You know, at first you start a new course, and it's kinda cool and chill, and you like it. And then, you go like *WAIT I BLINKED WWW..... WTF IS THIS THING???* *LINEAR ALGEBRA INTENSIFIES*

@acadoe

4 жыл бұрын

My thoughts exactly

@fruitbaskets7984

4 жыл бұрын

Lol... once he started about 3x3 matrix spaces....

@quirkyquester

3 жыл бұрын

i was lost for a while oops

@Marius-vw9hp

3 жыл бұрын

Thats when its nice to be able to rewind :)

Brilliant teaching, a gift to the world. Thank you prof. Strang and OCW

@mitocw

4 жыл бұрын

Thank you kindly

Watching this in 2019. Revising for Machine Learning. That guy is brilliant

@DeepakSingh-xt5io

5 жыл бұрын

same here

@lightspd714

5 жыл бұрын

Yep, same

@thangibleword6854

4 жыл бұрын

cemhier

@manujsharma1734

4 жыл бұрын

same here

@quantitativemethods8800

4 жыл бұрын

same here!

He's the Mr. Rogers of linear algebra.

I have an exam in like.. 5 hours and am exhausted, yet, as soon as I heard the joke about Clinton and Monica, I spat out my morning coffee and laughed so hard my entire dorm woke up. Thank you so much for not only teaching me everything I need to know, but also making it fun and making it stick!

We won't get a teacher like him possibly in our life time. 100 years fast forward people would be amazed to listen to his lectures and would marvel at the quality of education 100 years ago.

The joke at the end was likely very effective in making students interested and curious about the topic. What a genius!

Exam prep. lecture: At 1:00, he cover subspaces of 3x3 matrices, call it M. At 3:00, basis for M. At t = 15:00, basis for eigenspaces. At 24:00, he covers rank 4 matrices. At 29:00, feasible exam problem. At 39:00, graphs.

@AlexanderChirban

7 жыл бұрын

at 42:50, he drops a sick Monica Lewinsky joke

@lalithsharan4734

6 жыл бұрын

Oh, and also at 40:45, he invents LinkedIn

Do not get too excited, he is not wearing a different shirt, he is wearing a coat, you can see the over-loved shirt under it.

@renney77

4 жыл бұрын

Worth watching. My distance to Clinton is now three :D

@holden6059

5 жыл бұрын

Lol

@The5thBeatle2010

4 жыл бұрын

Then by watching a video of Bill Clinton your distance immediately drops to one

@0kBoomer

3 жыл бұрын

@@The5thBeatle2010 u must be fun at the parties

@Bridgelessalex

3 жыл бұрын

@@The5thBeatle2010 Very good counter-example!

@furkanatak4668

3 жыл бұрын

gilbert strang: my distance to Clinton is 2. random physics-student: well, that doesn't mean anything to me.

My right ear enjoyed this

@antoniolewis1016

7 жыл бұрын

It's meant to be listened to "right".

@younis24de

7 жыл бұрын

insert left nullspace joke here

on 31:09, the dimension of that subspace is 3, because there are 3 independent vectors: [-1 -1 1 1]', [-1 1 -1 1]', [-1 1 1 -1]'

I've been in love with Math my whole life. This course is so satisfying to me.

I recommend watching the final video in the 3B1B “essence of linear algebra” series before you watch this one

I can't explain in the words that how much your lectures have changed the way I study. I hope to see you one day and thank personally.

Excellent teacher. The little touches like reinforcing important concepts as he's moving on to new ones that make use of them is very helpful.

@NazriB

2 жыл бұрын

Lies again? Marine Soldier

I like Professor Strang's sweater. Reminds me of Mr. Rogers' Neighborhood.

Watched this lecture over and over, and every word starts making sense. Great for background knowledge of ML. Love this man so much!

@user-fh4xl3xz1f

Жыл бұрын

Watched these lectures in late 2019 before graduating. Now I am here 3 years later to freshen up my knowledge and to put all small building blocks in order. Precious material all of that.

I did not have 1 degree of separation from that woman.

@antoniolewis1016

7 жыл бұрын

Oh no he didn't alright. He had 0 degrees of separation.

@mikesmusicmeddlings1366

3 жыл бұрын

hahah nice

@Abhi-qi6wm

3 жыл бұрын

calm down jamal

30:00 R4 is being mapped to R1 (number of rows) so there are 3 (independent columns) dimensions collapsing to 0. They give us the 3 independent variables.

