My god this is probably the best lecture I have ever witnessed. Her voice is so calm and clear and she writes so clean and perfect. There is no way one could not understand.
She explains everything in one's heart. You will learn in logical harmony, nothing remains open and this along with tenderness and beauty. Thank you very much.
When I took a course on linear algebra at this time in 1976, we never looked at any geometric representations. It's no wonder I was so confused and frustrated!. This revelation comes 38 years too late, but better late than never. While I watched this video, I could feel my brain light up dendrites and neurons.
This is a new better of way of education. .You have doubts you can even put comments peope from net will help . evne use facebook to ask and clear doubts. New world of internet ..All you need is intrest ,wish and desire and ofcourse a bit of brain
In 1968 I took a Linear Algebra course at Brandeis and didn't understand a word of it. It was all [ defn, theorem, proof} repeated endlessly. I didn't understand the questions, or my answers, even when they were marked correct, Absolutely a waste of time. Meaningless !
Brilliant and yet simple explanation. Thank you very much! It looks like the row picture represents more the algebraic aspect and the column picture more the geometric aspect in a system of linear equations.
Dear MIT, You are the best. Your videos are easy to watch. We can fast forward, rewind, mix the videos and the videos are still making sense. Thank you for all these Masterpieces!
To simplify the calculations setting the x=1 as illustrated in slide of timing 4:36/16.35, I recommend to set both x and y on coordinate axes to initiate the geometry solution process. 2x + y = 3 (1) x - 2y= -1 (2) For equation 1, Set x=0, we get point a1 of (0, 3) Set y=0, we get point b1 of (1.5,0) Connect points a1 and b1, we get the first straight line. For equation 2, Set x=0, we get point a2 of (0,0.5) Set y=0, we get point b2 of (-1,0) Connect points a2 and b2, we get the second straight line. With Intersection of the two(2) lines, we have the geometry solution point s of coordinate (1,1), which sits on both straight lines. I'd done the drawing on a Word page, however, I don't know how to up-load the image.
Thank you for a very clear lecture. It wasn't rushed and nothing extraneous. Plus you have a pleasing speaking voice. I used to teach calculus and I had a tendency to rush what I was covering. I learned from your stye well as content. Thank you again. I hope to find more of your lectures.
Handwriting does help in mathematics - as you appreciate. Far better to be able to keep a record of progress made when you may read what has been written - and not struggle in fathoming the almost incomprehensible mess that accumulates during the course of videos produced by people such as The Khan Academy.
Good explanation ....I know the concept discussed here. The best thing in the video is the simple and lucid way of explanation.... Generally Math teachers always try to prove their dominance over the subject in front of students..
between 3:24 and 3:3:36 , it is said 'let s find out what is row picture', 'first let s look at row picture'. But then at 3:38 it is said 'please review what a row picture is.' That is confusing. The rhetoric implies that a demonstration is upcoming. Yet, at 3:38, the main idea is that it is asked that self-directed review of the concept that is not yet seen be reviewed; while announced beforehand that a presentation of it is upcoming.
when I was taking high school algebra in the 90s, we did not have these terminologies such as row&column pictures. We have done only solving system of equations (elimination or substitution), finding the coordinates, plotting the equations in the graphing paper, looking for the coordinate at which the equations intersect in our graph as proof, and then move on to another problem. The concept of vector was not taught to us we only have coordinates.
i & j are bases vectors and x & y the lengths of vectors which actually scale i & j.there4, When doing vector approach it is not professional to use x and y axes. X and y are the lengths of the vectors not the axes system. Use i and j for axes.
Thanks Prof Linan. There's one point in the video you should have said "column picture" but you said "row picture" instead. It doesn't really matter but I'm just pointing it out. I hope you make more videos. =)
I think there is an error at 7:55. The column space shouldn't have the axis labeled with a variable as the column vector is all the same variable. Strang, for instance, uses a no-variable axis. Thoughts?
while looking in the row picture we take (2{of first equation},1{of second equation}) as x coefficients and (1{of second equation},-2{of second equation}) as y coefficients but when we try to look at column picture she just interchanged the axis while drawing the vectors ; she took (2{of first equation} as x & 1{of second equation} as y) where 1{of second equation} represents x axis in the row picture.
