The best thing I like in his teaching style is that he picks everything from very basic. I have attended many lectures and classes, and I personally feel that people lack this; they make things more complicated unnecessarily. Kudos to MIT for this beautiful series. I remember one famous quotation by one of the best teachers that "Everything is simple and interesting if it is properly conveyed."
Lecture timeline Links Lecture 0:00 What are Vector spaces 1:05 Subspaces of R³ 2:33 Is the union of two subspaces a Subspace? 4:23 Column space 11:36 Features a Column space 14:46 How much smaller is the Column space? 15:48 Does every Ax=B have a solution for every B? 16:17 Which Bs allow the system of equations solved 19:39 Null space 28:12 Understand what's the point of a Vector space 40:24
I literally want to applaud after every lecture. If my linear algebra prof could communicate ideas this well, everyone in the course would definitely get an A.
Professor Strang is one of the best out there, you can have all the knowledge and skills for mathematics but some teachers, no matter how passionate or smart, are really bad. There is something about the way Professor Strang explains things which makes everything more understandable and interesting. I'm using Howard Anton's book at school + my teacher is THE WORST. Linear Algebra had been a nightmare up to the point where i found these videos. If i ever meet Professor Strang I'll hug him and wouldn't be able to thank him enough.
This teacher is amazing! Not just that he lightened me up with linear algebra but it made me really happy to see there are still people so passionate about their work. I just love it!
The fact that he is talking about spaces and somehow he is unable to manage tha space of the board is so very funny! I love this guy; the way he chooses his words is so proper, everything gets clear...Regards and respect from Greece mr Strang.
These students must be really spoiled for not clapping their hands after each of this brilliant mans lectures. I am even forced to do it sitting alone in my room! :)
This is really great and brilliant lecturer i never seen before. I like the methodology he is using and he knows how to engage his students. I can see now how linear algebra is applied. Thanks Gilbert Strang and MIT.
This teacher is excellent because you are able to follow along with the gears that are turning in his head. He actually reasons with you. I remember that the linear algebra teacher I had was hopeless and would merely bark canned lectures at you without a thought. Yeah, that guy wasn't a man of reason but a weight lifter, ex wood worker, simply there for a pay check.
Takes me back to my student days to experience that brilliance of the art of Mathematics! The way teaching was meant to be. The difference now being the "enjoyable aspect" of Professor Strang's obvious devotion to the subject. Brilliantly presented lectures on often abstruse aspects, with an inbuilt system of "creativity and innovation" for students. Surely remarkable.
he really is a fine teacher, mine just reads off from the book and i'am completely blank in the end. your lectures remind me the inspiration i had for choosing maths as my subject
I actually just took a formal linear algebra class at my university and it's crazy how the lectures are so similar. So I feel at least I'm getting a good education from my uni for a good price.
Terrific, terrific lecture, esp his way of using linear combination / "column picture" to solve equations. I have never heard of it, but it is so much easier! Thank you Prof. Strang/MIT for posting these lectures!
“Why don’t we learn all Linear Algebra in one lecture? - We just live so long ...” - Gilbert Strang is transmitting and implanting big ideas with Love. 🙏 Students are so lucky to be in his presence - of the real master. And we are lucky to watch it years later... 🙏
God bless you, just love each word comes out of his mouth. Very well explained. Spent a lot of time in books trying to understand the basic concept. The illustrations helped me to grasp the whole idea.
Linear algebra is used quite frequently in the real world. Especially when countless variables are being dealt with. Computer programs/software are great examples of this.
Timestamps 00:12 - Introduction to column space and null space 03:11 - Subspaces in R3 can be planes or lines containing the origin. 09:38 - When you take the intersection of two subspaces, you get a smaller subspace. 12:43 - Column space of a in R4 is a subspace by combining linear combinations of its columns 18:39 - Identifying vectors that allow the system to be solved 21:19 - Column space contains all combinations of the columns 26:53 - Column space is a two-dimensional subspace of R4 29:44 - Understanding null space and its properties in relation to column space 35:01 - The null space is a line in R3. 37:59 - Column space and null space are related through matrix multiplication. 43:24 - Subspaces have to go through the origin 45:58 - Column Space and Nullspace help understand systems of linear equations.
