It helps that the matrix is symmetric so that the eigenvalues are real and the eigenvectors are orthogonal. Not to knock, though. This is a beautiful demonstration. Generations of teachers have taught the determinant like it's just an arbitrary combination of numbers that somebody pulled out of thin air (to put it politely). The interpretation as a volume expansion is intuitive, and it also explains all those other interesting properties that the determinant has. For example, the det(A*B)=det(A)*det(B) - of course!. How about inverses? The inverse just gives you back the original unit cube, so det(inv(A))=1/det(A). And if A is singular? det(A)=0, so the cube gets squashed flat. So of course the singular matrix has no inverse, meaning that the squashed cube can't be reconstructed. Very cool :)
Yeah. I opted to make the video short and focus on intuitive arguments. I should have left a little more room for discussion, but maybe that's what the comment section is for?
@@LeiosLabs ok great ... I had actually figured out previously that a determinant of a two by two matrix was a surface .... But tell me when you say that this division NEW VOLUME divided with OLD VOLUME, then my question is : Is the old volume "1"? Thanks if you have time to answer me.. (I am studying linear algebra by myself)
@@TheNetkrot yes it pretty much is.BASICALLY IT DEPENDS ON THE BASIS VECTORS. Generally the standard basis vectors are unit vectors ( i cap,j cap,k cap).so the volume is 1
I always used to try to understand what I was doing during calculating the determinant in the class. Now I could understand what I was calculating. Thank you so much! I wish may I had the teacher like you who could make me feel these concepts in bones.
@LeiosOS same on my side. I just knew the déterminant of a 2*2 matrix would be the area of a parallelogram but I didn’t know it would be a ratio in higher dimensions. You have very interesting content
The point is just that you are taking a linear transformation of rank n, from a vector space of size n to itself, such that all the eigenvalues are real (and all eigenvectors have period 1) which means that the matrix representing the endomorphism is diagonalizable over R. Then the important property is that the determinant is an invariant and so it's the same considering the matrix of the endomorphism expressed with respect to the canonical base and with respect to one of the bases which "diagonalizes the matrix". Then you can finish knowing that the determinant of a diagonal matrix is the product of the elements on the diagonal (aka the eigenvalues). Just wanted to give an explanation on why it works, the video was great
I learnt much more in these three minutes than the entire semester class of linear algebra. It was really awesome and it gave me the feeling that I can see things instead of just solving mechanically
This is genuinely mind-blowing. I never truly understood what a determinant actually IS, I just took for granted that it somehow exists. Eye-Opening video. Thank you!
This is Eureka moment. Determinant, Eigenvector, and Eigenvalue. It's like after enjoying years of ham, bacon, and pork chops without knowing their relationship, one suddenly realizes they are all from parts of same animal. And this animal could give love and joy to the human as pet, and even a new life as heart valve. Great inspiration. Thanks.
Now Jacobian is a piece of cake. For coordinate tranformations, like the transformation from polar coordinates (r, φ) to Cartesian coordinates (x, y), the transformation does not change the volume but the unit, to keep the volume same, one needs scale factor known as Jacobian. And it is no surprise to know Jacobian is just a determinant.
After learning and using determinants, eigen values and eigen vectors for 5 years, finally understood what they mean!. this was some kind of enlightening moment for me, feels like now i have seen everything and know everything that i need to know lol. Thank you!!!
A note that might add to this video: aligning a volume V cube along eigenvectors, you get a scaled cuboid of volume V*det(M). Do arbitrary weird objects also scale in volume as det(M)? Yes: divide your object of volume V' into a large number of little cubes oriented along eigenvectors. After you perform your transformation, the little cubes will still be non-overlapping (assuming our matrix is full rank, that is, one-to-one), so you can just add their values to approximate the volume of the weird shape. As we increase the number of cubes, the volume before transformation goes to V', and after V'*det(M), just as expected.
I'm glad you liked it! A lot of time mathematical concepts are hidden behind some sort of cryptic formula or method when things could be explained much more intuitively.