I love this man the way he teach and connects the subject with reality. .. Last 5 min... 💚

Love the Gilbert Strang ability to make linear algebra very illustrative and theorems so simple. One more time, thank you! )

oh I can't stop this class, I watch it all night.

Thank you so much MIT and Professor Strang!

great professor ... amazing teaching method and he clearly has developed an astonishing intuition over the concepts, Bit Thumbs up from me!

Watching this in Denmark for exam preb. Could just have seen this course first, an amazing job. Thank you very much!

This is a masterpiece by DR. Gilbert Strang of MIT. In control engineering, which is part of electrical engineering linear algebra is a must. All students in this field of study, graduate and undergraduate must know linear algebra.

i never go to school thanks to and get to have way better lecture thanks to mit and the professors there.

People strolling in late to one of the greatest linear algebra lectures ever given.

Prof Strang can be quite the comedian - that comment about Lewinsky was hilarious.

This spans all my Spaces. Thanks to @lexfridman for pointing me in this direction again 30+ years after first taking and passing the course but not understanding anything.

Thanks professor strang. Your lectures are amazing.

Thank you for making rank 1 matrices so clear!

That differential equation analogy was impeccable

My Monica distance is too small 😂😭 Professor Strang comin’ at us with all that humor towards the end was fun 🧡🙌🏽✨

regarding the mono audio issues, if you are on W10 you can search ''ease of access'' -> ''audio'' and turn on ''mono audio''. it will work properly that way. (remember to turn it off afterwards)

@Fake-rf4fp

Ай бұрын

ty

As a mathematician I'm wondering why it's not mentioned at all, that the "nullspace" is also called "kernel", ker(A) = nullspace(A), where A is a matrix. The column space of a matrix A on the other hand is also called "Image" of A. Good lecture style. Thanks.

@andresmath

2 жыл бұрын

Probably because at this point the emphasis isn't on viewing matrices as linear transformations, but as objects of special interest in their own right.

This guy really is a good teacher.

At 14:20 the dimension formula is more intuitive if stated as dim(S) + dim(U) - dim(S∩U) = dim(S+U) i.e. " the dimension of S added to the dimension of U minus the dimention of the interscetion of S and U is the dimension of S+U" ... so much easier to visualize stated that way n'est-ce pas?

was the great seeing the humor come alive in this one.

Yo Mit thank Gilbert on my behalf hes the best teacher i have gotten to learn from thus far 😊

This lecture is the hardest one in 18.06.

@danieljulian4676

5 жыл бұрын

What a relief that is to hear!

My right ear learned a lot from this lecture.

Thank goodness for browser extensions. Setting this video to Mono makes it much easier to listen to. :D

I Really Like The Video From Your Matrix Spaces; Rank 1; Small World Graphs.

What a great professor

Is it me or this is more addictive than GOT

@ohno3929

Жыл бұрын

It’s definetely you

Could somebody please explain to me what fast way they used to find the dimension of the subspace S (all v's in R^4 with v1+v2+v3+v4=0)? in 30:20

@androidlg7311

7 жыл бұрын

I believe it is because you can write -v1 = v2 + v3 + v4 therefore they are not linear independent. This describe the null space of a matrix with v1 as pivot and the other three vectors as free variables

@swapnils6902

5 жыл бұрын

v1 can always be described in terms of the other, so it's dependent on those three. However, the rest three are independent, and thus, can span a space. Hence, thr number of dimensions of the null space is three.

@chaokong5527

4 жыл бұрын

The subspace S is actually the nullspace of A=[1 1 1 1]. Therefore the dimension of S is the dimension of N(A). dim N(A)=n-r, and the rank of A is obviously 1 since it has only one row, therefore N(A)=3. You can also solve the equation Av=0. Since A is already in its rref form [I F], the special solutions to Av=0 are column vectors of [-F I]^T, which is [-1 1 0 0; -1 0 1 0; -1 0 0 1]^T.

@randomdude354

4 жыл бұрын

Think about it this way: Once you have values for 3 variables, the fourth is not "free". In a way, the fourth variable is subject to what the other variables' values are. So, there are only 3 independent variables. It doesn't matter if they equal 0, 1, e, pi, or 42. In a way, the equation itself gives you some information and makes one of the variables redundant. In some contexts, you'll have this concept of "degrees of freedom". This is what it means.

@cocoma982

8 ай бұрын

I have the same problem understanding this part, and thanks for the question you ask and the discussion here!

After 15 year In 2020 You are amazing boos......

