Part 1: The Column Space of a Matrix
ฝัง
- เผยแพร่เมื่อ 4 พ.ค. 2020
- A Vision of Linear Algebra
Instructor: Gilbert Strang
View the complete course: ocw.mit.edu/2020-vision
TH-cam Playlist: • A Vision of Linear Alg...
Professor Strang explains why he now starts linear algebra classes by explaining column spaces and A = CR before A = LU. This captures the key idea of a basis for a vector space.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
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For links to Professors Strang’s related courses on OCW, visit the Related Resources page on the full resource site: ocw.mit.edu/2020-vision.
Prof. Strang certainly sensed the central role of Linear Algebra in the ML/AI development and produced this series to help those who want to tap into the area.
When everyone is complaining about the fast moving world and criticizing how short-sighted people have become, Prof. Strang is answering people's demand by distilling his 18.06 course into a even shorter introduction, with fully updated digitalized slides, AND it's FREE.
What can you say? RESPECT.
18.06, you mean. I thought for a moment there you meant Strang had a lectuere in the 19th century. I certainly agree with you though.
@@chaebeomsheen5875 should have put a period in there 18.06
余宗翰 that’s not a period, it’s a decimal point: 1, 8, 0, and 6 are all numbers.
houkensjtu “FREE” vs. ‘open source’ are different concepts. Printed course materials still must be purchased and I’m sure there’s an option for donations. Perhaps you’re familiar with ‘shareware’?
@@HighestRank You're right. Decimal point is the proper term.
The Legend is back!!!
The Legend never left.
Legends never die!
“ Can I tell you all right from the start?” Yes, Only you can, Dear Dr. Gilbert Strang, I miss your chalk board and your passionate handwriting, big matrices, lines and planes drawings. I think, you miss it too when you are trying to show lines and planes with your hands. Please let him write on white board or even better a wall and keep it there forever like an art! He is the best improvising, jazzing through the lecture, taking it in unpredictable direction and finishing on the most interesting moments, that you just can’t wait till the next episode, I mean lecture.
You inspire me to teach Linear Algebra your way. I am so grateful!
I miss him working on the chalk board as well.
Listening to this makes me so unbelievably happy, I could cry. Gilbert Strang is the math teacher we always wanted.
Clearly, Professor Strang is one of the best teachers- maybe the best teacher- on the planet. As a PhD student (not at MIT), I bought and read all his books. His book- Introduction to Applied Mathematics- is a marvelous reference for understanding the fundamentals of many areas of applied math. I spoke to him on phone in late 1990s regarding his book on Wavelets. He is also a very nice, caring person.
Protect Dr. Strang at all costs during this epidemic!!!
This guy is excellent. He's making me interested in this, and without even any fun visuals or anecdotes, but purely from his high energy.
Gilbert Strang's brand new video. This is a treat. I've been following him on MIT Open Course for about 10 years.
Omg 4 years ago i just start the same course in youtube, but i given up in half way .Cant imagine now i get the 2020 vision, and you just passion as the old day . May be i should start all over again and finish it this time, you are the fortunes of humanity.
First did Linear Algebra in the early 70's, never really understood it until I found his video's. An awesome teacher/lecturer.
Nice! New linear algebra course from Professor Strang!
▲
Oh, how nice to see our dear professor again in a new and modern environment, but most importantly, with the same zeal for teaching math. I wish you all the best, Professor Strang. Sincere greetings from Bosnia and Herzegovina.
He made my day whenever he says "Thanks"
I'm currently taking L.A. and physics at University and those playlists are a true blessing, paradoxically I'm learning much better now during lockdown then when I did at normal lectures,Prof Strang and Prof Lewin have become mentors of mine...you guys are revolutionizing education , i m sure that those lectures will reach one billions of views
Dr. Strang: On slide 4, perhaps observe that the third column is the sum of the first two, so when "filling in" between the first two, we don't go in any genuinely new direction, and therefore we have a plane, not all R^3. (Clearly the first two columns are not in the same direction because you can't rescale one to get the other, so we have at least a plane.) You discuss this on slide 5, _after_ you've already made use of it to claim a 2D column space. (And a big THANK YOU for these videos! They're excellent!)
With this man's help I'm filling in the small gaps in my computer science degree.
Very smart way to find Reduced row echelon form.
I watched the original videos two times and it helped me enormously in my engineering career. I really can’t thank you enough.
The immortal legend. His enthusiasm for the topic is so damn intoxicating
thanks for your devotion professor Strang.
Thank you so much for providing the courses!!!! Best linear algebra course ever!!!
Thank you Dr. Strang!
For those of you who miss understanding and rationale behind the very basics of matrices and vectors, I highly recommend "3 blue 1 brown" youtube channel. You won't regret it, the guy is a genius and explores the math the way it was explored by pioneers. You will like it.
