3. Multiplication and Inverse Matrices
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- Опубліковано 21 вер 2024
- MIT 18.06 Linear Algebra, Spring 2005
Instructor: Gilbert Strang
View the complete course: ocw.mit.edu/18-...
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3. Multiplication and Inverse Matrices
License: Creative Commons BY-NC-SA
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Math has consumed this man, and whats left of him is pure logic and extraordinary teaching skills.
I wanna become like that! But unfortunately instead of studying for JEE I am just wasting time on UA-cam.
@@psibarpsi you don't need iit to become a mathematician, iisc are way better than it, iit's only job is to produce cheap labor for MNCs
Consumed in a pleasant way :))
ua-cam.com/video/FffvCM0C3x8/v-deo.html
@@abhishekpatawari6871 No IITS are also producing great Indian startup heads like Flipkart, Infosys,and software startups and great scientists and teachers too like hc verma and many more
This guy teaches as if he is having an argument with himself in his mind & finally 1 part of the mind speaks out ..
Try 2x speed, it's amazing.
I love him!!! He answer the question 🙋♀️😀😀
He does it intentionally
Thats such a brilliant way of explaining. Thats also how i explain things to myself too. It works and listening to someone as intelligent as him teach this way is really enjoyable
ua-cam.com/video/FffvCM0C3x8/v-deo.html
I Love how he seems to discover things as he is about to write them on the board. But he is literally the who wrote the book on the subject. What a good teacher!
That’s the way a good teacher should be, let the student be allowed to go on the journey of discovery for themselves, or at least the illusion of it.
"But now for the most important step: WHY..." This is what sets apart a math teacher from a great educator. 90% of people teaching GJ elimination wouldn't bother to show you why it works, thus creating a legion of students who act more like computers than mathematicians. What Strang does should be considered the standard, and its deeply satisfying to learn from him because of his standard of decency; that is, to respect students' intelligence and curiosity.
Lecture timeline Links
Lecture 0:0
Method 1: Multiply matrix by vector 0:50
When allowed to multiply matrices 4:38
Method 2: Multiply matrix by COLUMN 6:12
Method 3: Multiply ROW by matrix 10:4
Method 4: Multiply COLUMN by ROW 11:37
Method 5: Block Multiplication 18:25
Inverse Matrices (Square matrices) 21:15
Invertible Matrix 22:0
Singular Matrix (No-inverse matrix) 24:39
Calculate Inverse of Matrix 31:52
Gauss-Jordan Elimination to solve Inverse of a matrix 35:20
Thank you good sir
thank you
22:00
You are a god amongst men.
You are a good man.
Words can't describe how important it is for a mathematics lecturer to have passion in providing clarity with his explanations.
He explain with heart and soul. Thanks for sharing. Big love
TRUE!
AND DICK
*He explains
@@sterkh66 ok
tifawine s teqbaylit ?
Thank God for the student at 38:58. I thought I was losing my mind for a minute there.
Hahaha, me too. I thought it was intentional, and was A (transpose).. :P
38:58
lol when i saw that I came to the comments to see if anyone had an explanation.
agree Haha
ha ha ha ! same here I rewinded so many times to hear again- what had I missed !!!
"Gauss would quit; but Jordan says keep going" 😂😂
I am able to differentiate both from now on
It should be called Jordan Elimination but people give credit to Gauss because Gauss knew about the method but didn't publish it. Jordan alone made all the advancements in the Matrix theory.
This man is awesome. Like every good teachers, he has the strength to refrain from getting angry at students for not understanding right away what comes from his fingers as music comes out relentlessly from a loudspeaker ! May he be live long and happy ! 🎨
26:37 this is for those who did not understand the first reasoning behind why the inverse was not possible.
A=[1 3;2 6]
X is supposed to be the identity matrix so that A^-1A=I
You can approach it this way:
A*(1st col. of X)= [1;0]
which is same as
1st col. of A* X11+ 2nd col of A*X12
but the problem is that since the first column and second column of A are dependent vectors (that is [1;2] and [2,6] differ only in length and not in the direction. It's impossible that a linear combination of them can result in a vector that has a different slope (as is [1;0])
I hope this helps.