+rajupowers Not just motivation, but also tradition factors. A degree earned by physically going to a university and study is regarded as more valuable to employers than a degree you obtained online. But in the future, this might change as technology evolves.
I wish I had access to something like internet when I was a kid, maybe I'd have had quit going to school, which was almost in vain, talking about math. The public education system in Brazil has collapsed for years :(
wow I like her!! she is from Tsinghua University, which is the top university in China, and those who can be admitted into Tsinghua can only be extremely intelligent or normal intelligent but extremely hard-working! And her English is so good! I hope I can go to MIT and meet her!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
I just watched this and also Strang and I see how it works using the vector form but I don't get how the entries of vector 1 can be plotted in the x , y plane as "x" and "y" coordinates if the entries of v1 represent exclusively the coefficients of the x variable in the two equations of the original system ? Is it arbitrary what 2D orthogonal axes are used?
what I find confusing is that when you draw the column vector of x you refer to both the x and y axis without explanation. The same with respect to y. An explanation step is missing. .
This is all interesting and rather obvious if you’ve ever studied mathematics. The big missing point is why is it valuable and important? What is the purpose of understanding the row and column pictures? Where can that knowledge be applied?
It is very interesting, for all students to know the mathematic field, linear algebra, who is use in different domain of our scientific and technic activity,it is very easy to understand all courses when we really concentred our attention good luck
When doing vector approach please please please do not use x and y axes! X and y are the lengths of the vectors not the axes system. Use i and j for axes.
Wow, she set work just one minute in. Ok, paused it, let's see: unknown_vector * Matrix2x2(Vector2(2, 1), Vector2(1, 2)) = Vector2(3, 1) So, obviously we need the inverse matrix. So we need the reciprocal of the determinant: 1/2*2-1*1 = 1/3. Now we transform the matrix: Matrix2x2(Vector2(2, -1), Vector2(-1, 2)) Now multiply by the scalar to get the inverse matrix: Matrix2x2(Vector2(2/3, -1/3), Vector2(-1/3, 2/3)) I assume that if I multiply the two matrices I get the identity. Meh, can't be bothered to check. Now we multiply the vector with the matrix. Ugh, I get Vector2(5/3, -1/3). No way she chose that and not two ints. I messed up somewhere as usual.
Ok, I just did matrix * inv_matrix (can't be bothered to write it out) and yeah, I get the identity. But I don't get 1, 1... I must have got my original matrix wrong from the formula... Oh no, the second one is a minus not a plus. Damn it! It's so dark and tiny on my screen. Oh, I give up.
So what is the exact use of column picture? Row picture and solving system of equations is giving me solution. But for Column picture she is using the solution arrived @row picture. So why do we need column picture. Is it just for verifying the answer
The lectures are in a separate playlist: ruclips.net/p/PLE7DDD91010BC51F8. We recommend you visit the course on MIT OpenCourseWare to see all the videos in context at: ocw.mit.edu/18-06SCF11. Best wishes on your studies!
My god this is probably the best lecture I have ever witnessed. Her voice is so calm and clear and she writes so clean and perfect. There is no way one could not understand.
I GOT CONFUSED AT THE END
She explains everything in one's heart. You will learn in logical harmony, nothing remains open and this along with tenderness and beauty. Thank you very much.
I'M GOING TO BE SICK ...
@@markconley5730YOU NEED HELP..SEARCH PROFESSIONAL SUPPORT..
When I took a course on linear algebra at this time in 1976, we never looked at any geometric representations. It's no wonder I was so confused and frustrated!. This revelation comes 38 years too late, but better late than never. While I watched this video, I could feel my brain light up dendrites and neurons.
Way I look at it .. better late than never! :) I am taking this 33 years after completing engineering .. thanks to Machine Learning and AI!
This is a new better of way of education. .You have doubts you can even put comments peope from net will help . evne use facebook to ask and clear doubts. New world of internet ..All you need is intrest ,wish and desire and ofcourse a bit of brain
I'm in the same boat. The geometric view makes linear algebra so much simpler.
