The Big Picture of Linear Algebra
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- Опубликовано: 5 май 2016
- MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015
View the complete course: ocw.mit.edu/RES-18-009F15
Instructor: Gilbert Strang
A matrix produces four subspaces: column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
when he emphasized "M-I-T blackboard" i felt that flex.
my professors only use regular blackboards.
In my school, they use a whiteboard
Yael Manlangit my professor used an overhead projector.
I hate black boards
Uzair Akram Don’t be racist.
He wasn't trying to flex. He was just trying to explain concretely. This happens when you spend a lot of considerable time in "maths world"
Prof. Strang gave his final lecture before retirement yesterday. He has touched the lives of millions of students. Please watch his final lecture just to see the emotion filled comments of his students from countries across the world. Thank you, professor!
China along has nearly 3 million views, he's a legend.
May the universe keep and always bless prof strong,
You see that part 5:20 when he rhetorically asked that question was as of he saw my soul...
Only today , have I fully intuitive understood the whole idea of matric maths...
I am an autodidact, polymath in the making that I am trying to switch to all tech and maths science as I see that this is the present and the future, and slowly I am getting all the "first principle" of things so I can be relevant in life....
Hopelly I get a scholarship to help my journey too...
I keep learning...
When you didn't study for your final exam and you have 16 minutes left
x"D !! Damn
I feel your pain
Me right now haha
Sames XD
lol
best part at 04:40
"not very thick, is it?
because it's just a line!"
x'D
Read Flatland by Abbott, if you haven't. Especially interesting after Linear Algebra.
Wink, Wink
wink wink
Gil being Gil
you're just as thicc as the null space
this lecture was in the null space of my brain but now it's in the column space
One thing that I really like about Gilbert Strang is that he has a very easy-going and friendly style.
He doesn't feel like one of those intimidating super-serious professors, he feels like someone who genuinely enjoys having a chat with the students and showing cool things about his courses, in a way that also happens to work as actual course lectures.
The nullspace is also known as the kernel, in case you see that somewhere.
Kernel as in computer programming?
@@normanhenderson7300 No, kernel as in mathematics: en.wikipedia.org/wiki/Kernel_(linear_algebra)
Thank you!
Thanks, I see that all the time. Example: A linear transform. T is injective iff Ker(T) = {0}
Seungchul Lee yeah it’s the set of all vectors which get mapped to the origin
Wow! I was using Strang's linear algebra book back in 1979 in college! Glad to see he's still going at it!
He’s on life support, though. You should have gone with Howard Anton
@@JohnSmith-qp4bt how do you know that?
Maybe the people at MIT should invent a blackboard that can fit an infinite plane?
You could fit more branching ideas if you use the hyperbolic plane. If you can map all the concepts to a tree graph, every branch could have more than enough angular space to expand upon the idea with.
If you want to illustrate Euclidean principles (or just keep text legible), you can stick your figures into a circle that internally scales linearly in a Poincare disk embedding.
Or you could also stick more Poincare disks inside your linear circular regions for asides. Or use apeirogons for enumerating countable sets, etc.
Sure, they have. Not only that but the infinite plane can accommodate both blackboards with infinite space left over.
atimholt nah. Sounds like it would be inefficient to write on and even more difficult to create, to a smart person efficiency trumps conjecture and uniqueness.
@@jacob9673 Computers make any math thing easy. UI’s just been stuck in a rut since Xerox.
A spherical blackboard should work :)
Wish I had seen this when I was taking Linear Algebra. Wonderful short lecture.
Prof. Strang is one of the best if not the best professor when it comes to linear algebra. He makes you visualise all the stuff. Thanks for MIT open courseware for providing this damn good content for free.
Yes it is wonderful to have an ivy league professor as a personal lecturer for free, just remarkable.
Not quite what I expected from the title, but he is great! Watching the video(s) is very entertaining and informative. Thank you!
I love Gilbert Strang's commitment!
He is a wonderful gentleman and a great prof.
He is indeed. An amazing professor. He is intelligent, mathematically conceptual and didactic.
Nice Tutorial. I read this in a Linear Algebra book, about 3 years ago. Fortunately I didnt forget them. Until I watch this Video, I understand the concepts and relationships between these spaces.
I love this Professor. He is amazing and I'm really grateful that he did this , and grateful that MIT hosts it. Thank you
This man will always hold a special place in my heart
this is whom I call a good teacher... brief, thorough and "BIG PICTURE" indeed
who
@@RobertMJohnson The guy in the video.
Best book I have on Linear Algebra is by Mr. Strang. Well worth the read!
He seems like he is very straightforward and to the point, which is great. Jumps right into it and doesn't waste time.