Thanks for the explanation. I didn't understand it well while watching the video.
Thank you for these lectures. I took Stage 2 Linear Algebra in 1980. These lectures make it sound simpler. I think Gilbert Strang is the Richard Feynman of linear algebra.
Apparently it's hard to see a teachers who understand the subject and explains it to student, but Dr. Strang just know exactly what he's doing, love it!
This teacher, oh boy I can`t remember his name... Well... I really appreciate how he uses intuition to teach Linear Algebra. I'd like to have had classes with him at the school. I would have saved many hours of my life studying for silly tests.
Sometimes I feel like studying Engineering made me less smart than I was as child. Sometimes I can really know that my teachers are trying to kill my creativity and coercing me to learn the hardway, nor the better way nor the easy way: just the way they had learned or the only way the can teach.
André das Neves Gilbert Strang... live long and prosper!
Same for me as a physicist.
André das Neves That's what I wanna say.
nothing could describe my sentiments more accurately than your exact words
foste para o técnico ahahahha
This professor is so wonderful! He presents each new bit of information in such a way that makes it seem intuitive to me and I just love listening to him lecture!
Me too. I hated overheads and markers on whiteboards.
Prof. Strang's style is amazing. He keeps you curios, suspended and involved the whole time.
I got an an A on my undergrad Linear Algebra class but I am still learning a great deal by watching these lectures and reading Mr. Strang's book. A great big thank you to Professor Strang and MIT.
This lecture highlights the essence of Linear Algebra which some textbooks would never be able to do so
Im almost ready to throw away my text and just study these lectures
Software Engineer graduated in Electrical Engineering in 2006 - I watch these for fun, seriously :)
This is pure art bro.
True, and it's not that of a geeky statement as well.
It just intrigues you to see what happens in the next lecture, just like any good netflix series.
ua-cam.com/video/titoevCaQcQ/v-deo.html
@@ahsanulhaque4811 lol you are so accurate, I thought of lectures that way when I watched the 18.01 videos
Some key points I learned from this section:
--- Five ways to perform matrix multiplication
--- When a square matrix is irreversible and why
--- The concepts of row space and column space
--- The Gauss-Jordan method to find the inverse of a matrix and why it works
my right ear feels more educated than the left ear
i have two screens so i simply using the right one instead.
Lmao
i switch my left and right earphones every 10 minutes
I put right into both channels
Man you made me laugh for a while
He is actually providing the algorithm behind the matrix operation used in coding. Amazing and extraordinary skills. now I'll be able to understand how to play with Maths.Thanks a lot Sir.
This teacher is awesome, thank you mister Strang, thank you MIT
@31:20 singular matrices take some non zero vector x to zero and there is no way A inverse can recover it, that's why A inverse does not exist. great explanation.
Your framing helps hammer the point home. Thanks.
@@tharsisharmonia9316
I could n't totally understand, what he means to say..by that statement.
Please elaborate ...
Got it..
Ax=0
Where A is singular matrix.
If suppose Ainverse exists
Then
Inv(A)Ax=0*Inv(A)
Ix=0
x=0
But x is non zero
Non zero =zero
Not possible
Hence Inv(A) does not exist.
@@vishwapriyagautam8227
A=[1 3;2 6]
X is supposed to be the identity matrix so that A^-1A=I
You can approach it this way:
A*(1st col. of X)= [1;0]
which is same as
1st col. of A* X11+ 2nd col of A*X12
but the problem is that since the first column and second column of A are dependent vectors (that is [1;2] and [2,6] differ only in length and not in the direction. It's impossible that a linear combination of them can result in a vector that has different slope (as is [1;0])
I hope this helps.
@@vishwapriyagautam8227 Thanks for this bro
I remember my college years when sometimes I really needed to go back in time and check what the teacher just said, when the material was hard for me to understand, or I needed to draw a better conclusion. It's great that we have videos nowadays.