Visceral indeed!
In 1968 I took a Linear Algebra course at Brandeis and didn't understand a word of it. It was all [ defn, theorem, proof} repeated endlessly. I didn't understand the questions, or my answers, even when they were marked correct, Absolutely a waste of time. Meaningless !
Brilliant and yet simple explanation. Thank you very much!
It looks like the row picture represents more the algebraic aspect and the column picture more the geometric aspect in a system of linear equations.
Such a beautiful explanation, that comparison with simple equation on real numbers really made this whole idea very intuitive.
Dear MIT,
You are the best.
Your videos are easy to watch.
We can fast forward, rewind, mix the videos and the videos are still making sense.
Thank you for all these Masterpieces!
Professor Linan Chen thank you for a beautiful explanation on the Geometry of Linear Algebra.
You are totally fantastic Linan ! thanks for a very clear presentation. I wish you were my math teacher.
To simplify the calculations setting the x=1 as illustrated in slide of timing 4:36/16.35, I recommend to set both x and y on coordinate axes to initiate the geometry solution process.
2x + y = 3 (1)
x - 2y= -1 (2)
For equation 1,
Set x=0, we get point a1 of (0, 3)
Set y=0, we get point b1 of (1.5,0)
Connect points a1 and b1, we get the first straight line.
For equation 2,
Set x=0, we get point a2 of (0,0.5)
Set y=0, we get point b2 of (-1,0)
Connect points a2 and b2, we get the second straight line.
With Intersection of the two(2) lines, we have the geometry solution point s of coordinate (1,1), which sits on both straight lines.
I'd done the drawing on a Word page, however, I don't know how to up-load the image.
This is the most creative presentation on linear algebra that I have ever seen.
why did this make so much more sense than the main lecture of the course.....the ocw videos are like hey if your not a genius hang on.
This is the one of the best geometric explanation of matrices and linear equations 👏👏👏👍👍👍
Wow, her calm voice and simple language is amazing and it makes learning a pleasure.
I love you teacher.
From a student in Cambodia.
Thank you for a very clear lecture. It wasn't rushed and nothing extraneous. Plus you have a pleasing speaking voice. I used to teach calculus and I had a tendency to rush what I was covering. I learned from your stye well as content. Thank you again. I hope to find more of your lectures.
Great introduction to the subject. Very clear linear equation to vector addition connection.
Well done.
What a gorgeous handwriting :D
Handwriting does help in mathematics - as you appreciate. Far better to be able to keep a record of progress made when you may read what has been written - and not struggle in fathoming the almost incomprehensible mess that accumulates during the course of videos produced by people such as The Khan Academy.
Great Job coming up with a super clear example of the concepts with very little room for confusion.
Good explanation ....I know the concept discussed here. The best thing in the video is the simple and lucid way of explanation.... Generally Math teachers always try to prove their dominance over the subject in front of students..
Wow this is great, I'm so happy I found this channel!
You just solved my conundrum. Thanks. You are sooo helpful.
between 3:24 and 3:3:36 , it is said 'let s find out what is row picture', 'first let s look at row picture'. But then at 3:38 it is said 'please review what a row picture is.' That is confusing. The rhetoric implies that a demonstration is upcoming. Yet, at 3:38, the main idea is that it is asked that self-directed review of the concept that is not yet seen be reviewed; while announced beforehand that a presentation of it is upcoming.
Linan Chen you are very good ! Thanks a lot !
when I was taking high school algebra in the 90s, we did not have these terminologies such as row&column pictures. We have done only solving system of equations (elimination or substitution), finding the coordinates, plotting the equations in the graphing paper, looking for the coordinate at which the equations intersect in our graph as proof, and then move on to another problem. The concept of vector was not taught to us we only have coordinates.
Column vectors forming a parallelogram by diagonalisation clearly indicating a piezo electric effect on any crystal by diagonalisation.