@MrSnitsarenko

3 жыл бұрын

20 years. The lectures were recorded in 2000

Reply to a question below on why we care about the 4 fundamental subspaces in the study of Ax=b: In professor Strang words (not in this lecture) "understanding the four fundamental subspaces elevates the understanding of Ax=b to a vector space point of view, specifically, to to a vector subspace level of understanding" It will pay off when learning different types of matrix factorizations, for ex, Single Value decomposition, SVD, where A=U∑V^T. Where U is composed of orthonormal vectors in the row space, V of orthonormal vectors in the Column Space, ∑ by the diagonal matrix of the squares of the singular values. Similarly, all other matrix factorizations QR, Q(LAMBDA)Q^T, etc benefit from their description in terms of subspaces. This understanding/insight is super powerful, even critical, in Optimization,Machine Learning, ill posed problems, etc Plz don't disregard this Vector Subspace approach, it's only apparently that it doesn't seem to shine when being learnt for the first time, but believe me, it's pure gold, and the clarity with which Professor Strang explains it is invaluable ...he said it himself, it bends your mind when extended to sets of matrices, and to sets of functions (the differential equation example, a prelude to Fast Fourier Transform) but the dividends you gain are huge

@elyepes19

3 жыл бұрын

And watch the next video lecture

Thank you.......you've shared the worth lesson...

The best joke of all occurs at 19:00. “Five minutes of 18.06 is enough to take care of 18.03.” And it’s not a joke, provided you have a thorough understanding of 18.06.

You've had a new shirt professor :D.

Wish we all had a professor like him

watching this in 2016. What's hillary's distance to Monica. LOL

@justicewillcome4247

7 жыл бұрын

LOL. It's funny.(￣▽￣)"

@roronoa_d_law1075

6 жыл бұрын

Who is Monica ?

@jamesteow36

6 жыл бұрын

en.wikipedia.org/wiki/Monica_Lewinsky

@roronoa_d_law1075

6 жыл бұрын

ty

@kreglfromworld

6 жыл бұрын

monica is like a mexican who does the job that americans feel to good for

If the rank is only one, it can't get away from us.

What will be matrix A if I want to express y''+y=0 in the form Ay=0 ?

Why this got only 300k views = 30k views per year from all over the world. There are millions of engineering, math and science students that need linear algebra each year.

The rule for the dimensions of Symmetric and Upper Triangulars only apply to 3x3 matrices?

awesome lecture!!!

At 31:44 in order to find the dimension of subspace S, Prof Strang takes A as [1 1 1 1] of which S is null space. My question is, we could have taken A as 4 * 4 identity matrix for which S would have acted as null space. In that case rank(A) is 4 and hence no free variable thus its null space which is S has (4-4) 0 dimension So why are we restricted to take A as [1 1 1 1] in which case the dimension of S comes out to be 3 as rank(A) = 1 ?

@siddharthabura4808

2 жыл бұрын

Not I guess, Identity matrix * column vector(v1,v2,v3,v4) = column vector(v1,v2,v3,v4) but not zero column vector, hence, taking identity matrix as A doesn't work.

When professor Stang says that A = uV^T, I suspect that u is the pivot column and V^T is the pivot row. Am I right? But nonetheless, the lecture is fantastic as usual and these truly are a gift. Thanks a lot professor Strang and MIT.

@hits6620

2 жыл бұрын

i suppose you're right

@vaibhavchhajer300

2 жыл бұрын

Yes for a rank 1 matrix it's pivot column and pivot row but I think when you write a rank 4 matrix as a sum of rank 1 matrices than that rank 1 matrics does not come from the pivot rows and pivot columns of the rank 4 matrix.

what does the matrix space actually means? and how to understand the all R^{3 by 3) matrix?

love the ending...

Shreyas e^(-ix) = (c1) cos(x) + (c2) sin(x) when c1 = 1 and c2 = -i

@healthfreak5438

9 жыл бұрын

thanks, can you also describe how this second order differential equation can be written out as Ay=0 while solving for y?

At 19:03, 5 minutes of 18.06 is enough to take care of 18.03. That's mathematics prof trash talk for ya.

For nearly three quarters of this lecture I was totally lost about each single word of it.