Yes. That got me wondering. What about repeated eigenvalues, or singular matrices..? Intuition tells me that singular matrices will yield a line or point after the transformation, i.e. 0 volume. And does that also mean we are unable to get back our original cube since no inverse can be found? Hm I am not so sure about repeated eigenvalues because sometimes I could find enough eigenvectors but other times when I can't, I'll just add a 't' in front of it (when solving ODEs). And what does THAT mean geometrically? Interesting stuff! Could you shed some light or share some sources that would? Thanks!
Hey guys, this video is meant to give an intuitive definition of the determinant. There are oodles of way to calculate it and I kinda assume that people watching this video have done a determinant calculation before. There are a few notes in the description, but I needed this video for certain videos in the future, so it was definitely worth doing. How did you guys feel about the "More info" tags that popped up? Were they too much? I think it's a good way to cite previous videos, but if you guys have a better way to do it, let me know! Thanks for watching!
youv'e done a good work,and i appreciate what understanding i took from you thank you, i would be watching more of your videos later more info is fine as long as it doesnt bother, so i approve :)
I'm glad! I tried to make this one a way to understand the determinant using more physical arguments, which some people appreciated, while others did not.
The way I see it, there are enough purely algebraic explanations and proofs regarding the determinant. What is severely lacking are intuitive notions which help guide computation. I have heard of the connection between the change in volume and its effects on the determinant before, but these specific visuals(which must have taken a bit of work) helped cement the idea even further, especially looking at the transformation with regards to the eigenvector basis.
since -3 is isolated in its own row and column for the determinant you could have just taken the determinant of the matrix at the top left times its cofactor at the bottom right (-3), giving you -3(1*1-(2*2)) = 9 Originally I though you would use the cofactor trick since the matrix was so nicely set up for it but since you didn't I thought I'd mention it
Thank you so much for that I was strugling with it for a long time. Will you please make a video on Physical or Geomatrical meaning of trace of Matrix...
So in other words the determinant of a Matrix is the volume of the transformed unit cube in that matrix space 👍 After many years I finally get it )) And now I get how it's useful, like normalizating a matrix vectors by dividing the elements by determinant? Like we do with vectors x,y,z/length
The first question that pops in mind is - Aligning the unit cube along the eigenvectors..... wait...what? How do we even know that the eigenvectors are all perpendicular to each other??? Doesn't it completely depend on the physical transformation being applied as to what 3 vectors will turn out to be eigenvectors???? Like stretching a plastic cube that transforms to a new shape... To be able to apply this type of restricted transformation, you should explcitly mention that - we are applying a restriction on the transformation now to match the volume of a regular transformation (with rotation involved) on the same cube. Hope you understand my point. Bill Smyth has already clarified it the the matrix is symmetric "so that the eigenvectors are all already orthogonal" but if i'm asked to stretch a Cube A and then take a Cube B and transform it, strictly following the orthogonality, such that the end volume is same as the Stretched Cube A's volume, ofcourse the product of eigenvalues will be the volume. This video explains a geometric interpretation, but lives on an assumption and a restriction to get something which then becomes only obvious.
I understand that this one is a little hard to follow and will avoid this format in the future. The idea of this video was to describe how to calculate the determinant in a new way for those who have been doing the calculation their whole lives.
It means the cube was not only rotated and scaled, but also mirrored by the transformation. (The transformation transforms a right handed system of vectors to a left handed one and vice verca.) The change in volume is actually given by the absolute value of the determinant.
Think of the cube shown in the video is above the surface. A negative determinant would indicate a cube below the surface mirroring the one with the positive determinant.
Yeah, PCA is on my radar. I'll bump it up the list, but no promises as to when it will be out (these videos take a while to make even though they are short).