That was an excellent explanation. Your explanation was perfectly paced. Thank you.
i & j are bases vectors and x & y the lengths of vectors which actually scale i & j.there4, When doing vector approach it is not professional to use x and y axes. X and y are the lengths of the vectors not the axes system. Use i and j for axes.
Thanks Prof Linan. There's one point in the video you should have said "column picture" but you said "row picture" instead. It doesn't really matter but I'm just pointing it out. I hope you make more videos. =)
both ears get educated. double bonanza. thank you MIT
I think there is an error at 7:55. The column space shouldn't have the axis labeled with a variable as the column vector is all the same variable. Strang, for instance, uses a no-variable axis. Thoughts?
OMG! I love you! Thank you so much!
Finally seeing myself in the mirror of concepts. THANKS A LOT.
As stated in Avenue Q when the subject of going back to college was being discussed, this T.A. sure does spark my interest.
The row picture and column picture by intersection with x and y axis forming a triangle and the area can be found out by formula.
i like this. it is good to see math on the youtube
The best lecture I had on linear algebra. Thank u so much
这口音听着真舒服! Nice video!!
while looking in the row picture we take (2{of first equation},1{of second equation}) as x coefficients and (1{of second equation},-2{of second equation}) as y coefficients but when we try to look at column picture she just interchanged the axis while drawing the vectors ; she took (2{of first equation} as x & 1{of second equation} as y) where 1{of second equation} represents x axis in the row picture.
...yes I have the same confusion
oh, thank you, words in the blackboard are very beautiful and the lecture is very simple and clear, thank you, Professor😊😊😊
Column picture is an amazing understanding, I have never seen it in Russia
Thank you professor. This is so clear.
I am reading CLRS for three days,but couldn't understand anything but this video cleared the concept
start - @ 6:30
+rajupowers then 11:00
+rajupowers then 13:00
Short and direct! Tks Linan Chen! =)
Mam, you are a very good teacher.
shes is so sweet and a good teacher
She's a great teacher!
Beautifully explained
Esta clase es como las que me daba mi maestro en la preparatoria, en México.
thanks very much. it's very easy to understand the key point of linear algebra.
ha ha very nice lecture i saw it. and i love she's voice because it's clear.
Her voice is very soft and comforting. 😊 This is better than ASMR.
Now I really find it so easy 😃
(s-a) +(s-b )+ (s-c )and s=a+b+c/2
What a wonderful class. It makes me want to do maths problems.
No need to go to school, learning online is enough., take an exam for grading.
+Nikola Romanos it's basically the motivation factor. When you go to school, you spend money and time and so you try to get something out of it
+rajupowers Not just motivation, but also tradition factors. A degree earned by physically going to a university and study is regarded as more valuable to employers than a degree you obtained online. But in the future, this might change as technology evolves.
I wish I had access to something like internet when I was a kid, maybe I'd have had quit going to school, which was almost in vain, talking about math. The public education system in Brazil has collapsed for years :(
Exactly my thought
Rodrigo Abreu - ruclips.net/video/LczlGKG-Ae4/видео.html
Excellent presentation!
wow I like her!! she is from Tsinghua University, which is the top university in China, and those who can be admitted into Tsinghua can only be extremely intelligent or normal intelligent but extremely hard-working! And her English is so good!
I hope I can go to MIT and meet her!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Thanks for the wonderful explanation!
Happy new year😀🎉🎉
She teach very well, and she s so cure too
Very good and interesting lesson !
Thank you !!! 😊
Immensely well taught!
More than nice. Excelent presentation, Thanks.
Quality is surprisingly good for 360p.
Thank you very much for the lecture!!
this is a so cool lecture
thank you so much you were very clear and you gave such a wonderFul explanation I wish you were my math teacher. Please keep up the good work
Thanks.
고맙습니다
I just watched this and also Strang and I see how it works using the vector form but I don't get how the entries of vector 1 can be plotted in the x , y plane as "x" and "y" coordinates if the entries of v1 represent exclusively the coefficients of the x variable in the two equations of the original system ? Is it arbitrary what 2D orthogonal axes are used?
what I find confusing is that when you draw the column vector of x you refer to both the x and y axis without explanation. The same with respect to y. An explanation step is missing. .