Grant Sanderson rocks!
Thank you! He is so damn good at explaining. He shows the right way to learn Linear Algebra.
I support this message
Thank you to Prof. Strang - a living legend!
Hope his eye might stay without getting so bad.
I still cannot believe I was using his book back in 1978 in my Linear Algebra class that is right next to me now on the shelf....just reached over and touched it!! Great to see you still out there Prof. Strang!
the best linear algebra prof ever !!!
A very helpful and condensed review series of linear algebra
Hero is back! Watching during Covid-19. I like Linear Algebra!
Good to refresh my knowledge in linear algebra even after my ML exam😄 Remember, linear algebra is the king in ML and calculus the queen
Agree😆
Wonderful explanations, thank you!
Its a really intense and distilled video and I don't think I understood everything but I still love it! I am going to use it as a start, especially to get some basic ideas, for which it is amazing, go to the longer videos and then come back!
Thank you very much
Thank YOU Professor Strang！🌹🌹
what i like in your videos not the way you explain them but the motivation to explain this course another time with a different way
you give me a motivation so not to get bored while watching
Cheer ! :D
Just listening to him makes my day
this is art not a course . just wonderfull
Thank you professor Strang 🙌
Wow new way; l just finished this course yesterday. I used your blackboard lecture to study this part and I liked it. Thanks.
@mitopencourseware, Thanks so much for providing us with a brand-new Lienar Algebra course! Really appreciate your effort to provide free, quality educational content during these difficult times. I have one question, though; is this course meant for first time LA students, or is this primarily for computer scientists who need refreshers on relevant topics? Thanks in advance.
❤️ from India. I learn " linear algebra " from your videos .
God bless you , sir
What a wonderful lecture. Thank you so much !
The legend returns
God bless Mr. Strang.
i love this perspective
Thanks Professor!
Love it love it
I like way you teaching. I have graduated and find job about mathematical software with your lessons.
Best linear algebra teacher
Feeling so alive so see a 85 years old teaching😢
Hats off to u Sir......high respect to you
Thank you so much for your help!
Best way to spend a quarantine
My full respect
Thank you MIT
This video cured my depression
I really like Prof. Strang's lecture. It is enlightening for beginners like me.
Maybe drawing on a blackboard will help to explain the concepts better than PowerPoint slides 😄
Thank you Sir ! Best Regards
5:36 :)) ''I ask..Ohh I 've just answered this question. Sorry!" lol
Such a funny and cute guy. :)) Wish you all the best boss
Thanks the lengend comes back! Why is every lecture less than 15mins now? Is this because of no writing on backboard anymore?
Hail prof. Strang
Dr Strang is the best thing ever happened to Linear Algebra for me ... because he is on youtube and that too for free
I still prefer Prof. Strang with chalk and board
I miss him working on the chalk board as well.
tks to my hero
Strang is the GOAt
4:40 Can someone help me understand why Ax (All linear combinations of columns of matrix A) fill out a plane in 3D-space?
What I understand:
- Like Mr. Strang explains, if you have two crossing lines that meet at (0, 0, 0) and fill out the space between them, you get a plane.
What I don't understand:
- How do you get from "(1, 3, 2)^T*x1 + (4, 2, 1)^T*x2 + (5, 5, 3)^T*x3" where each vector obviously has 3 coordinates to the understanding that every linear combination of them resides on the same plane?
Just those 3 particular vectors are coplanar (e.g., via triple product, y × z is orthogonal to both y and z, but if x ∙ (y × z) = det(x, y, z) = 0, then x is orthogonal to y × z, hence it is coplanar with y and z). In general that isn't true, for example, for I₃ = diag(1, 1, 1) we have I₃ x = x, which gives us the whole ℝ³ back.
@@user-jp9qi8po6m thank you.
This is the issue with explaining LA in an hour heh
One equation with 3 variables means 2 degrees of freedom: one pivot variable, 2 free variables->2 spanning vectors after scalar extraction
Wow looking at the response, it feels like people go out of their way to make things waaay more difficult, or to sound smart. The prof didnt really explain it very well and I had trouble with it at first. Visualising it helps tremendously.
Imagine two vectors in space starting from zero, with a small angle between them, Like say the angle between your index and your middle finger. Now imagine a third vector that adds those two original vector, (it of course also starts from zero, its a simple addition). What youll find is that with these three vectors, you've lost a dimension (the third) and are operating on only two dimensions, i.e a plane. This is entirely because that third vector adds the first two.
I have another silly but probably effective example, that will help illustrate why the domain becomes a plane instead of remaining 3d. Let me know
The third column is a linear combination of the first and the second (C3 = C1 + C2), which means that the columns of the matrix are not linearly independent. The first two columns represent two vectors which define a plane in R3. Since the third column can be expressed in terms of these two column vectors, it is on that plane.