Didn't know before that Eigenvector and Eigenvalue have their names from the German language. We call them Eigenvektor and Eigenwert. "Eigen" means something like "its own", "Vektor" means vector and "Wert" means value.
its very fun and easy to prove this with a 2x2 matrix and two vectors u and v that will undergo a transformation. Just calculate absolute value of det(u,v) to find the old area, then calculate the new area: absolute value of det(T(u),T(v)). Then you will easily see after some algebra steps that this new area is equal to absolute value of det(A)*old area
The last determinant where he got a 9 right? It was all inside one matrix so what was the original dimension and what are the new dimensions of the cube?
does it mean that the determinant of a matrix (in dimension 3x3) tell us how much can we magnify another matrix (also 3x3 representing a cube) if we multiply the first one by the second??? If this is it, it´s astounding awesome!!!
I understand the criticism. This video is probably one of my more controversial ones because it is trying to give an intuitive description of the mathematics instead of showing the math, itself.
Hey, I'm glad this was helpful! I actually took this video down for a while because people were saying it was too complicated. I'm glad to hear other people find this discussion useful!
even the number of elements are 9 in that matrix xd...awesome vid i was curious about the reason behind determinants and what they are used for...this video made it so much easier to understand cuz my teacher just plays around with properties what i lacked is the reason to use them...but i am still curious those elements in the matrix what do they represent in terms of the cube?
What happens when you apply a Matrix Transformation whose det=0 to a unit cube? What will be the resultant cube look like? Is it that there will be infinite possible resultant cubes with infinite shapes?
Great explanation! While I get the idea the determinate is the factor we scale the original one, but I'm still wondering how can a square matrix and its transpose have the same determinant intuitively? I can check the formula det(A) = det(A^T) by induction for a square matrix A, but how to understand the intuition behind it? Thank you!
@@budasfeet Eigen vectors are not orthogonal to each other unless the A matrix is symmetric, which is the case here in this example. Second, linear independence of of two vectors (a) doesn't depend on them being orthogonal, (b) and can still span the entire 2D space without being orthogonal.
This may have some VERY limited use in finding how an area (2 dimensional matrix) or a volume (3 dimensional matrix) has been stretched from its "unity" base. But what about higher dimensional matrices..................better still.........what if you're not dealing with areas in the 2 dimensional domain (or 3 dimensional)? What does knowing the determinant do for you, in those cases?...................if anything?
This video describes ways in which we can understand the determinant. One way is the area of a transformed cube after application of a transformation matrix to each unit vector.
does this generalize? is the determinant of a 2x2 increase in area and whatever it's called for 4 dimensional objects. What if the object you are transforming is not a cube? but some other arbitrary shape. Does it still work?
Hi Leios I have basic question dont think its silly question. why we need matrix ? what is the application ? in schooling we were thought have to perform matrix operations but not the application
Machine learning uses Linear Algebra (matrices and vectors) extensively! and this is only to name one application which is already transforming the world!
if we take a number of vectors in a vector space, they (at least in 2D and 3D) form a shape. Well, we seek an answer to the question: "by how much the area is scaled"? Therefore, it makes sense to associate to every transformation a unique scaler value, denoted by det(T). Noting that Det: a number of vectors ---> scaler field.
![BOWKYLION - วิงวอน (ex-change) [Official MV]](http://i.ytimg.com/vi/cLdnZWDaO-w/mqdefault.jpg)
It helps that the matrix is symmetric so that the eigenvalues are real and the eigenvectors are orthogonal. Not to knock, though. This is a beautiful demonstration. Generations of teachers have taught the determinant like it's just an arbitrary combination of numbers that somebody pulled out of thin air (to put it politely). The interpretation as a volume expansion is intuitive, and it also explains all those other interesting properties that the determinant has. For example, the det(A*B)=det(A)*det(B) - of course!. How about inverses? The inverse just gives you back the original unit cube, so det(inv(A))=1/det(A). And if A is singular? det(A)=0, so the cube gets squashed flat. So of course the singular matrix has no inverse, meaning that the squashed cube can't be reconstructed. Very cool :)
Yeah. I opted to make the video short and focus on intuitive arguments. I should have left a little more room for discussion, but maybe that's what the comment section is for?
