6. Column Space and Nullspace

MIT 18.06 Linear Algebra, Spring 2005
Instructor: Gilbert Strang
View the complete course: ocw.mit.edu/18-06S05
KZbin Playlist: • MIT 18.06 Linear Algeb...
6. Column Space and Nullspace
License: Creative Commons BY-NC-SA
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[@Heretic3030]

I like how he calls vectors or columns "this guy" and "that guy"

[@nickoleksyn3605]

Now I know why prof. Philippe Rigollet in his stats class does it all the time :)

[@yosansu]

Wow! This comment is old. You may be having kids now.😮

[@user-cz6dw8gv7n]

0:00 ~ vector subspace 11:38 ~ column space 28:12~ null space

[@Pentazoid111]

Thanks for ruining my anticipation for those topics

[@serden8804]

there is always an Asian making a favor for you

[@rogueshaman0911]

thank you for the splits it allows me to study them more efficiently.

[@ghsjgsjg53chjdkhjydhdkhfmh74]

와 같은 한국인👏👏

[@andrewy.8808]

thanks homie

[@MADLmaan]

those MIT blackboards are like Hogwarts...secret boards out of nowhere

[@enorfin1623]

see 8:45 for ref

[@akhilanr1233]

HAHAHHAA lol very true

[@genie9262]

12 years ago...damn...you're probably dead

[@TravelerVolkriin]

Getting. You. 🤣😳

[@jayvaibhawverma]

@@enorfin1623 Hahaha

[@ShoookeN]

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! :)

[@brentzhang6443]

+Edvin Moks Man that's really funny ;)

[@mrfchannel142]

so true!

[@dragoncurveenthusiast]

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.

[@atlantis_expedition_member4747]

Forget clapping. I'd perform a 21 gun salute after each lecture.

[@pruusnhanna4422]

+Dragon Curve Enthusiast: You're not the only one.

[@theindianscientist]

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."

[@pegasoos]

This guy taught me more than I learned when I studied Maths for four years.

[@dilnargheyret1465]

"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 )

[@alice_in_wonderland42]

2000* actually The videos are 21 years old. See copyright year

[@zynade9334]

@@alice_in_wonderland42 But the description says "Spring 2005"

[@surajpan8252]

At 39:33 🙂

[@raviiit6415]

*This is 2019 and videos made 15 years ago, so what still top resource for linear algebra on the internet*

[@DeadPool-jt1ci]

well its not like linear algebra changed within the last 15 years

[@DeadPool-jt1ci]

@Mr. Rootes oh definitely.At least from the ones i've seen

[@anonim5926]

@Mr. Rootes 3Blue1Brown Linear Algebra series is truly beatiful.You should watch them

[@solomonxie5157]

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

[@musicjunkie8228]

Xiè xie!

[@lq_12]

The Real MVP

[@aakashmalviya6231]

Thanks

[@user-go7gg6dd6v]

Thanks!

[@jurgenlekic1325]

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.

[@homoperse979]

if you learn linear algebra only from strang, it's not even boring

[@agrajyadav2951]

if u think linear algebra is boring, ur should get a brain.

[@PostSasso]

I bet you didn't even make it 10 minutes through the lecture.

[@belle060509]

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.

[@daniellek7536]

gosh same, strang has saved my semester tbh

[@Q.Mechanic]

Write him email thanking him, he'll like it :) .

[@dawsonb5699]

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.

[@nmx]

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.

[@Mohamed-zo6so]

he gives you the right vision of mathematical concepts. and that's important for problem solving.

[@faiskies_]

If i meet him, maybe tears will roll down in admiration and inspiration. Such a great guy and excellent teacher. Thank you professor!

[@ChristosChris3490]

Thank you Mr. Strang. You are an excellent prof. Thanks MIT too

[@Uditsinghparihar]

The world need more professors like you sir.

[@juanmanuelespinoza20]

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

[@wafs1393]

they prob dont care who he is

[@ishitajain965]

Linear Algebra was never as intuitive as Prof. Strang made it seem! Brilliant!

[@american-professor]

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"...

[@tzivastitziva]

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.

[@MrFili3333]

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.

[@georgiana1754]

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!

[@rudreshsingh307]

Finally those 5 lectures paid off..!!

[@ClaytonOT]

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.

[@akbarrahmatullah6701]

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.

[@tensorbundle]

He's a famous mathematician. Feeling privileged after watching his lectures.

[@rubabfatima3095]

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

[@monsieurbreakyourpc]

8:47 Gilbert Strong

[@tuna34joseph]

wawww great

[@bazzmx]

Teaching the Gainz-Jordan Linear Progression Method

[@stoneysauce]

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.

[@ralphmazzarella9081]

Great teacher. I'm 76. He makes a potentially abstruse subject simple.

[@Erik-jz9dk]

I used to think calculus was more fun than linear algebra, I was wrong.

[@sunritroykarmakar4406]

Calculus is much better my god linear algebra is sooo tough

[@Dagonemonkey]

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.

[@subash3]

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.

[@ziliestarrive]

This is super intuitive. Much better than my lecturer who just writes down rigorous definitions and expect us to understand the concepts.

[@jasonfranklin4614]

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

[@alexandramerlina]

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!

[@safiullahrahmani2807]

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.

[@sdcororaton]

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!