Excellent summary of those key topics in linear algebra. Also, thank you for the well written textbooks!
This guy is too amazing. Just when you think he didn't know how to raise the first board, he does it.
Thank you, prof Strang. Really hope to have you in our school to give a lecture on Linear Algebra.
from OCW 18.06, glad to see Professor Strang again.
havent done linear algebra in almost 7 years, this brought back a lot of the basics very fast
Here is one of the best lecturers in the world.
This lecture is a beautiful introduction to linear algebra.
I love it! Wonderful people like you make me want to study at MIT!
This was amazing. Thank you for the clear and concise explanation!
Just watched it, mind wondered off at the end, but was just pleasant to look and hear. Thanks!:) Legend.
Dearest Gil, It is beautiful and wonderful to watch you work. You are truly a gifted expositor in Mathematics. You bring richness to students of all ages. John M.
Amazing professor and well explained content!
"Oh just by beautifulness, general principles of elegance..."
-Professor Strang talking about the left null space
Brilliant overview of at least half of intro to linear algebra (18.06).
Phenomenal lecture as usual from Professor Strang. Thank you sir for your contributions and enjoy your retirement!
Thank you, prof Gilbert Strang.
once again, your teaching is wonderful~thank you
Thankiu very mocht Teacher Gilbert Strang for your excelent explanecion about of the Sistem’s Linear ecuations for any dimensional’s Spaces.
To the point explanation and I m also surprised with such recalling capacities of the things 🙏Thanks Sir!
Wow, amazing lecture, incredible.
This is an excellent explanation. I wish I had gone to MIT for the lecture and then entered into my alma...
Thank you Dr. Strang!!! You've helped me tremendously. I have never seen such a clear and concise explanation of linear algebra.
Just a couple notes for anyone still confused:
- column space is also known as range
- null space is also known as kernel
- the row space is sometimes called "null space perp(endicular)"
- the SVD computes a basis for each of these four spaces
- the row space is the set of all input vectors v for which Av =/= 0
- the left nullspace is the set of all vectors that A cannot produce in its output range
- vectors in the row space in the domain "get sent" to the column space in the codomain
- vectors in the null space in the domain get sent to the zero vector in the codomain
- In this example, the domain=R^2 and the codomain=R^3. If you haven't learned these terms yet, go look them up. Matrices are linear functions, so they can be described using the language of functions, such as domain, codomain, range/image, preimage, injective/one-to-one, surjective/onto, etc. Note that here I use range to mean the same thing as image, so in my terminology the range/image is a subset of the codomain.
"- the row space is the set of all input vectors v for which Av =/= 0
- the left nullspace is the set of all vectors that A cannot produce in its output range"
Plain wrong. The complement of a subspace is never a subspace because it never contains the zero vector.
Great teacher!!! Explained it perfect!!
Every youtube video on linear algebra I make a connection to something new.
I don't have any degree in math but I have been studying it for 2 years. I just bought a linear algebra book and I find this stuff so fun to do.
Which book is that? I'm also trying to study myself to have a shot at becoming data scientist
It seems legit to watch this video while learning Fredholm Alternative for compact operator. It always amazes me how mathematicians can go such far and I think there’s more generalized theory for normal operator.
It’s not just a blackboard. It’s an MIT Blackboard.
Watching it nearly two years after the exam.
Well at least it's nice to refresh these stuffs
Beautiful explanation, so fun to watch, thank you professor Strang
Great explanation as always! Thank you!
Must-see video for the rest of us, who were taught that linear algebra is about manipulating matrices.
Great and amazing prof. Thanks! so rewarding!
Officially mind blown with these patterns!!! How did I not observe any of those!! Need to improve observation skills
I have Gil's latest book ("Linear Algebra and Learning From Data"). It's just like listening to him talk.
Good to get such short lectures to introduce such difficult topics to me
When you go with a 1.5 OR 2 speed, dear Prof. Gil is quick and vigorous like a young man! It's the fascinating part of online learning.
!!!
Haha, you know how it works. Everyone can learn at its own pace. (Up to 2x speed ;))
@@David-km2ie Not necessarily just up to 2x speed :) I often watch videos at 3x or therabouts. Just a simple case of running [document.getElementsByTagName("video")[0].playbackRate = 3] in the browser console :)
Wade He old guys last longer.
@@han5vk you can just download an extension for that in chrome webstore
thank you for a clear vision on the big picture of linear algebra
His book on Linear Algebra is great fwiw
15:51. No: thank YOU, Sir!
Prof Strang just rock..
Wonderful, intuitive explanation.
So much information in so few words :)
Very clear and direct. Good teacher.
i began to find Linear Algebra an interesting Subject after watching this open course!
Thank you, professor!