That old man makes me crave linear algebra. What kind of sorcery is this?
I was coming here to post this exact comment! This man is ... wow!
33:01 after watching the 3blue1brown serie and being on this video on this serie I'm a huge fan of columns, and find they make much more sense than rows.
If you want to understand the geometrical meaning of linear algebra I recommend everyone to watch the 3blue1brown playlist, everything you use on linear algebra has a geometrical equivalent, why matrices that have determinant equal to 0 doesn't have an inverse, how linear transformations (matrices) modify space, how the basis vectors generates the space, how the determinant of a linear transformation is the factor by which the area of something in that space changes, and so forth.
That's exactly what I did! I love how 2blue1brown describes everything geometrically! I was having trouble understanding some of this stuff in these lectures, but it all makes sense now.
But, I still can't find the geometrical meaning of Minors, Cofactors, Adjoint, etc
for anyone confused about the explanation of the computation of the rows and columns of C:
> To compute a column of C, multiply all rows of A by a column/vector of B, aka sum of dot product of rows of A against column of B. The resulting column of C is a linear combination of the columns of A.
> To compute a row of C, multiply all columns of B by a row of A, aka sum of dot product of row of A against columns of B. The resulting row of C is a linear combination of the rows of B.
This is exactly how linear algebra should be taught. Instead of getting hung up on computations or abstractions, he shows the beautiful intuition behind it. Yes, it’s very important to be able to work in the abstract when you start getting to higher level math, but it’s so much easier to do that when you first understand it intuitively.
This is awesome. Absolutely no doubt this is the best i've ever seen. I wish I had known this course at highschool. Pure logic i love it
I learned Linear Algebra a few years back, but I’m enjoying these course lectures as a refresher. I think I’ve forgotten, but I’m gratified to see how quickly it comes back and how familiar it all is upon reviewing. And Gil is a wonderful educator.
Prof.Strang: "If you can tell me whats in that block , I'm gonna be quiet for the rest of the day" Wow !!
This shows his passion. *Hats off* Sir ...
"Lemme just do it the old fashioned way..."
blows my mind....
I study Computer Engineering at IPN in Mexico and I'm taking Linear Algebra after this summer. I can't believe I just found these videos, It's like taking a course at MIT! There's no way I could fail that course now. Thank you for sharing this helpful information.
how did it work out?
@@mind-blowing_tumbleweed Wow, it's been 9 years already, I didn't even remember writing this comment. I graduated in summer 2016 and as of now I work for a big tech. Regarding linear algebra, I aced the course with a 10/10.
Great , great lecture because Prfof Strang elucidates and focuses on the key ideas and the intuitive meaning - not just the mechanics as most books and tutorials do. Thanks Prof. Strang for sharing your great mind and your skills as a great educator!
There should be a way to directly thank this professor after the lecture. So lucid and clear. Hats off Sir.
i studied these things some 3 years ago and was pretty good at that. but never ever i have gone to such depths to understand these, this guy is a awesome.
it is weird that in this series there is no such advertisement like others. that helped a lot in focusing in the lecture
Salute and pranam to this great legend...I hope he lives till eternity
ua-cam.com/video/titoevCaQcQ/v-deo.html
Never seen it this way. He actually teaches the way things work, not just how to apply concepts. Wish I had seen it 10+ years ago. Thanks for sharing!
note: there are 5 ways to think of matrix multiplication.
1. cij=Sum[aik*bkj,{k,1,n}]
2. (Matrix*column=column), Columns of C are combination of columns of A.
3. (row*Matrix=row), Rows of C are combination of rows of B.
4. AB=sum of (columns of A)(rows of B).
5. We can cut matrix into blocks.
Never saw someone with so much logicality, understandability and clarity in teaching linear algebra, a true GENIUS of our times!
I'm taking this course (Linear Algebra) at the nacional university of Rosario, Argentina and this lectures were really helpful. We use Strang's book so this is perfect.
