The determinant | Essence of linear algebra, chapter 6

By | January 23, 2020


Hello, hello again. So, moving forward I will be assuming you have a visual understanding
of linear transformations and how they’re represented with matrices the way I have been talking about in the last
few videos. If you think about a couple of these linear
transformations you might notice how some of them seem to
stretch space out while others squish it on in. One thing that turns out to be pretty useful
to understanding one of these transformations is to measure exactly how much it stretches
or squishes things. More specifically to measure the factor by which the given region
increases or decreases. For example look at the matrix with the columns 3, 0 and
0, 2 It scales i-hat by a factor of 3 and scales j-hat by a factor of 2 Now, if we focus our attention on the one
by one square whose bottom sits on i-hat and whose left
side sits on j-hat. After the transformation, this turns into
a 2 by 3 rectangle. Since this region started out with area 1,
and ended up with area 6 we can say the linear transformation has scaled
it’s area by a factor of 6. Compare that to a shear whose matrix has columns 1, 0 and 1, 1. Meaning, i-hat stays in place and j-hat moves
over to 1, 1. That same unit square determined by i-hat
and j-hat gets slanted and turned into a parallelogram. But, the area of that parallelogram is still
1 since it’s base and height each continue to
each have length 1. So, even though this transformation smushes
things about it seems to leave areas unchanged. At least, in the case of that one unit square. Actually though if you know how much the area of that one
single unit square changes it can tell you how any possible region in
space changes. For starters notice that whatever happens to one square
in the grid has to happen in any other square in the grid no matter the size. This follows from the fact that grid lines
remain parallel and evenly spaced. Then, any shape that is not a grid square can be approximated by grid squares really
well. With arbitrarily good approximations if you
use small enough grid squares. So, since the areas of all those tiny grid
squares are being scaled by some single amount the area of the blob as a whole will also be scaled also by that same single
amount. This very special scaling factor the factor by which a linear transformation
changes any area is called the determinant of that transformation. I’ll show how to compute the determinate of
a transformation using it’s matrix later on in the video but understanding what it is, trust me, much
more important than understanding the computation. For example the determinant of a transformation
would be 3 if that transformation increases the area
of the region by a factor of 3. The determinant of a transformation would
be 1/2 if it squishes down all areas by a factor
of 1/2. And, the determinant of a 2-D transformation
is 0 if it squishes all of space onto a line. Or, even onto a single point. Since then, the area of any region would become
0. That last example proved to be pretty important it means checking if the determinant of a
given matrix is 0 will give away if computing weather or not
the transformation associated with that matrix squishes everything into a smaller dimension. You will see in the next few videos why this is even a useful thing to think about. But for now, I just want to lay down all of
the visual intuition which, in and of itself, is a beautiful thing
to think about. Ok, I need to confess that what I’ve said
so far is not quite right. The full concept of the determinant allows
for negative values. But, what would scaling an area by a negative
amount even mean? This has to do with the idea of orientation. For example notice how this transformation gives the sensation of flipping space over. If you were thinking of 2-D space as a sheet
of paper a transformation like that one seems to turn
over that sheet onto the other side. Any transformations that do this are said
to “invert the orientation of space.” Another way to think about it is in terms
of i-hat and j-hat. Notice that in their starting positions, j-hat
is to the left of i-hat. If, after a transformation, j-hat is now on
the right of i-hat the orientation of space has been inverted. Whenever this happens whenever the orientation of space is inverted the determinant will be negative. The absolute value of the determinant though still tells you the factor by which areas
have been scaled. For example the matrix with columns 1, 1 and 2, -1 encodes a transformation that has determinant Ill just tell you -3. And what this means is that, space gets flipped over and areas are scaled by a factor of 3. So why would this idea of a negative area