@@LeiosLabs ok great ... I had actually figured out previously that a determinant of a two by two matrix was a surface .... But tell me when you say that this division NEW VOLUME divided with OLD VOLUME, then my question is : Is the old volume "1"? Thanks if you have time to answer me.. (I am studying linear algebra by myself)
@@TheNetkrot yes it pretty much is.BASICALLY IT DEPENDS ON THE BASIS VECTORS.
Generally the standard basis vectors are unit vectors ( i cap,j cap,k cap).so the volume is 1
@@krishnasaikanigiri971 thanks for this
Yes.....
I always used to try to understand what I was doing during calculating the determinant in the class. Now I could understand what I was calculating. Thank you so much! I wish may I had the teacher like you who could make me feel these concepts in bones.
Yeah, that was the point of the video. I am glad it was helpful!
Your 3 minutes video just changed how I view matrix.
I'm glad it helped!
@LeiosOS same on my side. I just knew the déterminant of a 2*2 matrix would be the area of a parallelogram but I didn’t know it would be a ratio in higher dimensions. You have very interesting content
i have discovered we are living in a matrix, nothings real mate
Please tell me too
That's a pretty good movie.
The point is just that you are taking a linear transformation of rank n, from a vector space of size n to itself, such that all the eigenvalues are real (and all eigenvectors have period 1) which means that the matrix representing the endomorphism is diagonalizable over R. Then the important property is that the determinant is an invariant and so it's the same considering the matrix of the endomorphism expressed with respect to the canonical base and with respect to one of the bases which "diagonalizes the matrix". Then you can finish knowing that the determinant of a diagonal matrix is the product of the elements on the diagonal (aka the eigenvalues).
Just wanted to give an explanation on why it works, the video was great
I learnt much more in these three minutes than the entire semester class of linear algebra. It was really awesome and it gave me the feeling that I can see things instead of just solving mechanically
This is genuinely mind-blowing. I never truly understood what a determinant actually IS, I just took for granted that it somehow exists. Eye-Opening video. Thank you!
This is Eureka moment. Determinant, Eigenvector, and Eigenvalue.
It's like after enjoying years of ham, bacon, and pork chops without knowing their relationship, one suddenly realizes they are all from parts of same animal. And this animal could give love and joy to the human as pet, and even a new life as heart valve.
Great inspiration. Thanks.
Yeah! Honestly, I struggled with the same concepts until I looked into it. I'm glad it was helpful!
Now Jacobian is a piece of cake. For coordinate tranformations, like the transformation from polar coordinates (r, φ) to Cartesian coordinates (x, y), the transformation does not change the volume but the unit, to keep the volume same, one needs scale factor known as Jacobian. And it is no surprise to know Jacobian is just a determinant.
Good analogy sir, have a +1!
Ham and bacon come from the same animal? :o
Fled From Nowhere lol I didn’t know either
A lot of people don’t understand Mathematics because of lack of explanation like this!
teachers tell you to memorize the formula, legends explain the logic behind the formula
After learning and using determinants, eigen values and eigen vectors for 5 years, finally understood what they mean!. this was some kind of enlightening moment for me, feels like now i have seen everything and know everything that i need to know lol. Thank you!!!
A note that might add to this video: aligning a volume V cube along eigenvectors, you get a scaled cuboid of volume V*det(M). Do arbitrary weird objects also scale in volume as det(M)? Yes: divide your object of volume V' into a large number of little cubes oriented along eigenvectors. After you perform your transformation, the little cubes will still be non-overlapping (assuming our matrix is full rank, that is, one-to-one), so you can just add their values to approximate the volume of the weird shape. As we increase the number of cubes, the volume before transformation goes to V', and after V'*det(M), just as expected.
this is beautiful! I've taken linear algebra courses in college but there's so much meaning and intuition behind it that I've yet to discover!
I'm glad you liked it! A lot of time mathematical concepts are hidden behind some sort of cryptic formula or method when things could be explained much more intuitively.
Yes. That got me wondering. What about repeated eigenvalues, or singular matrices..? Intuition tells me that singular matrices will yield a line or point after the transformation, i.e. 0 volume. And does that also mean we are unable to get back our original cube since no inverse can be found? Hm I am not so sure about repeated eigenvalues because sometimes I could find enough eigenvectors but other times when I can't, I'll just add a 't' in front of it (when solving ODEs). And what does THAT mean geometrically? Interesting stuff! Could you shed some light or share some sources that would? Thanks!