These lectures will live on for as long as linear algebra lives and so will Gil Strang. Now, that's some way to become immortal.
This is also USA. A very impressive part of it. Sharing wisdom. Thank you professor Strang, and thank you MiT.
Thanks very much I finally have a clear pictures of inverse of matrix, what it means, how it was produced, etc..
You are such a good teacher, if one day I can go to MIT I'll definitely visit you.
omfg this course is absolutely amazing. I've never seen matrix operations being explained with such pure logic.
It helps, also, if they have a unique personality, like this guy does. I watched all these linear algebra videos and it was a pleasure watching this professor in action because it was entertaining. No disrespect.
One of the best teachers I've had pleasure to learn from - only online, but thanks to quality of content and ease of acces it feels almost as good.
THANK YOU MIT!
The best thing in the first 3 lectures is the new way you look at a matrix multiplication as linear combination of either rows or columns which makes more complicated topics easier to understand.
I knew gauss jordan and i applied it several times but i never got the meaning of it.But man you finally explained the method.
Hats off to you man
Great lecture, and what I can commit is just 'like it' and 'comment'.
Without any payment like other MIT students, This is huge opportunity.
Thanks for sharing this greatness with world. As we grateful, let's do study hard. :)
Absolute masterclass. I'm trying to learn some linear algebra so I can teach myself quantum mechanics, and I think I may have stumbled upon one of the greatest resources out there.
thank you MIT for open courseware
From the bottom of my heart I want to thank MIT and the professors for creating such beautiful and elegant videos. They are hands down the only reason I understand this topic because my professor unfortunately can explain it as clearly as you guys have presented it or at least the way he presents the lecture isn’t my style.
Most of us trying to learn math are often not made aware of the fact that Math is a tool for understanding how the world works and it's not just the computations that make up the god damn thing. I am amazed by this way of teaching, in which the professor asks the question why and gives it much more importance than all the nitty gritties of how. At 33:44, the professor even points it out. I don't know why all the professors can't seem to see the learning process in the same way. Hats off to this amazing teacher!
i love the sound of the chalks
I love u :)
Very feminine.
@@roronoa_d_law1075 Thirsty ass nigga
What the fuck are these replies damn.
N. Rivers Chalk Dust Torture.... great song. Haha
I love the way he is just being himself with Math, giving the pure logistic to student. I was really desired for a lesson like this.
Having such a person makes me love maths
While discussing 4th way of matrix multiplication, gives subtle hints so that it'll strike you that matrix multiplication is actually about the inner product of the column vectors of the 2 matrices (a^T×b).
What an absolute legend.
I wish I knew about this professor before.
The best explanation. I remember I did gauss jordan elimination back in university but never understood why even are we doing it.. thank you mit for posting it.
From hating Linear to loving it .... this man is a gem ❤
What helped me understand the statement at 9:10 was by using an example. Say A=[1 2 and B = [2
3 4] 3]. A*B can be though of as 2*[1 + 3*[2
3] 4]
@9:18 columns of C is a linear combination of rows of A and Columns of B. C11 = (A11*B11)+ (A12*B21)++ (A13*B31)...
Somehow, I managed to forget all that. Now, I have watched this lecture 3 times with an interval of one week. After that I should never forget how to find inverse :)
Fu.. I had to watch it 4'th time :(
I am having the same problem
I hate myself I cannot remember. I have friends over 25 and still remember what they studied at school. I don't know how come they donot forget
Knowledge is like the leaves of a tree. If your branches are weak, the leaves will wither away. Work on your branches and the leaves will be strongly connected to the branches.
Do a few problems guys that might help.
One easy way to understand matrix multiplication using Machine Learning Classification strategy: 1. Rows of C are combinations of rows of B: just treat it as classification with one example but mutiple classifiers; 2. Columns of C are combinations of columns of A: just treat it as classification with multiple examples but only one classifier; 3. sumOf(Columns of A * Rows of B): jut treat it as classification with multiple examples and mutiple classifiers but should be added up with all features in data examples.