scaling factor be a natural way to describe orientation flipping? Think about the seres of transformations you
get by slowly letting i-hat get closer and closer
to j-hat. As i-hat gets closer all the areas in space are getting squished
more and more meaning the determinant approaches 0. once i-hat lines up perfectly with j-hat, the determinant is 0. Then, if i-hat continues the way it was going doesn’t it kinda feel natural for the determinant
to keep decreasing into the negative numbers? So, that is the understanding of determinants
in 2 dimensions what do you think it should mean for 3 dimensions? It [determinant of 3×3 matrix] also tells
you how much a transformation scales things but this time it tells you how much volumes get scaled. Just as in 2 dimensions where this is easiest to think about by focusing
on one particular square with an area 1 and watching only what happens to it in 3 dimensions it helps to focus your attention on the specific 1 by 1 by 1 cube whose edges are resting on the basis vectors i-hat, j-hat, and k-hat. After the transformation that cube might get warped into some kind
of slanty slanty cube this shape by the way has the best name ever parallelepiped. A name made even more delightful when your
professor has a nice thick Russian accent. Since this cube starts out with a volume of
1 and the determinant gives the factor by which
any volume is scaled you can think of the determinant as simply being the volume of that parallelepiped that the cube turns into. A determinate of 0 would mean that, all of space is squished
onto something with 0 volume meaning ether a flat plane, a line, or in
the most extreme case onto a single point. Those of you who watched chapter 2 will recognize this as meaning that the columns of the matrix are linearly
dependent. Can you see why? What about negative determinants? What should that mean for 3 dimensions? One way to describe orientation in 3-D is with the right hand rule. Point the forefinger of your right hand in the direction of i-hat stick out your middle finger in the direction
of j-hat and notice how when you point your thumb up it is in the direction of k-hat. If you can still do that after the transformation orientation has not changed and the determinant is positive. Otherwise if after the transformation it only makes
since to do that with your left hand orientation has been flipped and the determinant is negative. So if you haven’t seen it before you are probably wondering by now “How do you actually compute the determinant?” For a 2 by 2 matrix with entries a, b, c,
d the formula is (a * d) – (b * c). Here’s part of an intuition for where this
formula comes from lets say the terms b and c both happed to
be 0. Then the term a tells you how much i-hat is
stretched in the x direction and the term d tells you how much j-hat is stretched in the
y direction. So, since those other terms are 0 it should make sense that a * d gives the area of the rectangle that our favorite
unit square turns into. Kinda like the 3, 0, 0, 2 example from earlier. even if only one of b or c are 0 you’ll have a parallelogram with a base a and a height d. So, the area should still be a times d. Loosely speaking if both b and c are non-0 then that b * c term tells you how much this parallelogram is stretched or squished in the diagonal direction. For those of you hungry for a more precipice
description of this b * c term here’s a helpful diagram if you would like
to pause and ponder. Now if you feel like computing determinants
by hand is something that you need to know the only way to get it down is to just practice it with a few. There’s not really that much I can say or
animate that is going to drill in the computation. This is all tripply true for 3-rd dimensional
determinants. There is a formula [for that] and if you feel like that is something you
need to know you should practice with a few matrices or you know, go watch Sal Kahn work through
a few. Honestly though I don’t think those computations fall within
the essence of linear algebra but I definitely think that knowing what the
determinate represents falls within that essence. Here’s kind of a fun question to think about
before the next video if you multiply 2 matrices together the determinant of the resulting matrix is the same as the product of the determinants
of the original two matrices if you tried to justify this with numbers it would take a really long time but see if you can explain why this makes
sense in just one sentence. Next up I’ll be relating the idea of linear transformations
covered so far to one of the areas where linear algebra is
most useful linear systems of equations see ya then!