Thank you. Studying this now in conjunction with the singular value decomposition of an m x n matrix and least squares - trying to gear up for generalized linear models. This was very understandable, but complex enough, and I'm getting an inkling that it's fundamental to linear algebra - in fact I heard it described as the fundamental theorem of linear algebra, i.e. important enough, that's it's very interesting.
Make sure to learn how the SVD decomposes a matrix into bases for all four subspaces (partitions of the left and right singular vectors). This will really make this all click for you, from a modern, computational perspective. Most presentations of SVD don't even mention this
@@q44444q Thank you. I'm right at that point.
Very important topic! Only Gilbert Strang’s book and course has it early in the course, that is the only way to teach it. Thank You! It is unfortunately missed in most of the other textbooks early in the course, where it should be, and showed randomly in different chapters, so students never get a complete picture. But in MIT OCW by Gilbert Strang it is done right at the right time in the right place! Right On Gilbert Strang! Respect! Thank You!
Professor Strang is amazing. The best.
5:20: The way he says "You wanna know why?" Like he was about to say something really naughty 😂
The way linear algebra is taught is absolutely baffling. If every course began with just this simple 15 minute overview, I'm sure it would spare a lot of students a year of hell.
Kindly raise the first board to give more descriptions for our better understanding of electrodynamic principle projections along perpendicular null space .
the null space, now available wherever vectors are sold.
I attended his linear algebra class in the early 00s before the OCW. Such a legendary professor.
It's kind of amazing that there are so many professors out there that don't succeed in explaining this to people, but someone that watched this video could take the information learned and explain it to a fellow student successfully in a short amount of time. I will admit it is possible that youtube videos such as these are doing no more than filling in gaps of professors, instead of informing the student 100 percent more than they were before they watched it.
Well, Strang's entire MIT linear algebra course is also online in the form of "videos such as these", so it's in fact informing the student 100% and not leaving any gaps!
My class had a teacher with a thick accent, was very softspoken, and always blocked the board when writing down concepts while explaining them. I literally learned like less than 10% of the stuff he was teaching. If it wasn't for these lectures, I would have never made it through that class. I didn't even need to show up to class since the grade was 50% midterm and 50% final. I'm paying for an education that I'm not getting. I'm getting it from these videos.
Dear Sir, you could easily teach other teachers how to teach.
There are a few like you, but not as many as we need.
Dear MIT,
Out of the ones on the internet, MIT has the most gifted teachers as teachers, before being experts in their areas.
As far as I am concerned.
So please continue to do what you are already doing.
Now I need someone to give me the big picture of the big picture so I can understand what he's talking about.
Thank you so much sir! I enjoyed this beauty of math while I learned via your wonderful teaching! God bless you!
The reason it is called the left null space is because it is typically obtained by analyzing vA = 0 which is essentially the same thing he did to calculate the left null space but he presented it differently with transpose of A
Thank you. Could you tell me why the null space MUST be perpendicular to the row and column spaces? Thanks!
Thanks! This is really a great video :))
I love you ❤️
Keep the good stuff, precious you
Excellent summary of a subject I've long been curious about. Not sure if Professor Gilbert is kidding or not though.
Thank you for clearing up the concepts that our lecturers messed up!
This lecture would have made Axler much easier to understand! Never thought I’d say this, but I’m excited to pick up the book again.
thank you professor Strang
The math he did at 4:00 blew my mind
At the end of the video, shouldn't left null space go together with row space and null space with column space?
Wish i was given the opportunity to be taught by him!
I took Abstract Algebra, and at least 2 other math courses that required Linear Algebra as prerequisites, without actually taking Linear Algebra. Watching this 15 minute video helped me realize what I missed. Thank you!
At 6:35 he says "zero combination of the rows" but that's incorrect. It's zero combination of the columns; isn't it? Additionally, what are m and n. I'm assuming the matrix A dimensions, but they were used without being defined!
"Only has 1 puny vector..."
Love it
Insightful: row space is perpendicular to null space, so is column space to left null space.
what a beautiful metasummary of linear algebra, kudos to prof. Gilbert for the amazing mini lecture.
This directional changes from clockwise to anticlockwise with null space projections in a way represent an electrodynmics combinations as inspired by Professors explanation.
An amazing teacher!
really good teacher but what about linear transportation , determinant etc
Great Visualisation... ❤
the legendary Gilbert Strang yeah...
I love this guy. Best math teacher ever
3:50 whats happening, "one of the third one of the second" and so on, how can you make a 3d vector like that.. how can you multiply a 3d vector to a 2d plane like that?
Knowledge and teaching are beautiful
Internet is amazing. I watched this man teaching linear from 2009 to 2016 to 2020.