This is all interesting and rather obvious if you’ve ever studied mathematics. The big missing point is why is it valuable and important? What is the purpose of understanding the row and column pictures? Where can that knowledge be applied?
Good Job Professor
Awesome teaching!
I like her handwriting
Bravo, excellent explaination.
muy bien explicado!
Mit always the best
fabuloso, gracias
Lecture was good but in the row pic she mentioned the intersection two lines makes values of unknowns easily but not in column picture
Jeeva Lion in column space, you can always projected b into the two columns in the column space, that will be the solution
I do not understand the relationship between x and y in the equations and the X- and Y-axis in the column picture. Is there one?
...same for me
Thanks alot I appreciate
Suddenly i enjoying linear algebra
y=ax +b.That is basic linear formula.Caslculating what is known and what not we can demistified that formula.
I like her more than linear algebra
simp
You're not supposed to!
It is very interesting, for all students to know the mathematic field, linear algebra, who is use in different domain of our scientific and technic activity,it is very easy to understand all courses when we really concentred our attention good luck
16:27 So A inverse is
| 1 -2 |
| 2 1 |
?
and you multiply it by
| 3 |
|-1 |
?
Correct?
sorry :)
+TheiLame That's the transpose, not the inverse.
+Michael Bear That's the transpose, not the inverse.
Thank you
Thank You!
MIT OCW is awesome!
When doing vector approach please please please do not use x and y axes! X and y are the lengths of the vectors not the axes system. Use i and j for axes.
Right. i & j are bases vectors and x & y the lengths of vectors actually scale i & j
great info. thank you.
Wow, she set work just one minute in. Ok, paused it, let's see:
unknown_vector * Matrix2x2(Vector2(2, 1), Vector2(1, 2)) = Vector2(3, 1)
So, obviously we need the inverse matrix. So we need the reciprocal of the determinant:
1/2*2-1*1 = 1/3.
Now we transform the matrix:
Matrix2x2(Vector2(2, -1), Vector2(-1, 2))
Now multiply by the scalar to get the inverse matrix:
Matrix2x2(Vector2(2/3, -1/3), Vector2(-1/3, 2/3))
I assume that if I multiply the two matrices I get the identity. Meh, can't be bothered to check.
Now we multiply the vector with the matrix.
Ugh, I get Vector2(5/3, -1/3). No way she chose that and not two ints.
I messed up somewhere as usual.
Oh, the vector was Vector2(3, -1)... Not 1. It's hard to see on my phone. Ok, amen.
Ok, I just did matrix * inv_matrix (can't be bothered to write it out) and yeah, I get the identity. But I don't get 1, 1...
I must have got my original matrix wrong from the formula... Oh no, the second one is a minus not a plus. Damn it! It's so dark and tiny on my screen.
Oh, I give up.
Matrix2x2(Vector2(2, 1), Vector2(1, -2))
1/2*-2-1*1 = -1/5
Matrix2x2(Vector2(-2, -1), Vector2(-1, 2))
Matrix2x2(Vector2(2/5, -1/5), Vector2(-1/5, 2/5))
Vector2(7/5, -1)
Ok, now I really do give up.
one question tortured me an entire day. How can you make coefficients into vectors? Any underlying magic?
@AlphonsePride Thanks ..
Can someone explain why she labels the axes as x and y in the column picture?
I mean the mistake is covered up by the fact that [v1 v2] is symmetric but it is a mistake conceptually, no?
So what is the exact use of column picture? Row picture and solving system of equations is giving me solution. But for Column picture she is using the solution arrived @row picture. So why do we need column picture. Is it just for verifying the answer
Column picture part was good.
This refers to "the first lecture" but this is the first video in the playlist. Is there a video of the "first lecture" somewhere?
The lectures are in a separate playlist: ruclips.net/p/PLE7DDD91010BC51F8. We recommend you visit the course on MIT OpenCourseWare to see all the videos in context at: ocw.mit.edu/18-06SCF11. Best wishes on your studies!
Thanks!
Verry good lecture Thanks
thank you that was very helpful
Thankyou teacher