linear system by RREF (Reduced Row Echelon form) as the following:
Firstly, you have to put it as an Augmented Matrix, then you have to apply RREF on the Augmented Matrix.
Then, in the last Augmented matrix which you will reach, you get infinitely many solutions if you have a row of zeros.
At some time in the future, I will solve it for a 2x2 linear system, and for now, I really recommend to you to see these videos which I have explained as you mentioned but for 3x3:
Gauss Jordan elimination (RREF) for Ax=b, infinitely many solutions
th-cam.com/video/FN_Lsqexlsk/w-d-xo.html
Confirm the infinitely many solutions for Ax=b
th-cam.com/video/Xd67U--5dgA/w-d-xo.html
Gauss Jordan elimination (RREF) for Ax=0, Infinitely many solutions
th-cam.com/video/IaN1NyNitCk/w-d-xo.html
Let me know if you need any more explanations 😇
Thanks your Vedio help me a lot
You are so welcome and glad it helped 😇
it was completely clear and simple ☺️thanks for your efforts 💜💜
i just want to ask where you are from??
Glad to hear that & you are so welcome 😇
It makes me feel better when I say I'm from this planet since we're all human and we have no difference, and I like helping & supporting all kinds of life equally.
So..i agree with you about that we are all humans and there is no difference, but I asked couse i found your English is understandable and clear to me as a non English speaker...and I
watched others but I could not understand what they were saying couse of there accent
That's it
I wish you got it 😌
@@نورالهدىاحمد-ش5ت racist lol
@@نورالهدىاحمد-ش5ت
Occasionally, I find native English speakers to be fast and hard to follow, causing students to be unclear about the lesson, especially international students.
It makes me happy to hear that and I'll do my best to make it as simple, understandable, and interesting as possible. Additionally, sometimes if it's hard, I feel like I really want to know, like if I'm playing a game and set it to hard 😆
Good luck👍
@@نورالهدىاحمد-ش5ت مبين انه هربي عربي
youre amazing thank you!
Glad you liked it & you're so welcome 😇
thank you for the video 🙂
You're welcome 😊
How A+X has unique solution in set of m×n matrices? ....bcz it is one equation and it will always have unique solution ?????
Could you please illustrate what do you mean by A+X? or for example, at what time I mentioned A+X?
I can explain more about m×n matrices. Each row in an augmented matrix represents an equation, and if there is only one answer for this augmented matrix that satisfied these rows it is called a unique solution.
Thank you for this video
You're so welcome & Glad you liked it 😇
wow thank you a good lesson
You so welcome 😇
Great👏 concept gotten
Glad you liked it & great to hear that 😇
Habibi very good teacher mashallah
Thank you so much & Glad you liked it
Thank you sir
Thanks for watching & Good luck 👍🏻
Wow well understood
I'm happy to hear that & glad you found it helpful 😇
What level is this for gcse?
Thanks for your question! This video covers topics from Linear Algebra that are typically taught at the College or University Level, so it's more Advanced than GCSE. However, in some countries, the introduction to matrices, including operations like Addition, Subtraction, Multiplication, and Solving Systems like Two Equations with Two Variables, is part of the High School Mathematics Curriculum.
If you're looking to understand the basics of Matrix Operations, you might find my other videos in the *Linear Algebra playlist* helpful. These include
*Scalar Multiplication of Matrices*
th-cam.com/video/SK1jBwFh4t4/w-d-xo.htmlsi=Cf2mOTFBGfjMCldD
*Adding and Subtracting Matrices*
th-cam.com/video/4WS9xCU07C0/w-d-xo.htmlsi=HTVnlWn15xefi66X
*Multiplying Matrices*
th-cam.com/video/-_1QKxx_sus/w-d-xo.htmlsi=CrVenkYrVb1aob0i
*Solving Two Linear Systems Using the Inverse*
th-cam.com/video/ZOFOx3g8Oq0/w-d-xo.htmlsi=DsoG0HgBsqfgELaA
*Linear Algebra Playlist*
th-cam.com/play/PLPm9fyDNbwj9zUIHKYfNwkfeWWBamk-q_.html
These topics might provide a good foundation and are definitely worth checking out if you're interested in Foundational Mathematics.