100 thoughts on “The determinant | Essence of linear algebra, chapter 6

  1. mademoiselle2181 Post author

    cannot be more curious about how prof Eliashberg pronounces 'parallelepiped'

    Reply
  2. Simon Tuomi Post author

    Nice introduction for linear algebra with "Hello hello again/LOL, oh again".

    Especially on a video about tranforming spaces which are described by the things on the space, which exist on the thing they describe what it is by doing their thing.
    At least from the feel i got from the previous vid.

    Reply
  3. M. Chavoshi Post author

    Wow! I am 24 and all my life I didn't know is the scale of area. How come it wasn't taught to me this way?!!!😐

    Reply
  4. Daniela Anaya Post author

    Muchas Gracias por compartir este video. Es super claro, y creo que son muy buenos los ejemplos.

    Reply
  5. → to the knee Post author

    I'm not understanding how these videos are like 10 minutes each. I think at first "cool, I have 10 minutes of awesome animations and intuition." And then, seemingly 30 seconds later, the outro music comes on and I feel like I've been swindled!

    Reply
  6. Louis Post author

    I've known for years what a determinant is, yet today I learned what a determinant IS. Wtf is wrong with my linear algebra teachers? Why did we learn so many abstract formulas and methods and not the meaning of all this?

    Reply
  7. Dudley Hunt Post author

    Knowing that someday the students might leave the university and understand the esscence of linear algebra fills you with

    DETERMINATION

    Reply
  8. Scaryder92 Post author

    As everybody else already pointed out, the quality of your work is outstanding and whoever has the possibility to support it should do it. A "thank you" in the comment section is not enough to repay for your effort!

    Reply
  9. vandan agrawal Post author

    Michael from Vsauce and you need to teach me in real life. You are the type of teacher who best fits me, dude. I am done with those teachers who when asked sth out of the syllabus, are either blank, give a completely wrong explanation cuz of confusion, or who simply say "Don't think so much on this topic. Just do this in the exam and you will fetch enough marks." These type of teachers kill a student's interest in the subject and make education a dull and sad factory process.

    Reply
  10. The Colonel Post author

    You would be shocked by the number of mathematicians who know HOW to figure out the determinant, but have no clue what it actually represents. Oh, BTW and am talking at the University level.

    Reply
  11. Carlos Post author

    The Right Hand Rule or like Dr. Marchini at the Univ. of Memphis calls it: Physics' Gang Sign…

    Reply
  12. Iliya Shofman Post author

    A random question: det(M1 *M2) = det(M1)*det(M2) = det(M2)*det(M1) = det(M2*M1).

    But matrix multiplication is anti-commutative. So how can A = M1*M2 and B = M2*M1 have the same determinant?

    Reply
  13. Dominik Pucuła Post author

    7:44 Anno Domini – Before Christ? That's a piece of beautiful math!

    Reply
  14. Victor Flavius Post author

    A lot of people said this didnot exist in their textbooks. That is true. Because intuitive understanding is often ignored by a lot of textbooks. Intuitive understanding usually describes the origin of the math problems. I believe the founder of linear algebra must have mentioned the content in this video.
    MATH IS A MODEL FOR THE REAL PHYSICAL WORLD AND IT IS USUALLY USED IN THE COMPUTATION OF PHYSICS. This is why most early mathematicians were also physicists.

    Reply
  15. Gas Post author

    3 years in a math degree and i finally understand what a determinant is x)

    Reply
  16. Blessing Hwacha Post author

    Am I the ONLY one who went on to Google Yakov Eliashberg..?

    Reply
  17. aditya vedam Post author

    The Actual Answer that I feel which can be in 'one sentence' MIND YOU can be only the following

    Look product of Two Matrices M1 and M2 will be always another Matrix whose determinant will a Number….which is the same as obtained from the product of their determinants which is just product of two numbers…..

    For e.g Let det(M1) be A and det (M2) be B then A.B will be always C which is a number and not a matrix

    hope this serves as a one sentence answer

    Reply
  18. Bana Post author

    Your videos are so informative and helpful, especially because I prefer understanding concepts of calculations instead of just using them! Thank you!

    Reply
  19. Tech Tins Post author

    I studied at Kingston University and Middlesborough, Honours Physics Degree and computer graphics MSc. Used linear algebra extensively. Lots of 3d Animations, inverse kinematics and other complex numerical problems solved. But only now have a really understood what a determinant represents. Crazy I know but there you go. Great video. Your good, write a book, I will buy it as I am sure many others would to.

    Reply
  20. Neko Salad Post author

    the answer to your question in one line(i hope its correct) :
    if determinal is a (1×1) matrix dedicated to only one (n x n) matrix then multiplying two (1×1) matrices equals the transformation of two (nxn) matrices. there you go one line.

    Reply
  21. Marcel Knezevic Post author

    Why do i pay thousands of dollars to go to university?

    Reply
  22. Matthew Siu Post author

    Every linear algebra course should require this series as a primer. Having this background makes things so much clearer.