[@_HJ_K]

These lectures are so old (but they are truly gold) I guess some of the students back there have become professors themselves

[@vedatkurtay5488]

we've used it without proving it but that's okay we only llive so long, let's skip that proof. :))

[@facelessenemy3755]

+vedat kurtay its introduction to linear algebra, if you want proofs read his fourth edition of linear algebra and its application.

[@vedatkurtay5488]

I just rephrased his saying buddy chill out :))

[@Hwella555]

I need to prove to further understand the structure and mathematics. If you have the time, it's beneficial to some people

[@dalkomchoco]

39:30

[@MrAlpiz]

What a coincidence to see you here, hocam! Sevgiler, saygılar... -a student from your Tuesday PS :)

[@aleant]

AMAZING Lecturer! Easy steps to follow and talks slow enough to understand. Thank you MIT!

[@olgashatunova2385]

“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... 🙏

[@antonbanks8303]

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.

[@nikobaka5272]

These are best lectures I have ever find in entire KZbin.

[@NehadHirmiz]

Dr. Strang thank you for being such an amazing lecturer

[@sdavid1956]

I love this professor because really is a "teacher "in his soul. Deserves Respect and Appreciation...❤❤❤❤

[@niftkislay]

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..

[@georgesadler7830]

This is another brilliant lecture on column and row space. These topics are very important in linear algebra for current and future learning.

[@jayarani3416]

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.

[@SilvanaBuilesG]

Congratulations to this great Professor! Bravo!!!

[@Q.Mechanic]

The way Dear professor smiles at the end is so beautiful. I love you dear teacher. God bless you and :) . Love from Kashmir

[@2222Soham]

Vector space feels a lot more interesting after this class...the column rank is dealt in a much brief manner out here though..

[@aznpiccplayer123]

Finally someone who can explain image/range/column space clearly!!!

[@SatyaKomatineni]

Beautiful lecture, this one, and the entire series.

[@pubgplayer1720]

Even at old age he is razor sharp. I've seen old lecturers get confused; this man is extremely sharp. Great lecturing and teaching.

[@NithinVasisth]

we only live so long...lol, amazing professor!!

[@asaflevif]

jejej YOLO strang version!! :P

[@genidor]

Oh, how I wish my LA prof had been this good! Prof Strang is indeed a highly skilled teacher.

[@sdcororaton]

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!

[@awesomeous20]

He explained in 1 lecture what took my professor 3.... very good teacher

[@rileypotter1673]

Nice. How's the next decade treating you?

[@xploi]

Finally the subtitles are on sync!! This is great!!

[@maousam1389]

Glad to see the classroom with students:)

[@hj-core]

The course is so good. Most of the time, Prof. Strang tells us why we do this instead of just how to do this.

[@peterbonnema8913]

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.

[@calincojocaru3168]

Thank you, Professor Strang.

[@Leonugent2012]

Professor Strang might be the best teacher I’ve ever seen

[@ilhamazad]

All those watching prof strang's lectures during quarantine, hit a like!

[@g.t.werber4476]

Thank you very much for these lectures. They are very useful!

[@mounirkanane8083]

"I shouldn't say absurdly simple, that was a dumb thing to say" - Gilbert Strang. This humility is what makes him an excellent teacher.

[@tuha3524]

I love this lecture so much. Thank you a lot prof.

[@tulasijamun3234]

Prof. Strang is amazing because he weighs his words...doeesn’t fill the leecture with combinations of thhe same ideas inn different words.

[@psk_1998]

These lectures > TV series/movie . Be proud of yourself for watching these

[@HAAH999]

This course is a mine of gold

[@shubhamtalks9718]

This man is an artist.

[@meghamangla1983]

Professor u saved me.Thanks for your lectures. Our college teacher is the worst in teaching linear algebra.

[@shahzaibmalik9948]

You are the best professor ever....Great great I want to keep saying great

[@and1fer]

My left ear feels lonely....

[@sjs7007]

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.

[@hubert1946]

sjs7007 q

[@loucololosse]

You can also activate mono sound in your computer settings (windows).

[@kun8873]

@@loucololosse Thanks!!! EASY and USEFUL

[@nathanalgren288]

My right ear feels more enlightened...

[@pritishster]

Love this guy! Thank you so much!

[@he2he]

@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.

[@he2he]

@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.

[@falcord]

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.

[@UncleBards2]

what a don! i wish my lecturer was this guy, he makes it so simple

[@nautiyogi8386]

Teachers like this are born once in 400 years.

[@amaldave9601]

Wow, I love this guy :D

[@arteks2001]

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

[@kriszhli]

I love how the left audio channel is reverb only.

[@akbeh]

It helps a lot Polytechnique Montreal students, cuz we re using Gilbert Strang book translated and course

[@indumathi5182]

You are the God of Linear Algebra

[@lin1450]

Such a beatiful mind... Thank you so much!!!

[@karthiksukumaran85]

Started it for Machine Learning. started loving linear algebra.

[@mosesbasing7729]

Me too

[@yiyu9519]

love this course

[@zhaoc033]

Great lecture. I am so happy that i found this.:)

[@DANIELELIASVARELA]

wtf this guy is a monster teaching one of the most abstract disciplines of engineering school

[@Kadozier]

oh thank god for this first 18.06 test Friday

[@ksrajavel]

39:30 LOL : "We only live so long, we just skip that proof."

[@karenkordab5495]

all respect!king of algebra!!

[@thenewdimension9832]

Like a God of linear algebra so far I have seen ❤❤❤❤

[@bfl9075]

He made it easy to understand!

[@adrianroam95]

Great lecture!

[@avidreader100]

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.