@Thinkingoutloud-ek6xw

2 жыл бұрын

@glyn hodges hey man, could you please tell me how the dimension is 6 for symmetric and upper triangular. Not getting it

@Thinkingoutloud-ek6xw

2 жыл бұрын

@glyn hodges yep I see that now. Thanks a lot mate. It's quite interesting but quite complex at the same time

Dont know if its an error , maybe im wrong but at 14:00 prof strang says that dim( S intersect U ) + dim( S + U ) = 3 + 6 = 9 , but he prior to that he said dim( S + U ) = 9 so I got confused by this because now we have that 3 + 9 = 3 + 6

I have a question here, for me a point in 3d is zero dimensional, but if some living entity is smaller than that point some microscopic living thing, for it the point will have some dimension, so, is it so that the vector spaces that i am studying, I can visualise in my existing space, if someone is there in some different space they will do some other vector space?The solution will remain same in different spaces for a particular problem or example??

@elyepes19

3 жыл бұрын

Zero vector and zero point are abstract conventions, so for your thought experiment of small living beings, a point, and a zero vector, will also be smaller to them, and to anyonelse who adopts this abstract formalism

He's so sweet, funny and a bigger teacher

I have a doubt at 12:47 if I have a matrix in M which is [0,0,0; 0,0,0; 1,0,0] which is neither a part of Symmetric matrix or upper triangular matrix, then how come on doing S+U I get all 3x3 matrices?

@jacksonweekes

7 жыл бұрын

Its the sum- if you wanted to get the matrix you describe you could have (using simple basis) S=[0,0,1;0,0,0;1,0,0] and U=[0,0,1;0,0,0;0,0,0] and get M = S + -U

@Bridgelessalex

3 жыл бұрын

Recall that the scalars can be negative

For the subspace in r4 with all the components of a vector adding up to zero, how is that a subspace? What if I had the vector with components [-4,-6,7,3], those add up to zero but if I were to add up any member than this wouldn't be a subspace.

@prat7431

7 жыл бұрын

If you have two vectors such that the components of each vector add up to zero, then the components of the sum also add up to zero. For example, take [-4,-6,7,3] and [1,0,-1,0]. In both cases, components add up to zero. The sum of the two vectors is [-3,-6,6,3], the components of which also add up to zero.

@Nobody-Nowhere-Nothing

7 жыл бұрын

Pratham Bhat Thank you!

fascinating

Can anyone please explain how the dimension of symmetric matrix is 6? (at 6:43). I understood we have have 3 diagonal basis, but I do not understand how we get the other 3... [100;000;000], [000;010;000], [000;000;001] is all I can think of now... how the other basis can be a symmetric?

@hinmatth

7 жыл бұрын

a[100;000;000]+b[000;010;000]+c[000;000;001]+d[010;100;000]+e[001;000;100]+f[000;001;010] Together we have 6 :))

@dragoncurveenthusiast

7 жыл бұрын

well, in a symmetric 3x3 matrix there are 6 elements you can choose independently. You can think of it as choosing the three in the diagonal and three in the remaining upper triangle (meaning the upper triangle without the diagonal), because the 3 in the lower triangle then have to match the ones in the upper triangle (meaning the upper triangle without the diagonal). there are many sets of 6 matrices that form a basis for this space, but the most simple set would be: [100;000;000], [000;010;000], [000;000;001] (the ones you mentioned, defining the diagonal) [010;100;000], [001;000;100], [000;001;010] (filling the remaining upper triangle and the lower triangle in a symmetrical manner)

This guy is the Mr. Rogers of linear algebra.

Completely confused by 10:57. Can somebody explain what's the difference between union and his sum "+"?

@vijaybm9305

5 жыл бұрын

Union represents a matrix thats in either S or U whereas their sum can give a matrix thats not in either of them. You can check it by a simple example.

@Mike-mu3og

5 жыл бұрын

@@vijaybm9305 Got it, it's actually _really_ summing up all S and U combinations. Thanks

06:58 Anybody knows that why the dimension of symmetric matrix and upper triangular matrix are 6?

@kwanwookim1443

7 жыл бұрын

Thank you very much!! I like to clarify. 3 by 3 matrix has 9 components which are a11, a12, a13, a21, a22, a23, a31, a32, a33. Do you meanthat a12, a13, a21, a23, a31, a32 have 1 respectively?

@jankopiano577

7 жыл бұрын

In a symmetric matrix you have six independent values, 3 on the diagonal: a11, a22, a33, and 3 that have reflections: a12=a21, a13=a31, a23=a32. In upper triangular it is six because there are 6 values that can vary and the lower 3 are fixed to zero.

Monica joke is just awesome!!

Thank you sir.

Does anybody know what movie or book title he is talking about in the last minutes of the lecture at 43:47 ?

@turokg1578

Жыл бұрын

six degrees of seperation lol. he literally calls it out.