Thank u professor, what an excellent teacher u are..great, brilliantly conveyed every single notion of linear algebra in a lucent way, i feel fortunate to watch your lecture series which made me to love linear algebra and understand the concepts. I wish my teacher also should have watched your lectures once.
What are Vector spaces 1:05 Subspaces of R³ 2:33 Is the union of two subspaces a Subspace? 4:23 Column space 11:36 Features a Column space 14:46 How much smaller is the Column space? 15:48 Does every Ax = b have a solution for every b? 16:17 Which b's allow the system of equations solved 19:39 Null space 28:12 Understand what's the point of a Vector space 40:24
Terrific, terrific lecture, esp his way of using linear combination / "column picture" to solve equations. I have never heard of it, but it is so much easier!
HELP! I think Strang might have got WRONG around 05:40. I think P U L is a SUBSPACE of P as P & L itself is a subspace of P. Think like this: let p & l be a vector from P & L respectively. than u=p+l belongs to P U L and u lies within P as p is within P and l is also within P. Also c*p & c*l belongs to P & L respectively where c is scalar as P&L are subspace. so c*u=(c*p + c*l) belongs to P U L. Finally, zero vector lies in both P & L. so Zero vectors belongs to P U L. So P U L is subspace.
I came back to this after seeing a Domain/Codomain description of subspaces in row perspective/column perspective tied to the rank-nullity theorem. This is so much clearer than the introduction I had to this material. I would love to see his description of the link between row rank/col rank
Any subspace must contain vector (0,0,0), otherwise, if you do w*0 the answer would not belong to the subspace. If the plane doesn't go through the origin, it's not a subspace.
The question he actually meant/wanted to ask is the one he asked at 26:00 which is essentially the same as the one you pointed out and would equal your second interpretation. Because if all columns are INdependent the subspace would be 3D (in R4) but if the 3e column would be Dependent the subspace would be 2D (in R4) and the 3e column would just be a variation (linear combination) of the 1e and 2e columns. By the way, the only 4D subspace in R4 possible is R4 itself.
Leonardo di caprio's father once told him ," if you want to see a great actor look at Robert de niro" I tell you , if you want to see a great teacher look at professor Strang and remember his face
@SynthMelody The union is not a subspace. The union is bigger than P or L so it definitely can't be a subspace of either of them. and by adding a vector from P with a vector from U you can get to a point that is neither in P nor in U or in other words by adding two points from P∪L you can get to points outside of P∪L (somewhere in R³). But to form a subspace you have to be able to add any vectors from that subspace and the result has to be in that subspace.
@SynthMelody Maybe a simpler example helps. We take the X axis as one subspace and the Y axis as another subspace. So the union of those two spaces is all vectors on either axis, but nowhere else. For example (1 0 0) is on the X axis and (0 1 0) is on the Y axis. But the sum of the (1 1 0) is not on either of those lines. It's outside of it, so the union can't be a subspace, as otherwise you'd stay inside it when you add two vectors.
how are they plane , i think that [1 0 0] is a line starting from origin that goes to 1,0,0 and also it passes through origin so this should be a vector space.
At 17:30, he says we have four equations with three unknowns. How does a logic Ax = b may not have a solution flow from that? For three variables, three equations are enough. A fourth equation would appear dependent, or make a solution impossible. Is that the point? While b could have been any thing had there been four variables, but constrained when there are only three variables. Edited: I get it. b can not be any thing arbitrary. Unless it is in the column space of A, there can be no solution.
My teacher in our college just told us how to calculate Ax=b, but didn't explain the reason. How exactly does the teacher in the video understand linear algebra is. Conclusion that:1. Ax=b has solutions only when b is contained in column space of A. The solution may be multiple. 2. Consider all the x, which satisfy Ax=0, all the linear combination is called null space.
