6. Column Space and Nullspace

[@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.😮

[@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.

[@황현태-d9d]

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

[@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.

[@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

[@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.

[@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.

[@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.

[@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

[@Upgradezz]

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

[@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

[@吳建興-r5r]

Thanks!

[@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 🙂

[@faiskies_]

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

[@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

[@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.

[@Mohamed-zo6so]

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

[@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

[@ChristosChris3490]

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

[@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.

[@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"...

[@tensorbundle]

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

[@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!

[@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

[@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.

[@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.

[@ziliestarrive]

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

[@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... 🙏

[@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.

[@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.

[@ralphmazzarella9081]

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

[@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.

[@rudreshsingh307]

Finally those 5 lectures paid off..!!

[@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!

[@nikobaka5272]

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

[@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!

[@aznpiccplayer123]

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

[@sdavid1956]

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

[@HamizAhmed-uk4de]

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.

[@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.

[@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..

[@aleant]

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

[@Upgradezz]

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

[@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.

[@psk_1998]

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

[@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.

[@awesomeous20]

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

[@rileypotter1673]

Nice. How's the next decade treating you?

[@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.

[@Leonugent2012]

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

[@_HJ_K]

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

[@SilvanaBuilesG]

Congratulations to this great Professor! Bravo!!!

[@nautiyogi8386]

Teachers like this are born once in 400 years.

[@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

[@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

[@DANIELELIASVARELA]

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

[@georgesadler7830]

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

[@genidor]

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

[@meghamangla1983]

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

[@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!

[@monsieurbreakyourpc]

8:47 Gilbert Strong

[@tuna34joseph]

wawww great

[@bazzmx]

Teaching the Gainz-Jordan Linear Progression Method

[@SatyaKomatineni]

Beautiful lecture, this one, and the entire series.

[@maousam1389]

Glad to see the classroom with students:)

[@xploi]

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

[@emmahart7054]

"at least that was the artist's intent when he drew it" lol love these lectures

[@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.

[@NehadHirmiz]

Dr. Strang thank you for being such an amazing lecturer

[@shubhamtalks9718]

This man is an artist.

[@2222Soham]

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

[@zorroesclito5286]

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

[@indumathi5182]

You are the God of Linear Algebra

[@thenewdimension9832]

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

[@UncleBards2]

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

[@HAAH999]

This course is a mine of gold

[@NithinVasisth]

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

[@asaflevif]

jejej YOLO strang version!! :P

[@tulasijamun3234]

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

[@rajkamalingle9144]

@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

[@shubhamide]

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.

[@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 :)

[@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.

[@shahzaibmalik9948]

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

[@sakthimadhankumar3254]

Best Professor in the world ngl.

[@Oshanii]

"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

[@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.

[@LnX53]

he emphasizes everything so good so that even the most idiots can understand... Great man !!

[@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

[@ksrajavel]

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

[@karthiksukumaran85]

Started it for Machine Learning. started loving linear algebra.

[@nest99100]

3 days before finishing linear algebra... I finally understand linear algebra

[@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.

[@thehyphenator]

39:30 "That's OK, we only live so long, skip that proof" LOL

[@sleeponadeathbed]

Clear, concise, coherent so who disliked this

[@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.

[@kielmeister]

What a good prof. At my school I have all sorts of questions but my prof doesn't take them. I wouldn't even need to ask in this class,

[@SphereofTime]

7:40 Union und intersection of

[@akbeh]

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

[@g.t.werber4476]

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

[@KaosRichie]

i hate the way i get educated i could get alot smarter by just following these awesome lectures online i dont even have to ask anything

[@Gccalin]

Thank you, Professor Strang.

[@ilhamazad]

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

[@bipulmanipokhrel5749]

I have a little question though. In the union between the plane P and the line L, explained from 05:00. There are some cases in which the union can be a subspace. This case occurs if the L is a subspace of P, meaning if L lies in P and goes through the null vector (0,0,0). Correct me If I am mistaken!

[@yt.abhibhav]

No, You are correct but in mathematics every words means and if you go through that segment again you would see that prof here said will that "always" be a subspace.