At 17:35, Shouldn't either of cos(x) or sin(x) should be the basis because either of sin or cos can be generated from the other. (like cos(x) = sin(π/2 - x)

@shiestynayak9711

2 жыл бұрын

if we have only two here like cos(x) and sin(x) you should be thinking of one being the multiple of other, if x is being used too then you are right , but we went for the general idea of one being some multiple of other without introducing x. which is not true without knowing the x

19:02 SAVAGE!!! lol

But adding an identity matrix plus upper triangular gives us again upper triangular so in this special case we obviously dont get all 3×3 matrices or 9 dimensions? Or am i missing smthing?

@suryavanshikartik

7 жыл бұрын

same doubt

@dennisjoseph4528

4 жыл бұрын

Not all symmetric matrices are identity matrices. There can even be matrices like [0,1,0;1,0,0;0,0,0] that exist in the symmetric subspace. The point is all linear combinations of symmetric and upper triangular matrices do give you every 3*3 matrix

is there some mistakes in 3+9=6? i think dimensions can't be added directly.

@dragoncurveenthusiast

7 жыл бұрын

It's 6+6 = 3+9 13:40 Though I think it makes more sense as 6+6-3 = 9 the sum of dimensions of the subspaces minus the dimension of their intersection is the dimension of their sum dim(s1) + dim(s2) - dim(s1∩s2) = dim(s1+s2) my intuition: the added dimensions of the subspaces minus the part you counted double, because it's in both subspaces, is the dimension of the sum of the subspaces.

The distance between Monica and Clinton is actually two nodes connected by zero edges. And the linear equations between hillary and monica is beyond the realm of mathematics.

facebook was built in 2004, this was recorded in 2005....i wonder if he knew this biggest application of graphs.....

Jumper game on point.

How can be the dimension of a 3*3 new vector space be 9, can anyone explain this to me ???

@kevinnejad1072

4 жыл бұрын

The space is combs of 3x3 matrices. You can have 9 3x3 matrices, thus 9 basis. Combs of those 9 basis fill up the space with 9 dimension.

@sushanthch4310

4 жыл бұрын

@@kevinnejad1072 Thank you very much , now I understand .

The jokes near the end: Hilarious.

How do we know that cos and sin are the only possible unique/independent solution to the differential equation @19:00

@danieljulian4676

5 жыл бұрын

If you really want to know, you'll probably have to learn something about complex numbers, variables, and functions. If you want to construct a different field, you might be able to show that there are yet different solutions, but you'd better make it so that we can have differential equations. It's a harder problem than you thought, isn't it?

At 30:00, Is it silly to consider the vector v as a matrix and to look for the rank ??? I get why the dimension of the subspace is 3, because the 4th component does not bring anything new to the space created by the first 3. BUT I cannot say why it would be wrong to take v as a matrix and say the rank is one !!!!! It seems I am mixing the dimension of the image created by a single vector, with the dimension of the image created by combination of all vectors v of S. If someone can help :)

@santiago8509

2 жыл бұрын

Yep. You mix dim(N(v)) with dim(v). Apart from that, your thoughts are right, like rank(v)=1 for any particular v.

why dimension of {0} is 0 ? I think it could be 1. Because there is a vector [0]. You can multiply it by c, it still keep answer to be zero. Any explain ?

@aliquis4460

4 жыл бұрын

Dimension of a subspace = the dimension of the space spanned by the basis vectors. So since the basis of {0} is [0], it's dimension is 0 ([0] takes up the point 0,0,0, and a point has dimension 0). [0] is not the same as [1] or [3], etc., I think. [0] is more like a way to say "this is a nothing-vector".

@qinglu6456

4 жыл бұрын

The dimension of a zero vector space {0} is defined to be zero because the zero vector space {0} does not have a basis. The only element in the zero vector space {0} is the zero vector [0], but it does not form a basis. As a basis, vectors in the basis should be linearly independent, that is, the only possible linear combination of them that equals zero vector is that their coefficients are all zero. However, since 1*[0]=[0], here 1 is not zero, so [0] is not a basis.

Why? Why talk about matrix space?

How come at 12:44 the sum of S and U is all 3×3 matrices?

@jimweiner2937

3 жыл бұрын

I didn't see it right away either. But then it hit me. Take any 3x3: [a b c; d e f; g h i]. But that's equal to [0 d g; d 0 h; g h 0] (symmetric) + [a (b-d) (c-g); 0 e (f-h); 0 0 i] (upper triangular). QED.

my right ear feels wholly educated

Amazing

Thank u mit

this guy is a superman in mathematics

Amazing lecture😂

Is this course usefull for machine learning ?