Part 2: The Big Picture of Linear Algebra
ฝัง
- เผยแพร่เมื่อ 4 พ.ค. 2020
- A Vision of Linear Algebra
Instructor: Gilbert Strang
View the complete course: ocw.mit.edu/2020-vision
TH-cam Playlist: • A Vision of Linear Alg...
Multiplication by A transforms the row space to the column space. Professor Strang then reveals the Big Picture of Linear Algebra where all four fundamental subspaces interact.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
For links to Professors Strang’s related courses on OCW, visit the Related Resources page on the full resource site: ocw.mit.edu/2020-vision.
In university i had an impression that linear algebra is sort of least interesting subject of maths, maybe not too complicated in general but certainly not cool. When MIT started OCW, ive overlooked LA videos for a long time. Eventually and by an accident i looked into a couple of those from first banch. What a surprise! My biggest respect to Gilbert. Back in 2014 i was in Boston for few days and three times was walking into MIT into his office to thank him in persons, yet alas, he was out of office. Thank you anyway for these great lessons. If someone from MIT reads this, please pass my thanks to Mr Strang (Russia)
4:48 "You can die happy now." -- Prof.Strang, Spring 2020
4:47 "There you go. You can die happy now".
😅😅
Well. X marks the spot. 😂 😆 Lol
hahhaha XD XD XD XD
@@brandomiranda6703 let’s not fuss over the chalk😎
I checked the comments as soon as I heard him say it lol
Thank you MIT OCW and Gilbert Strang for this series. I have already taken a linear algebra class, but this introductory summary is a great refresher and more detailed/concise than any others on youtube at the moment.
The grand sequel that we didn't deserve yet didn't know how much we needed it
Somehow it doesn't seem right to see him without a chalk in his hand
He's getting old now. I think anything that helps him teach easier in his condition is perfect enough.
This is just brilliant. So much knowledge is condensed into this 11mins video. Respect!
So cool perspective on LU!
A great thanks to professor strang! Amazing Guy!
Lol this is so much more intuitive and easy to remember what you want to do than those rules they teach you in school so you can compute stuff and they tell you've learned it
Thanks a lot !! Professor!
I would say the permutation part needs a little more explanation. But still, what a great lecture!
great! similar as kernel and image of homophrism
Orthogonal sounds cooler :-), I love him
Can someone explain what is the meaning of the arrows? What do they signify?
08:53 I see what you did there, Professor :)
Will there be more?
Thankyou
Is the correctnes of A=LU proved by induction?
Nice vid
I have never taken linear algebra before!
I can't explain why I find matrices so exciting.
Perhaps because they are somewhat of a mathematical representation of dimensional realities.
Sorr for being pedantic, but I am not sure I appreciate why A = CR is nicer...Is it mainly because it contains some basis for the column space in C and some basis for the row space in R? I failed to appreciate this new start! haha :(
Wait I’m lost now. It’s been years for me. What’s the null space again?
Didn't study very hard but I tend to think of them as of the kernels of linear mapping
It's the set of all the x values that you can put into a function to make it equal zero. It's like the linear algebra equivalent of solving an equation for y=0.
The set of vectors that get mapped to 0 by the matrix
Friendly and Wondrous
1:52 that sums up my issue with maths, I had to google orthogonal at the start and after seeing the definition I thought, "so perpendicular?" he went on to explain that was correct but it's this ostentatious philosophy that creates a barrier to high level maths, I'll use this word or that symbol because it looks or sounds cool... Maths is the art of obfuscating simple concepts with notation and symbology. People with no decades long foundation in the subject, building up knowledge of the symbols and there often multiple uses and meanings, have a hard time teaching themselves maths because of this.
I definitely sympathize with your frustration about math jargon. However, while I generally agree that when working over real vector spaces in 2 or 3 dimensions, the word "perpendicular" is best to use rather than "orthogonal," it gets a bit more difficult to picture those nice geometric 90 degree angles as you delve into higher dimensions (though that geometric intuition is still extremely useful).
However, using precise mathematical language, two vectors are orthogonal if their inner product equals zero. If you haven't seen the term "inner product" before, just think of it as a generalization of the dot product that you may have seen in precalculus or algebra class For example, when you pick two vectors in three dimensional space and take their dot product, if this dot product is zero then we say the two vectors are orthogonal. It can be seen geometrically that this is equivalent to the vectors being perpendicular.
When you go on to study more linear algebra, you will learn that there are other mathematical sets than 3 dimensional space that act like "vector spaces," and other notions of "inner product" rather than the dot product in 3D. These generalizations may seem difficult or abstract, but they actually extend linear algebra far past simply solving systems of equations.
The lesson here is that mathematics gets more and more precise in its language and definitions as you go deeper into the subject (for good reason). This can create great confusion when new topics are being motivated to students who really don't need to be familiar with the high level stuff going on behind the scenes. A high-school algebra teacher should probably just say "perpendicular" since their students are most likely going to still be hanging on to that geometric intuition to tackle new topics. A college professor in an intro linear algebra course should most definitely introduce the term "orthogonal" in class, because even if their students do not go on to study higher math, they will most likely see the term continue to pop up.
When I first learned LA I thought of perpendicularity based on my experience in euclidean 3d space, but after a while I saw orthogonality as meaning independent: just as two perpendicular vectors are independent with no components in common. Maths is more like a shorthand for complex logical ideas. Yes it can be confusing but it would be really hard to advance otherwise. Saying 'perpendicular' at the beginning might seem ok but in non-Euclidean spaces it becomes deceptive and obfuscating itself.
Thank you sir
Linear Algebra review day7
I can die happy now pepeSadge
Energizing! 11st
Upvote me if you still think writing on the board is better than this.
I used to watch Linear algebra and Diff Eq from Strang to pass the class.
I am confused, how can a vector X be orthogonal to n n-dimensional vectors at once? Isn't X only allowed to be the zero vector in this case?
Once X is orthogonal to an entire space, it is orthogonal to all vectors in that space. So, for example, a vector on axis z would be orthogonal to all 2-dimensional space.
I dont think this lecture was good, if you have a hard time understanding check the lecture with the same title he did six years ago it's much better
Thirteenth Comment , 😂🤭
It's somewhat annoying to see this called the "big picture". It's more like the small picture in a very rudimentary freshman-level understanding of linear algebra, which is much more than solving linear equations. IMHO you don't really have much idea about what linear algebra is all about until you realize there are many more functions of matrices than just A -> Inverse, for example exp(A) for a matrix A, sin(A), f(A) for f:R->R and A self-adjiont, f(A) for f:C->C analytic and A arbitrary, ect. Then there are functions of more than one matrix, which is much more complicated and interesting. If you want to see linear algebra in its most natural physical setting, see the course notes by John Preskill for Caltech Physics/CS 219 on quantum information and computation.
I’m doing good to follow the pointer. It’s bigger than quantum.
I can only assume that he gives extra credit for posting a positive comment below. He is just as bad as teaching this subject matter as his book is. Ive have some great professors before that break down stuff so that anyone can understand. No doubt Prof Strang is brilliant and knows his stuff. But when it comes to teaching it, I dont think he is as good. I learned nothing new from this video.
Sometimes you have to consider several teachers. Usually at the bottom line they will agree.
Second comment!
Third comment!
First comment!