The Big Picture of Linear Algebra 

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!

when he emphasized "M-I-T blackboard" i felt that flex. my professors only use regular blackboards.

When you didn't study for your final exam and you have 16 minutes left

best part at 04:40 "not very thick, is it? because it's just a line!" x'D

this lecture was in the null space of my brain but now it's in the column space

The nullspace is also known as the kernel, in case you see that somewhere.

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.

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.

Wow! I was using Strang's linear algebra book back in 1979 in college! Glad to see he's still going at it!

Wish I had seen this when I was taking Linear Algebra. Wonderful short lecture.

Maybe the people at MIT should invent a blackboard that can fit an infinite plane?

I love Gilbert Strang's commitment!

This man will always hold a special place in my heart

I love this Professor. He is amazing and I'm really grateful that he did this , and grateful that MIT hosts it. Thank you

Not quite what I expected from the title, but he is great! Watching the video(s) is very entertaining and informative. Thank you!

Best book I have on Linear Algebra is by Mr. Strang. Well worth the read!

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.

He seems like he is very straightforward and to the point, which is great. Jumps right into it and doesn't waste time.

from OCW 18.06, glad to see Professor Strang again.

Here is one of the best lecturers in the world.

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 Gilbert Strang.

He is a wonderful gentleman and a great prof.

this is whom I call a good teacher... brief, thorough and "BIG PICTURE" indeed

Excellent summary of those key topics in linear algebra. Also, thank you for the well written textbooks!

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.

Thank you, prof Strang. Really hope to have you in our school to give a lecture on Linear Algebra.

This lecture is a beautiful introduction to linear algebra.

Amazing professor and well explained content! "Oh just by beautifulness, general principles of elegance..." -Professor Strang talking about the left null space

havent done linear algebra in almost 7 years, this brought back a lot of the basics very fast

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 😂

Phenomenal lecture as usual from Professor Strang. Thank you sir for your contributions and enjoy your retirement!

This was amazing. Thank you for the clear and concise explanation!

Brilliant overview of at least half of intro to linear algebra (18.06).

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.

I love it! Wonderful people like you make me want to study at MIT!

Just watched it, mind wondered off at the end, but was just pleasant to look and hear. Thanks!:) Legend.

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.

15:51. No: thank YOU, Sir!

Must-see video for the rest of us, who were taught that linear algebra is about manipulating matrices.

Now I need someone to give me the big picture of the big picture so I can understand what he's talking about.

Wow, amazing lecture, incredible.

once again, your teaching is wonderful~thank you

I have Gil's latest book ("Linear Algebra and Learning From Data"). It's just like listening to him talk.

Thankiu very mocht Teacher Gilbert Strang for your excelent explanecion about of the Sistem’s Linear ecuations for any dimensional’s Spaces.

Beautiful explanation, so fun to watch, thank you professor Strang

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.

Wonderful, intuitive explanation.

His book on Linear Algebra is great fwiw

Very clear and direct. Good teacher.

Prof Strang just rock..

So much information in so few words :)

I attended his linear algebra class in the early 00s before the OCW. Such a legendary professor.

Great teacher!!! Explained it perfect!!

Officially mind blown with these patterns!!! How did I not observe any of those!! Need to improve observation skills

Watching it nearly two years after the exam. Well at least it's nice to refresh these stuffs

Great explanation as always! Thank you!

Good to get such short lectures to introduce such difficult topics to me

thank you for a clear vision on the big picture of linear algebra

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!

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.

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.

Thank you, professor!

Great and amazing prof. Thanks! so rewarding!

I love how messy his board gets as he talks, each lecture is a demonstration of high entropy.

Kindly raise the first board to give more descriptions for our better understanding of electrodynamic principle projections along perpendicular null space .

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

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.

thank you professor Strang

This is a very important lecture.

the null space, now available wherever vectors are sold.

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.

i began to find Linear Algebra an interesting Subject after watching this open course!

Knowledge and teaching are beautiful

Excellent summary of a subject I've long been curious about. Not sure if Professor Gilbert is kidding or not though.

I love this guy. Best math teacher ever

Wish i was given the opportunity to be taught by him!

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.

To the point explanation and I m also surprised with such recalling capacities of the things 🙏Thanks Sir!

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!

This lesson is a masterpiece 💜

The math he did at 4:00 blew my mind

Great Visualisation... ❤

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.

Thanks! This is really a great video :))

I love Gilbert Strang

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.

confusions untangled with simplicity. Thank you so much. You are a gift to humanity

At last. A clear picture. Thank you professor.

I love you ❤️ Keep the good stuff, precious you

Awesome lecture!

Made it all make sense.... The beauty.

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!

the legendary Gilbert Strang yeah...

I feel special when he winks at me. ;)

Every youtube video on linear algebra I make a connection to something new.