I like how he calls vectors or columns "this guy" and "that guy"
Now I know why prof. Philippe Rigollet in his stats class does it all the time :)
Wow! This comment is old. You may be having kids now.😮
0:00 ~ vector subspace
11:38 ~ column space
28:12~ null space
Thanks for ruining my anticipation for those topics
there is always an Asian making a favor for you
thank you for the splits it allows me to study them more efficiently.
와 같은 한국인👏👏
thanks homie
The best thing I like in his teaching style is that he picks everything from very basic. I have attended many lectures and classes, and I personally feel that people lack this; they make things more complicated unnecessarily. Kudos to MIT for this beautiful series. I remember one famous quotation by one of the best teachers that "Everything is simple and interesting if it is properly conveyed."
I love how he uses different ways of looking at the same thing to help drive concepts home (from the very first lecture). Strang is a fine teacher.
Lecture timeline Links
Lecture 0:00
What are Vector spaces 1:05
Subspaces of R³ 2:33
Is the union of two subspaces a Subspace? 4:23
Column space 11:36
Features a Column space 14:46
How much smaller is the Column space? 15:48
Does every Ax=B have a solution for every B? 16:17
Which Bs allow the system of equations solved 19:39
Null space 28:12
Understand what's the point of a Vector space 40:24
Xiè xie!
The Real MVP
Thanks
Thanks!
May God bless this teacher.These are those kind of people that makes you to love learning anything even if that thing might be boring.
if you learn linear algebra only from strang, it's not even boring
if u think linear algebra is boring, ur should get a brain.
I bet you didn't even make it 10 minutes through the lecture.
This guy taught me more than I learned when I studied Maths for four years.
those MIT blackboards are like Hogwarts...secret boards out of nowhere
see 8:45 for ref
HAHAHHAA lol very true
12 years ago...damn...you're probably dead
Getting. You. 🤣😳
@@enorfin1623 Hahaha
I literally want to applaud after every lecture. If my linear algebra prof could communicate ideas this well, everyone in the course would definitely get an A.
*This is 2019 and videos made 15 years ago, so what still top resource for linear algebra on the internet*
well its not like linear algebra changed within the last 15 years
@Mr. Rootes oh definitely.At least from the ones i've seen
@Mr. Rootes 3Blue1Brown Linear Algebra series is truly beatiful.You should watch them
If i meet him, maybe tears will roll down in admiration and inspiration. Such a great guy and excellent teacher. Thank you professor!
Professor Strang is one of the best out there, you can have all the knowledge and skills for mathematics but some teachers, no matter how passionate or smart, are really bad. There is something about the way Professor Strang explains things which makes everything more understandable and interesting.
I'm using Howard Anton's book at school + my teacher is THE WORST. Linear Algebra had been a nightmare up to the point where i found these videos.
If i ever meet Professor Strang I'll hug him and wouldn't be able to thank him enough.
gosh same, strang has saved my semester tbh
Write him email thanking him, he'll like it :) .
"we only live so long, we just skip that proof" -- Prof. Strang 2009
(and I low-key wish this was the case for all math tests )
2000* actually
The videos are 21 years old.
See copyright year
@@alice_in_wonderland42 But the description says "Spring 2005"
At 39:33 🙂
This teacher is amazing! Not just that he lightened me up with linear algebra but it made me really happy to see there are still people so passionate about their work. I just love it!
he gives you the right vision of mathematical concepts. and that's important for problem solving.
The fact that he is talking about spaces and somehow he is unable to manage tha space of the board is so very funny! I love this guy; the way he chooses his words is so proper, everything gets clear...Regards and respect from Greece mr Strang.
Thank you Mr. Strang. You are an excellent prof. Thanks MIT too
These students must be really spoiled for not clapping their hands after each of this brilliant mans lectures. I am even forced to do it sitting alone in my room! :)
+Edvin Moks Man that's really funny ;)
so true!