    Reply
  23. Dung Nguyễn Post author

    so this is what it feels like to be taught something… havent felt it in a long long time

    Reply
  24. Praval Mishra Post author

    Some of the best works on youtube are on this channel. 😀
    Can you also create VR stuff for explaining 3D concepts in more detail (ofcourse for those who use VR)

    Reply
  25. THE Mithrandir09 Post author

    @3Blue1Brown I just love this video! In our courses we only learned the algorithm to calculate a determinant but never got taught what it meant. Could you maybe do a video on all properties of a matrix and provide intuitions like in this video(e.g. definite-ness, well formed(for inverse) etc..). 🙂

    Reply
  26. spelunkerd Post author

    Order matters when multiplying matrices, that is A*B does not always equal B*A. However, evidently when you multiply determinants of sequential matrices, the scalar result does not share that property, order no longer matters…. Why?

    Reply
  27. Gert-Jan Roodehal Post author

    Amazing video and explanation! Also, I;m very happy to see I'm not the only student only now figuring out what the determinant actually is 🙂

    Reply
  28. Michelle Holden Post author

    such a well-done video with a slight hint of humor and straightforward explanations

    Reply
  29. Evanescence_1981 Post author

    8:44 This… This is beautiful. This right here is beautiful. You can literally understand it in less than a minute and it's not mentioned ANYWHERE in the textbooks I've read. This is the answer to the question "WHY AD-CB?" that every student studying linear algebra has had. Truly beautiful, and helpful.

    Reply
  30. Ankit Sharma Post author

    Determinant of AB is equal to det. A into det. B
    One side we calculate a combined transformation and then calculate scale of transformation on other side we calculate scale of two different transformation and then multiply them.

    Reply
  31. aditya vedam Post author

    The One Sentence answer is the following..as per my opinion..
    He said that the determinant is just the factor by which a given matrix is scaled so by that definition..
    We arrive in one sentence that
    'The factor or the Number by which ..the product of two Matrices A and B is scaled is same as the product of scaled Matrix A and scaled Matrix B by the same factor or number viz..k where k is just a number..
    Hope this is the answer….
    Hit the Like button if you agree

    Reply
  32. Katrina Rogers Post author

    "warped into some kind of slanty-slanty cube" (6:07) greatest thing I've heard all day 😂

    Reply
  33. Clairvoyant81 Post author

    I always remember that someone told that the area of a parallelogram is the length of the cross-product of its sides.
    With that in mind, it becomes fairly easy to remember how to compute the determinant of a 2×2 matrix.
    If you know the 2×2 determinant, the 3×3 determinant is a matter of taking each scalar in the first row of the matrix and multiply it with the determinant of the remaining 2×2 matrix when you "strike out" the current scalars row and column. Add up the results and you're there.
    Much easier to remember than trying to keep the entire formula in your head.

    Reply
  34. Suseendran Padmanaban Post author

    If the rotation happened in 45 degree, the determinant area will half come down and half up. Then the determinant value is still 1, but it is positive or negative or zero?

    Reply
  35. Ran Shi Post author

    Your videos saved my time, why didn't my prof say exactly what you just said? Your explanation made much more sense, god bless you. — From a dying first-year engineering student

    Reply
  36. Jerry Liu Post author

    Does this definition of determinant only works on the standard basis? What about the bases besides standard one?

    Reply
  37. Tazaria Post author

    I disagree with you on the idea that actual computations aren't part of the essence of linear algebra. It's still mathematics and computing stuff is part of that. In the case of determinants it's probably the easiest to learn about matrix reduction first.
    Apart from that though, this series is brilliant!

    Reply
  38. hucker233 Post author

    It is funny, I learned determinants years ago and they were just taught as a mathematical property where bad things happened when they are zero and for some transforms it is nice when they are 1, but it was just a number. Now it is obvious what is going on. If I ever meet my college teachers again I will let them know they let me down.

    Reply
  39. Aayush Gupta Post author

    By far the best mathematics series I have ever watched. The how's and why's were never taught to us at college and this is just an eye opener. Thank you for these explanations.

    Reply
  40. Alomgir Miah Post author

    Really surprised by this intuition. Never thought of determinants this way. You are the best!!!

    Reply
  41. Elisabeth M.D. Post author

    Because Multiplication is distributive???
    I don’t know don’t judge

    Reply
  42. Andy Shen Post author

    I think the shortest sentence for solving the last question: Think M2 as a unit Matrix and det(M2) is the unit area after first transformation.