Amazing video!! I never ever imagined determinants and eigenvectors this way... Thank you so much 👌👌
I'm glad it was useful!
This is the first time for me to be able to clearly and visually understand the relationship between determinants and eigenvalues
This is good that you give a clear concept with a reasonable reality based example... I really enjoying you:)
I just started leaning vectors and this makes me wanna scream but it does help me to understand in a way so thanks.
I'm glad it was somewhat useful. Sorry if it was a little complicated!
Finally I'm able to understand way more on what I'm working with on my linear algebra class. Thank you!
came here to understand determinants, now I also understand eigenvectors and values even more. Wow thanks
Hey guys, this video is meant to give an intuitive definition of the determinant. There are oodles of way to calculate it and I kinda assume that people watching this video have done a determinant calculation before. There are a few notes in the description, but I needed this video for certain videos in the future, so it was definitely worth doing.
How did you guys feel about the "More info" tags that popped up? Were they too much? I think it's a good way to cite previous videos, but if you guys have a better way to do it, let me know!
Thanks for watching!
Looks good! keep up the good work
youv'e done a good work,and i appreciate what understanding i took from you
thank you, i would be watching more of your videos later
more info is fine as long as it doesnt bother, so i approve :)
Amazing good ....very good...are you a mathematician ?
please make subtitles in Ukrainian
Thank you, this was amazing. You have taught me about something in minutes which I couldn't learn from hours of lectures.
A perfect toturial, a terrible background music. Instructer, a good lecture does not need music, because mathematics itself is a beauty.
Very well explained, and kudos for the visualization of the concept!
Thanks! I am glad you found it useful!
I find this algorithm for the computation very intuitive.
Ty
After so many years, i finally understand. Thank you very much
Your voice is similar to the welchlab tutor , he is absolutely amazing , Especially the way he opened the complex number world in my eye.
I love that guy! His videos are great!
This one video was enough for me to subscribe (after glancing at the other videos you have). Thanks a bunch!
I'm glad! I tried to make this one a way to understand the determinant using more physical arguments, which some people appreciated, while others did not.
The way I see it, there are enough purely algebraic explanations and proofs regarding the determinant. What is severely lacking are intuitive notions which help guide computation. I have heard of the connection between the change in volume and its effects on the determinant before, but these specific visuals(which must have taken a bit of work) helped cement the idea even further, especially looking at the transformation with regards to the eigenvector basis.
Thanks, made me link the determinant from the eigenvalue matrix with the determinant of the matrix !!
Here we are told to mug up that product of eigen values is the determinant of a square matrix. Thanks for telling why as well.
Well, done. Swift, concise, yet clear. *thumbs up
I'm glad you liked it!
Few words , much more understanding .
just amazing !
I'm glad it was useful!
Simply exceptional! This is the video i wanted to see!!
Glad you liked it!
since -3 is isolated in its own row and column for the determinant you could have just taken the determinant of the matrix at the top left times its cofactor at the bottom right (-3), giving you -3(1*1-(2*2)) = 9
Originally I though you would use the cofactor trick since the matrix was so nicely set up for it but since you didn't I thought I'd mention it
seriously such a beautiful video with good description
I'm glad you liked it!
yeah mind blowing videos u have,which made people like me curious
THANK YOU SO MUCH LEIOS , IT MADE MAKING REVISION OF MATRICIES AND EIGENVALUES MUCH MORE INTUATIVE AND ENJOYABLE !!! :) :) :)
Yeah, it's super cool!
Thank you so much for that I was strugling with it for a long time. Will you please make a video on Physical or Geomatrical meaning of trace of Matrix...
Thank you so much.your explanations are so beautiful.
What 4 years of engineering couldn't teach ... you did it in 2.51 minutes ❤
You helped me in getting sense of my high school matrix!
Thank you very much for your detailed explanation and the channel in general!