I was shouting out "No!" at one point to answer one of his questions. luckily no one else was in the room at that moment.
Forget clapping. I'd perform a 21 gun salute after each lecture.
+Dragon Curve Enthusiast: You're not the only one.
This is really great and brilliant lecturer i never seen before. I like the methodology he is using and he knows how to engage his students.
I can see now how linear algebra is applied.
Thanks Gilbert Strang and MIT.
"I shouldn't say absurdly simple, that was a dumb thing to say" - Gilbert Strang. This humility is what makes him an excellent teacher.
Linear Algebra was never as intuitive as Prof. Strang made it seem! Brilliant!
this professor is just amazing; I guess the guys attending are watching in absolute awe, melting in their seats, and that's why they remain in silence
they prob dont care who he is
He's a famous mathematician. Feeling privileged after watching his lectures.
This teacher is excellent because you are able to follow along with the gears that are turning in his head. He actually reasons with you. I remember that the linear algebra teacher I had was hopeless and would merely bark canned lectures at you without a thought. Yeah, that guy wasn't a man of reason but a weight lifter, ex wood worker, simply there for a pay check.
If I had a Linear Algebra professor like this back in the day I wouldn't have been studying it right now 10 years later "from scratch"...
Takes me back to my student days to experience that brilliance of the art of Mathematics! The way teaching was meant to be. The difference now being the "enjoyable aspect" of Professor Strang's obvious devotion to the subject. Brilliantly presented lectures on often abstruse aspects, with an inbuilt system of "creativity and innovation" for students. Surely remarkable.
he really is a fine teacher, mine just reads off from the book and i'am completely blank in the end.
your lectures remind me the inspiration i had for choosing maths as my subject
Great teacher. I'm 76. He makes a potentially abstruse subject simple.
Not only is he going faster than what the syllabus calls for, but he managed to do that without loosing me. Dr. Strang is very good at what he does.
I actually just took a formal linear algebra class at my university and it's crazy how the lectures are so similar. So I feel at least I'm getting a good education from my uni for a good price.
This is super intuitive. Much better than my lecturer who just writes down rigorous definitions and expect us to understand the concepts.
Terrific, terrific lecture, esp his way of using linear combination / "column picture" to solve equations. I have never heard of it, but it is so much easier! Thank you Prof. Strang/MIT for posting these lectures!
These are best lectures I have ever find in entire UA-cam.
“Why don’t we learn all Linear Algebra in one lecture? - We just live so long ...” - Gilbert Strang is transmitting and implanting big ideas with Love. 🙏 Students are so lucky to be in his presence - of the real master. And we are lucky to watch it years later... 🙏
I love this professor because really is a "teacher "in his soul. Deserves Respect and Appreciation...❤❤❤❤
Finally someone who can explain image/range/column space clearly!!!
loving this Gilbert guy! his 6 lectures has taught me more than 2 months of linear algebra at Chalmers University did! thumbs up and thank you MIT!
AMAZING Lecturer! Easy steps to follow and talks slow enough to understand. Thank you MIT!
God bless you, just love each word comes out of his mouth. Very well explained. Spent a lot of time in books trying to understand the basic concept. The illustrations helped me to grasp the whole idea.
The way Dear professor smiles at the end is so beautiful. I love you dear teacher. God bless you and :) . Love from Kashmir
Finally those 5 lectures paid off..!!
May god bless every seeker with a guru like him. Respect and good wishes from India..You are awesome sir..May you have a long life and good health..
Linear algebra is used quite frequently in the real world. Especially when countless variables are being dealt with. Computer programs/software are great examples of this.
Timestamps
00:12 - Introduction to column space and null space
03:11 - Subspaces in R3 can be planes or lines containing the origin.
09:38 - When you take the intersection of two subspaces, you get a smaller subspace.