    Reply
  43. Γιάννης Ματθαιουδάκης Post author

    youre tone of voice and pacing… you are like the Obama of professors! cant thank you enough.

    Reply
  44. Aalap Shah Post author

    det(M1*M2) = det(M1)*det(M2)
    Multiplying two matrices is equivalent to applying two transformations? And the total determinant of the whole transformation (area or volume or higher dimensions) i.e. ratio of transformed objet to original must be equal to each transformation applied?
    I haven't seen any responses yet, just testing my understanding.

    Reply
  45. Aalap Shah Post author

    For a 3×3 matrice [1 2 3, 4 5 6, 7 8 9], when I find the determinant, it is Zero (in either direction). I am trying to visualize what that means. I think it means the volume collapses to Zero. Doesn't that mean the columns are dependent, i.e. at least two of (i + 4j + 7k), (2i + 5j + 8k) and (3i + 6j + 9k) are in exact same direction? But they aren't. Where did I go wrong?

    Reply
  46. Ea Dap Post author

    Imagine not going to university and studying STEM just with quality YouTube videos and books. Self-education + free resources (like YouTube, Khan Academy) + books are the future of education IMHO.

    Reply
  47. keerthan m kumar Post author

    While computing determinant what's the reason behind alternative positive and negative sign

    Reply
  48. alrayyaniQtr Post author

    College don’t care about your understanding. They just want you to put in the work. Minimize what you have to learn as little as possible. check the boxes. Get the grades. and boost the college average scores.

    The fact that most kids never knew what determinant meant is a testament to that

    Reply
  49. Cric168 Post author

    Thank you 3Blue1Brown , thank you very much .

    Here is what I learned at 10:02

    i mean one x axis unit
    j mean one y axis unit

    while in transformation
    a b
    c d
    a and c means your 1i trans into ai ( 1i change to size ai ) and cj ( up or down c units )
    b and d means your 1j trans into bi ( left or right b units ) and dj ( 1j change to size dj )

    so concluded
    a : x axis unit size = each 1 x unit resize to a units
    d : y axis unit size = each 1 y unit resize to d units
    c : x axis slope c/a ( 1 x unit, it is now a units, up or down c units )
    b : y axis slope b/d ( 1 y unit, it is now d units, left or right b units )

    hence about det[ a b c d ] = ad – bc area size ,
    that means 1 unit x y area trans into ad – bc area size

    here
    ad means area size factor ( a times in x axis and d times in y axis )
    b and c means slope factor ( b and c are told as above , but we have no idea about bc when talking about a 2D coordinate and but may has its meaning in other system )

    so why ad – bc ?

    the actual calculation is showed in this video at 10:02

    I would thought as
    ad means you resize 1 unit area into ad size ( a times in x axis and d times in y axis )
    -bc means ad area should reduce bc size( because the effect of x y rotated ) ,
    aftermath you got the real area

    and it is the determinant of [ a b c d ]

    Reply
  50. Prathamesh Majgaonkar Post author

    After completing this wonderful playlist, where do I go next?

    Reply
  51. isaac irigoyen Post author

    i dont know who you are but i love you, ive been struguling to pass linear algebra scince last year, and with 5 videos so far ive understood much more than on the hole semester of class

    Reply
  52. Sgurd Meal Post author

    Why do text books hide what the determinant really is? Or at best quickly say "yeah its how much areas are scaled I guess…". It hides the real understanding behind the determinant, which is ass backwards when you're trying to teach math.

    Reply
  53. Prathamesh Majgaonkar Post author

    6:44 The vectors are linearly dependent because the x and z coordinates for î and k̂ are the same. While they are 0,0 in case of ĵ. They all lie on the same plane. It takes 3 points to define one plane (axiom). Therefore, they are linearly dependent… I hope I am correct.

    Reply
  54. Icelick Post author

    That explains why you gotta multiply by the Jacobian determinant when using a change of variables on an integral.

    Reply
  55. Weitao Tang Post author

    I got A on my linear algebra course by memorizing the equations, however, I realized I had learned nothing after watching this video.

    Reply
  56. JWentu Post author

    wow. how did I spend so many many years without fully realizing this??