Let me know if you have any other questions 😇
thanks a lot
Thank you for watching 👍🏻
What is a, b, c here
a, b, and c are the values of the result of Ax=b after applying Reduced Row Echelon Form (RREF).
The following video link shows the results after applying RREF, which are
[-3
6
-4]
(a=-3, b=6, c=-4)
th-cam.com/video/YbDqFr04EOg/w-d-xo.html
at time 12,08
which has a unique solution.
In following video link shows the results after applying RREF, which are
[-7
6
1]
(a=-7, b=6, c=1)
th-cam.com/video/jr_OFD3xjkE/w-d-xo.html
at time 12,25
which has no solution.
In following video link shows the results after applying RREF, which are
[1
1
-2]
(a=1, b=1, c=-2)
th-cam.com/video/FN_Lsqexlsk/w-d-xo.html
at time 10,35
which has infinitely many solutions.
You can call them a1, a2, a3 instead of a, b, c because b here is different from Ax=b.
I wish you the best of luck, and if you have any more questions, feel free to ask👍
Thanks man
You're welcome!
Ofcourse the video was awesome and ofcourse it was very helpful
Thank you so much & I'm thrilled to hear that you found the video awesome and helpful 😇
Thanks sir
You're so welcome
thank u bro
You are so welcome & thanks for watching 😇
Good.
Thanks for watching 😇
I don't get were the 1,2,2 come from
I assume that the result vector is equal to (1,2,2) because I wanted to illustrate what is unique, no solution, or infinitely many solutions with an example.
You can hear that I have said let's say a=1, b=2, and c=2 which means that I have assumed the result vector is equal to (1,2,2)
Please, let me know if you want me to explain this more, and good luck 😇
Thanks
Thank you too for watching 😊
Hello, I still do not very understand Why 1) X3 is equal to 2?
Hello bro,
Here's how I'm gonna illustrate it:
Let's write number 1)
[1 0 0|1
0 1 0|2
0 0 1|2]
This can be represented as three equations:
(1*x1) + (0*x2) + (0*x3) = 1
(0*x1) + (1*x2) + (0*x3) = 2
(0*x1) + (0*x2) + (1*x3) = 2
Since anything multiplied by zero is equal to zero, so we can write these equations as
1*x1 = 1
1*x2 = 2
1*x3 = 2
Since anything multiplied by one is equal to the same thing, so we can write these equations as
x1 = 1
x2 = 2
x3 = 2
As you can see, we have only one solution, which is called a unique solution. Like us, each of us is unique (we have unique fingerprints & I think everything about us is unique) 😊
Good luck & feel free to ask anything!
@@Mulkek isnt it 1x3 is =3??
@Daniel Leo 1*x3 means 1 multiplied by x_3.
I have used the same notation as @How lam cherry Chung.
So, 3 in x_3 is a subscript for x. Therefore, 1*(x_3)= x_3.
I hope this is what you asked for.
I haven't seen your comment before now. I wish you the very best, and if you have any more questions, feel free to ask 👍
what if ax=0
It will be the same idea since I put Ax=b and b it can be any number including zero.
However if b=0 you will just have two possible solutions:
1. The system has a unique solution which means only one solution.
2. The system has infinitely many solutions.
So, you will not have a no solution since we mentioned in the video it has to be "the left of the line has a zero row and the right of it a number not equal 0".
To be more clear if you have Ax=0
9:25 2. no solution
it will be the same as
10:50 3. infinitely many solutions
so, since you have a zero row, then, it will be infinitely many solutions.
I recommend you to see the following video which I have compared Ax=b Vs Ax=0.
th-cam.com/video/K4SKNWDbuXc/w-d-xo.html
Let me know if you need more explanation & Good luck 👍
I enjoyed it
Glad you like it & Good luck 😇
احسك عربي صح ولا😂؟
حتى انا قلت جي والله يبين من اللكنه 😂
Underrated
2-7=y+7=22 made it easier for you lol
Shut up this ain’t middle school algebra
Tq
You are most welcome
🤫🤫🤫🤫
🙂
Helo
Hello
@@Mulkek hello.
Time consuming
0 teaching skill sorry sir
Not true, his instructions and examples helped me a lot. Just listen better
You are bot even paying for this course just be nice and appreciate his time
very helpful sir, thank you so much
Glad it helped & you are most welcome
great thanks
You are welcome, and good luck 😇
Thanks
Glad you liked it 😇