12:43 - Column space of a in R4 is a subspace by combining linear combinations of its columns
18:39 - Identifying vectors that allow the system to be solved
21:19 - Column space contains all combinations of the columns
26:53 - Column space is a two-dimensional subspace of R4
29:44 - Understanding null space and its properties in relation to column space
35:01 - The null space is a line in R3.
37:59 - Column space and null space are related through matrix multiplication.
43:24 - Subspaces have to go through the origin
45:58 - Column Space and Nullspace help understand systems of linear equations.
He explained in 1 lecture what took my professor 3.... very good teacher
Nice. How's the next decade treating you?
8:47
Gilbert Strong
wawww great
Teaching the Gainz-Jordan Linear Progression Method
Even at old age he is razor sharp. I've seen old lecturers get confused; this man is extremely sharp. Great lecturing and teaching.
Thank u professor, what an excellent teacher u are..great, brilliantly conveyed every single notion of linear algebra in a lucent way, i feel fortunate to watch your lecture series which made me to love linear algebra and understand the concepts. I wish my teacher also should have watched your lectures once.
Congratulations to this great Professor! Bravo!!!
What are Vector spaces 1:05
Subspaces of R³ 2:33
Is the union of two subspaces a Subspace? 4:23
Column space 11:36
Features a Column space 14:46
How much smaller is the Column space? 15:48
Does every Ax = b have a solution for every b? 16:17
Which b's allow the system of equations solved 19:39
Null space 28:12
Understand what's the point of a Vector space 40:24
we've used it without proving it but that's okay we only llive so long, let's skip that proof. :))
+vedat kurtay its introduction to linear algebra, if you want proofs read his fourth edition of linear algebra and its application.
I just rephrased his saying buddy chill out :))
I need to prove to further understand the structure and mathematics. If you have the time, it's beneficial to some people
39:30
What a coincidence to see you here, hocam! Sevgiler, saygılar... -a student from your Tuesday PS :)
These lectures > TV series/movie . Be proud of yourself for watching these
Professor Strang might be the best teacher I’ve ever seen
I used to think calculus was more fun than linear algebra, I was wrong.
Calculus is much better my god linear algebra is sooo tough
The course is so good. Most of the time, Prof. Strang tells us why we do this instead of just how to do this.
These lectures are so old (but they are truly gold)
I guess some of the students back there have become professors themselves
Terrific, terrific lecture, esp his way of using linear combination / "column picture" to solve equations. I have never heard of it, but it is so much easier!
Professor u saved me.Thanks for your lectures. Our college teacher is the worst in teaching linear algebra.
HELP! I think Strang might have got WRONG around 05:40. I think P U L is a SUBSPACE of P as P & L itself is a subspace of P.
Think like this: let p & l be a vector from P & L respectively. than u=p+l belongs to
P U L and u lies within P as p is within P and l is also within P.
Also c*p & c*l belongs to P & L respectively where c is scalar as P&L are subspace. so c*u=(c*p + c*l) belongs to P U L.
Finally, zero vector lies in both P & L. so Zero vectors belongs to P U L.
So P U L is subspace.
Vector space feels a lot more interesting after this class...the column rank is dealt in a much brief manner out here though..
I came back to this after seeing a Domain/Codomain description of subspaces in row perspective/column perspective tied to the rank-nullity theorem. This is so much clearer than the introduction I had to this material. I would love to see his description of the link between row rank/col rank
This is another brilliant lecture on column and row space. These topics are very important in linear algebra for current and future learning.
Any subspace must contain vector (0,0,0), otherwise, if you do w*0 the answer would not belong to the subspace.
If the plane doesn't go through the origin, it's not a subspace.
My left ear feels lonely....
For others suffering with this issue : You can download and then play it using VLC after selecting Audio Channel as right. Will sound fine then.
sjs7007 q
You can also activate mono sound in your computer settings (windows).
@@loucololosse Thanks!!! EASY and USEFUL
My right ear feels more enlightened...
Beautiful lecture, this one, and the entire series.