Thank you Lord that I saw these lectures now. It's a shame that is late, but it happened none the less. God bless you Professor Strang
Great summary in the first part of this lecture! Simple examples. Best linear algebra teacher ever! 👏🙏🌏
Being quite about matrix multiplication about rest of the day is what i need at min 20. This guy know how to make a good joke and being helpful with it. Thanks Strang!
If a country has teacher like him, that country is bound to prosper...
The column by row example at around 14 mins really hit home with the whole linear combination point. Something so simple can flick a switch in your head so it all makes sense, so kudos.
Not even Gauss could see instantly it works. Nicely said.
Beautiful ways to look at matrix multiplication! Extremely useful
Tip - Turn mono audio On in the windows 10 audio settings for better audio
C_mp was super helpful! That's a pretty useful technique to figure out how the end result should look before seeing the final matrix.
In windows 10 you can force mono sound:
Go to settings [win+i] > Accessibility > Sound; and here you have a toggle for combining both channels.
The Mister Rogers of Linear Algebra. (The soothing tone, the rhetorical questioning)
I have never been this excited to watch a teacher's lecture. Thank you sir.
I'm Fayçal from Casablanca in Morocco and I do thank you so much for this courses, it makes me see matrices clearly :)
My favorite MIT professor (among many greats).
Matrix multiplication by four methods:
1. Cij: (ith row of A).(jth column of B)
2. Columns of C are a combination of columns A. The columns of B tell how they are linearly combined.
3. Rows of C are a combination rows of B. The rows of A tell how they are linearly combined.
4. Sum of [(column of A).(row of B)]
really like this teaching method. give me new insight about matrix
I have huge respect for him, he teaches in a way to make the subject interesting.
Things we are not supposed to know:
1- The Fight Club
2- Sex
3- Determinants (25:50)
DR. Strang thank you for another great lecture on matrices and their inverses.
I love how every lecture so far they managed to come up with a new and creative way to record sound. It's cute buy annoying.
It's feeling like I m in MIT and great respect for prof.Strang
One of the most useful lectures! By far...
These recitations are actually very good. It is helpful to quickly revise the concepts.
Wow just amazing,in 2 and a half minute, u generalized my concept and the use of the dot product was astonishing, hats off professor🙇♂️🙇
These lectures should be stored properly so that whenever aliens do come, they can learn linear algebra the right way.
At t = 21:00, Strang covers the definition of inversion (and at 32:00 he gives an example): first suppose A is an n x m matrix. Then if you have a left inverse L that acts on A, it will produce an identity matrix, call it I_n; similarly, If you have a right inverse R, when A acts on R it will also produce an identity matrix, call it I_m. So a Matrix is non-singular (sometimes mistakenly called regular) only if m = n, A is square, and det(A) (does not) = 0. In particular, if the matrix A have full rank = n, all its columns and rows would be independent, so A would be invertible.
This man is a beast!!! I love when he says, OK 👌 He’s extremely clear and detailed! I wish I had professors like him🙏
If this is what classes were back at the 2000's, I wonder how good they have become as of now 2010
Or now, as of 2020 o_O
@@smellofmetal Or now, as of 2050 O_o
@@thesickbeat o_O
I paused the video at 33:59 and tred to solve it for myself. Using a, b, c, d as the entries of A^-1 :
For the first entry of I(1, 1) = 1*A + 3*C (=) 1*A + 3*C = 1. For I(1, 2) = 1*B + 3*D (=) 1*B + 3*D = 0, and so on.
Create a new matrix using the coeffients of each of A^-1's entries and solve from there.
this is the clearest tutorial video!!!!! (screams in excitement!!!
huge respect SIR. Thanks a lot.
from BANGLADESH
For the first time in my life that understand what the heck is Gauss-Jordan idea! Thanks Mr. Strang
I went to a GOOD university and met some GOOD people, but I did not GOOD GOOD study, day day up. Watching these videos just because I missed my undergrad life so much
Passionate teacher who walks you through the sessions.