    Reply
  57. SaboTage Post author

    6:15 Параллелепипед !!!!!!!!!! Представьте, что я ваш учитель) imagine that I am your Professor)))))

    Reply
  58. Akash prajapati Post author

    Man you are……

    just awesome
    I was never taught to determinant like this And I don't the why too

    Really you are juuuuuuust awesome

    Reply
  59. Akash prajapati Post author

    I think you are taught to all these cool stuff by some one
    Or are you figured it out on your own.
    Whatever

    You are just awesome

    Reply
  60. dojinho Post author

    The diagram at 8:43 is just sublime. I'm a mathematician, and I had never heard this physical interpretation of the determinant. Only the definition in terms of how to compute it. Brilliant! 🙂

    Reply
  61. p gajbhiye Post author

    In one sentence..

    "In mathematics direct is equals to indirect"

    And mathematically same statement.

    Reply
  62. prum chhangsreng Post author

    Did I accidentally familiarize myself with determinant in a 10mn video?

    For ** sake, determinant always feel like some random junk cuz i cant visualize what it is. I only know it through mathematics abstraction and thing is. Ik i wud start to learn sth after watch ur video for a while and i end up learn so many thing

    Reply
  63. Abdul-Kareem Post author

    I have re-watched this video after 2 years, maybe. How nicely explained. I am meeting determinants for the third time in my life, but I had kind-of forgotten their behavior. Thanks again, Grant.

    Reply
  64. Emanuele Chierici Post author

    Det (M1M2)=det(M1)det(M2) Because it is a homomorphism or you might say linear application in english

    Reply
  65. Casskario Post author

    The first sentence blows my mind…
    Nobody ever said the determinant is the factor of how the given region changes… .

    Reply
  66. Михаил Галухин Post author

    6:14 – чёрт, how strange to hear parallelepiped in English -__-

    Reply
  67. Thyron Dexter Post author

    One sentence: The composition of transformations is changing areas exactly as much as applying the first transformation, then the other.

    Reply
  68. Harvey Hensley Post author

    9:00 "Go watch Sal Khan work through a few" :-}

    Reply
  69. Miguel Flagstaff Post author

    Public schools have failed and need to be shut down.

    Reply
  70. pounet2 Post author

    Good video (as always).
    For the justification of the computation of the determinant though I think you could have done a little more.

    You almost give the answer with
    det(AB)=det(A)det(B)
    which is easy to understand.

    It is clear that the determinant of a diagonal matrix is the product of the diagonal.
    Then you can apply Gauss algorithm to transform your matrix into a diagonal one (you don't really care to get a formula for not inversible matrices as you can change it just a little to make it inversible).

    Applying the Gauss algorithm to a matrix consists to multiplie it (on the left hand side) by simple matrices for which it it easy to compute the determinant.

    In my opinion, it is the best way to show how to compute the determinant. It illustrates three major ideas:
    1/ if you want to compute something complex, start with simple cases
    2/ Try to decompose the computation in simple steps to be able to reduce the computation to the trivial case.
    3/ Use of invariance, here
    det(A)=det(BA)/det(B)

    True, one of the problem is that you don't have presented the Gauss Algo at this stage… so I would present the computation of the determinant after the linear systems.

    Reply
  71. David Johnston Post author

    So there’s two sides to the sheet and the negative value corresponds to the other side?…

    Reply
  72. Jatin Sanghvi Post author

    You could have shown determinant and matrix multiplication for 1-D shapes too. Though it would be dumb in some sense, determinant of 1-D matrix is the number itself, or multiplying two 1-D matrices is same as a 1-D matrix containing product of numbers from individual matrices, but it could probably make things even more accessible for audience, just like the 3-D examples did.

    Reply
  73. Sean Ghaeli Post author

    It's hard to put into words how useful this is. No one dares mention where the formula comes from and to be honest it makes sense why the derivation (for me) wasn't trivial enough to figure it out right away. cheers.

    Reply
  74. mhill88ify Post author

    Jesus Christ Grant, I basically handed you the formula and then asked you to explain it to me if I don't pick it up…you the man.

    Reply
  75. arun karthick Post author

    I am a bit confused here. For example, if a transformation is something like [-1 0 0 -1], then it is flipping orientation, but its determinant is still +1. So, I am not sure whether determinant how determinant explains orientation. Sorry if I have misunderstood the concept of orientation.

    Reply

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