Oh, how I wish my LA prof had been this good! Prof Strang is indeed a highly skilled teacher.
we only live so long...lol, amazing professor!!
jejej YOLO strang version!! :P
Dr. Strang thank you for being such an amazing lecturer
wtf this guy is a monster teaching one of the most abstract disciplines of engineering school
The question he actually meant/wanted to ask is the one he asked at 26:00 which is essentially the same as the one you pointed out and would equal your second interpretation. Because if all columns are INdependent the subspace would be 3D (in R4) but if the 3e column would be Dependent the subspace would be 2D (in R4) and the 3e column would just be a variation (linear combination) of the 1e and 2e columns. By the way, the only 4D subspace in R4 possible is R4 itself.
Leonardo di caprio's father once told him ," if you want to see a great actor look at Robert de niro"
I tell you , if you want to see a great teacher look at professor Strang and remember his face
Teachers like this are born once in 400 years.
Like a God of linear algebra so far I have seen ❤❤❤❤
* I can solve Ax=b for all b that is in the column space of A.
* Nullspace is all the vectors x that solve Ax=b where b=0
Glad to see the classroom with students:)
Finally the subtitles are on sync!!
This is great!!
Prof. Strang is amazing because he weighs his words...doeesn’t fill the leecture with combinations of thhe same ideas inn different words.
what a don! i wish my lecturer was this guy, he makes it so simple
You are the best professor ever....Great great I want to keep saying great
This man is an artist.
@SynthMelody The union is not a subspace. The union is bigger than P or L so it definitely can't be a subspace of either of them. and by adding a vector from P with a vector from U you can get to a point that is neither in P nor in U or in other words by adding two points from P∪L you can get to points outside of P∪L (somewhere in R³). But to form a subspace you have to be able to add any vectors from that subspace and the result has to be in that subspace.
I love how the left audio channel is reverb only.
17:10 "you can see from the way I am speaking what the answer is going to be..."
--wish my profs spoke like him...
@SynthMelody Maybe a simpler example helps. We take the X axis as one subspace and the Y axis as another subspace. So the union of those two spaces is all vectors on either axis, but nowhere else. For example (1 0 0) is on the X axis and (0 1 0) is on the Y axis. But the sum of the (1 1 0) is not on either of those lines. It's outside of it, so the union can't be a subspace, as otherwise you'd stay inside it when you add two vectors.
Best Professor in the world ngl.
"we only live so long, we can skip that proof"
wow maybe if my LA prof had the same outlook i would actually remember something from the classes
@43:40 In this case, it will be a plane (not line) that doesn't pass thru origin as [ 0 -1 1] and [1 0 0] are LI vectors
how are they plane , i think that [1 0 0] is a line starting from origin that goes to 1,0,0 and also it passes through origin so this should be a vector space.
Started it for Machine Learning. started loving linear algebra.
At 17:30, he says we have four equations with three unknowns. How does a logic Ax = b may not have a solution flow from that? For three variables, three equations are enough. A fourth equation would appear dependent, or make a solution impossible. Is that the point? While b could have been any thing had there been four variables, but constrained when there are only three variables.
Edited: I get it. b can not be any thing arbitrary. Unless it is in the column space of A, there can be no solution.
My teacher in our college just told us how to calculate Ax=b, but didn't explain the reason.
How exactly does the teacher in the video understand linear algebra is.
Conclusion that:1. Ax=b has solutions only when b is contained in column space of A. The solution may be multiple.
2. Consider all the x, which satisfy Ax=0, all the linear combination is called null space.
You are the God of Linear Algebra
"at least that was the artist's intent when he drew it" lol love these lectures
22:47 - "Think of a solution, then figure out what b turns out to be." I see what you did there, very clever.
he emphasizes everything so good so that even the most idiots can understand... Great man !!
This course is a mine of gold
39:30 LOL : "We only live so long, we just skip that proof."
Thank you very much for these lectures. They are very useful!
Best of bests
39:30 "That's OK, we only live so long, skip that proof" LOL
All those watching prof strang's lectures during quarantine, hit a like!