The determinant | Essence of linear algebra, chapter 6

The determinant | Essence of linear algebra, chapter 6


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 comments

  1. Answer of the Question:

    suppose a,b are sides of a two squares A,B respectively, then
    det(AB) == det(A)*det(B) equals to (a * b) ^ 2 == a^2 * b^2

  2. Charles Dodgson (A.K.A. Lewis Carroll) was actually a mathematician and came up with a really easy way to calculate determinants.

  3. i always struggle a bit in maths, but i made an attempt at answering the quiz question at the end: det(M1*M2) will describe the "overall" factor by which the original area will have been scaled. But det(M1)*det(M2) is describing the "succesive" steps/factors by which the original area has been scaled and is so eventually describes the "overall" factor too!
    Is this a valid explanation? I know it's not one sentence but I hope the it makes sense

  4. This video pushed me to find another way to calculate determinants on 2×2 and 3×3 matrices, based purely on geometry.

    On 2×2:

    if a=(a1,a2), b=(b1,b2) and our matrix is:

    |a1 a2|
    |b1 b2|

    then:

    |Det| = square_root( (|a|*|b|)^2 – <a,b>^2) , whereas <a,b> is the dot product between a and b

    ——————————————————————————————————————————————————

    On 3×3:

    if a=(a1,a2,a3), b=(b1,b2,b3), c=(c1,c2,c2) and our matrix is:

    |a1 a2 a3|
    |b1 b2 b3|
    |c1 c2 c3|

    then:

    |Det| = square_root( (|a|*|b|)^2 – <a,b>^2)*<c,p>/|p|

    whereas p = j * ( 1 , A , (b1+b2*A)/(-b3) )

    whereas A = (a1*b3-a3*b1)/(a3*b2-a2*b3) , and j is any real number

    Let me know if you find this way easier than the typical 3×3 det algorithm

  5. 2:20 the squares on the right extend beyond the blob.

    Finally after many hours of flawless graphics, there is a minute mistake. So, he is human after all 🙂

  6. In high school, they just teach us numbers and operations on them
    It's a waste of time
    seriously, 5 minutes from 3Blue1Brown is better than the whole high school.

  7. A unit victor is length 1, So a 1×1 matrix is a box and the determinate of a 1×1 is 1, but the determinate of a 1×1 shear is an area greater than 1, as to of the sides of a shear have a length greater than 1. were have I gone wrong?

  8. After watching this, I start to believe there's some global conspiracy among linear algebra academic society. Seriously, I don't understand why my teacher and the authors of textbooks never mentioned this explanation of a determinant, when it is so simple and intuitive.

  9. Wow! I thought about the flipping stuff myself, and can tell if a flip occurs (i.e. if i-hat moves to the left of j-hat) just by a quick glance at the matrix (basically, it happens if the diagonal coordinates i2 and j1 are both strictly greater than at least one of i1 or j2)! See my comment a couple videos ago. Playing with these ideas really does help!

  10. To extrapolate determinants into larger contexts, requires the emotional-logic of Intuitionistic mathematics, which are about four times more complex, and capable of expressing networking systems logics. A recent Big Data meta analysis of evolution revealed the insight that metabolism actually follows how fast an organism can grow, and not the other way around. Hence, Boyle's law can extrapolated to express the modified Bayesian probabilities vanishing into indeterminacy that describe the brain, while Relativity has turned out to express the same mathematics as thermodynamics, making it possible to combine it with quantum mechanics, using the new discovery that the Golden Ratio is not entirely random, but expresses a multidimensional equation.

  11. I have taken numerous linear algebra courses, and have a degree in maths. I have NEVER heard determinant explained this way, ever. This video is BRILLIANT! Thank you!

  12. I'm a native russian speaker and I have no idea what's special if a person with russian accent will say the “parallelepiped” word. Btw its russian pronounce differs just in one letter and sounds like “parallelepeeped”.

  13. Hands down the best video for explaining the determinant of a matrix, I could not understand its meaning in university, no one explained this, but this answere it beautifully in 10min, thanks!!

  14. The word parallelepiped comes form the Greek "Παραλληλεπίπεδο". It's Greek to you but not to me. It means an object whose sides are parallel to each other

  15. How about linear differential operators like divergence, gradient etc ! Even I am a postgraduate student, I still don't understand what they represents.

  16. In One Sentence – "If you scale the sides of any rectangle twice, its area is same as if you are multiplying the areas of rectangles formed by individual scaling."

  17. 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.

  18. 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!

  19. 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?

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

    DETERMINATION

  21. 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!

  22. 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.

  23. 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.

  24. 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?

  25. 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.

  26. 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

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

  28. 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.

  29. 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.

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

  31. Love the video love the channel. I'm not in any classes or taking any exams. I'm learning Numpy so I can learn deep learning python and just because it's cool I"m a nerd and I never did this before. SO much to learn, you present well though, much more clearly than wikipedia on the subject, which was confusing but I also love that site. Sometimes you already have to have a good understanding to read certain pages to get a better understanding, and that was the case here. I also love your piano sounds.

  32. 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)

  33. @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..). 🙂

  34. 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?

  35. 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 🙂

  36. 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.

  37. 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.

  38. 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

  39. 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.

  40. 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?

  41. 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

  42. 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!

  43. 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.

  44. 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.

  45. 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.

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

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