Thank you very much
Without advertisements 🙏👌😍
I love this guy
A=CR is so brilliant that it can be used to build the simplest and shortest proof of *row rank = column rank*.
very nice !
best teacher ever 😁
Step by step video solutions for civil engineering questions
why is the linear combination from minute 5 only taking up a plane? I would think because it has 3 vectors of 3 components, it would fill r3... I am new to this study though so some clarification would be really helpful
ok he answered this! we don't need to evaluate the 3rd column because it is just the sum of the first two so it will be on the same plane.
@@microndiamondjenkins566 Are you still single because you are very pretty and I am single if is there any scope of .......you know what I am talking about
I like that you're explaining column spaces first, but i tought it was a little confusing to jump into an example straight away on slide #4 since it's unclear which parts of what you're explaining apply to this matrix, matrices like it, or all matrices. i think it would have been better to have a quick summary of how different kinds of matrices result in the different kinds of column spaces, and then give examples of line, plane & R^3 to illustrate the differences.
Excellent first lecture. That CR factorization was new to me.
There is just one detail I don't understand. Probably I am just a bit tired.
We have r independent rows in the R matrix. The row space of A lies in the span of those rows.
But how do we know the rows of R don't span a larger space than the row space of A?
Hello, Can Some one point to me where is the PPT that Prof Strang is using in the videos???
It doesn't look like the PPT was made available. What we do have available is here: ocw.mit.edu/2020-vision. Best wishes on your studies!
6:04, why are there only 2 lines that meet not 3? Wouldn't each of the column vector represent a line, in which case the 3 vectors define the whole R^3?
did you get the answer to this? I have the same question.
@@harshitjuneja7768 I believe I understand it now. A single column vector represent a point not a line. A line has to be defined by 2 vectors, one indicating any point along the line, and the other the direction.
@@harshitjuneja7768
Imagine two vectors in space starting from zero, with a small angle between them, Like say the angle between your index and your middle finger. Now imagine a third vector that adds those two original vector, (it of course also starts from zero, its a simple addition). What youll find is that with these three vectors, you've lost a dimension (the third) and are operating on only two dimensions, i.e a plane. This is entirely because that third vector adds the first two.
I have another silly but probably effective example, that will help illustrate why the domain becomes a plane instead of remaining 3d. Let me know
Where can I find all your videos?
th-cam.com/users/MITsearch?query=Gilbert+Strang
@@mitocw Thanks
Ok I will come back after I finish calculus 1, 2, 3 & linear algebra class😢
You can just start to watch this series of linear algebra. You know he always makes it easy :)
정말 이렇게 늙고 싶다. 너무 멋지시다.
my undergrad linear algebra course used his text. he was a cult figure among math majors.
in a just world, and a civilized society, Gilbert Strang would be a household name, and cardi b would be unheard of..
cardi who?
@@ronaldjensen2948 amen!!
12:53 Do anyone have examples for ill-conditioned C and R?
Kind of confused
Same question! Don't know what 'difficult to deal with' means
Kinda feels sad realizing that Prof. Strang has grown so old :/
I think it's need to be more edited For visualization.."
You can find unbeatable visualization here - th-cam.com/video/uQhTuRlWMxw/w-d-xo.html
Legendary!!! Which presentation software does he use?
I’m sure he is just using Microsoft PowerPoint with a theme applied to the slides.
He is using this en.wikipedia.org/wiki/Beamer_(LaTeX)
LaTeX files were saved as PDF and placed into PowerPoint.
This is like yoga in linear algebra
My distracted ass just wanting to hook up a PS4 to that massive television
2020 vision. Clear vision. Good joke. 20/20
Thought this course was gonna be on factorising quadratics, expanding triple brackets, completing the square etc... then I read the channel name "MIT"... probably some university level stuff 0of
We have a full range of mathematics. Here is a list of what we have: ocw.mit.edu/courses/find-by-topic/#cat=mathematics. Best wishes on your studies!
LaTex!!!!!
8:09 ok
Ok this video is so helpful.
I did NOT know you can understand rref like that... Just wow.
Linear Algebra review day6
Bring back the chalk board
I do not understand slide five what so ever.
I don't like how the video keeps switching between a version of the slides with Dr. Strang and a version without him. Very distracting :(
What's the point of switching the views around if we can already clearly see the material from the screen Dr. Strang is using?
MIT please minimize unnecessarily context switches in future videos. It's mentally taxing for some of us.
Thanks!
Не, лекции Мануйлова лучше будут)
Jumping from live to slide, and back again, is a distraction.
Who disliked...?
Probably someone who is not precise with their cursor.
Respect. Chalk and board is better
Please just give him a black board with a chalk and watch the magic
I miss him working on the chalk board as well.
He's kind of confusing
Not a very clear explanation this time to know where everything starts and where everything ends.