Iam not understanding it completely..but made me to realize there is much more to learn in linear algebraa...thank u very much sir.
This has answered SO many questions, thank you!
Thanks 😊😘
So in other words the determinant of a Matrix is the volume of the transformed unit cube in that matrix space 👍 After many years I finally get it )) And now I get how it's useful, like normalizating a matrix vectors by dividing the elements by determinant? Like we do with vectors x,y,z/length
we learned linear algebra for 1 semester and now i finally know what all of these things mean in 3 min.
I memorized the property that determinant is product of eigenvalues without knowing why, and this really explains it, Thank you!
Yeah! It's one of those things that's a little difficult to grasp intuitively!
incredible, I was wondering for so long what was the meaning of a det. Ty
really a great video, just changed the point of viewing matrix.
This video is really great! Thank you :D
This is so beautiful I wanna cry
I'm glad it was useful!
I was never taught this when we learned about determinants. We were only taught how to find one, not what it actually was.
Exactly. That was why I made the video
Thanks for the great explanation. 👍
please get rid of the background music
Phenomenal explanation
Well that was startlingly easy; why did nobody explain it that way in school? I'd have "got" matrices a lot quicker that way!
Yeah. It's an interpretation, but not the best interpretation in all cases.
determinant show what statistically like mean values ,deviation ,standard deviation, correlation coefficient etc
Ur a very unique teacher
The first question that pops in mind is - Aligning the unit cube along the eigenvectors..... wait...what? How do we even know that the eigenvectors are all perpendicular to each other??? Doesn't it completely depend on the physical transformation being applied as to what 3 vectors will turn out to be eigenvectors???? Like stretching a plastic cube that transforms to a new shape... To be able to apply this type of restricted transformation, you should explcitly mention that - we are applying a restriction on the transformation now to match the volume of a regular transformation (with rotation involved) on the same cube.
Hope you understand my point. Bill Smyth has already clarified it the the matrix is symmetric "so that the eigenvectors are all already orthogonal" but if i'm asked to stretch a Cube A and then take a Cube B and transform it, strictly following the orthogonality, such that the end volume is same as the Stretched Cube A's volume, ofcourse the product of eigenvalues will be the volume.
This video explains a geometric interpretation, but lives on an assumption and a restriction to get something which then becomes only obvious.
I'm not understanding anything.
I understand that this one is a little hard to follow and will avoid this format in the future. The idea of this video was to describe how to calculate the determinant in a new way for those who have been doing the calculation their whole lives.
Your video taught me more than my Linear Algebra class on this subject.
so nice and elegant!
How would you interpret negative determinants then? In this particular example.
Basically the cube moving in the other direction, if that makes sense.
It means the cube was not only rotated and scaled, but also mirrored by the transformation. (The transformation transforms a right handed system of vectors to a left handed one and vice verca.) The change in volume is actually given by the absolute value of the determinant.
Think of the cube shown in the video is above the surface. A negative determinant would indicate a cube below the surface mirroring the one with the positive determinant.
Sweet and simple
what is the transform that you have applied to get the new volume?
The determinant matrix. I used it as a transformation matrix.
Marvellous
Love these short videos! Subscribed, what sort of videos do you have coming up? I'd love something with regards to Principal Component Analysis?
Yeah, PCA is on my radar. I'll bump it up the list, but no promises as to when it will be out (these videos take a while to make even though they are short).
Didn't know before that Eigenvector and Eigenvalue have their names from the German language. We call them Eigenvektor and Eigenwert. "Eigen" means something like "its own", "Vektor" means vector and "Wert" means value.
Yeah, this is definitely a german thing.
its very fun and easy to prove this with a 2x2 matrix and two vectors u and v that will undergo a transformation. Just calculate absolute value of det(u,v) to find the old area, then calculate the new area: absolute value of det(T(u),T(v)). Then you will easily see after some algebra steps that this new area is equal to absolute value of det(A)*old area
tutorial was very helpful, thank you :)
I'm glad you liked it! I tried to keep it short and explain the determinant intuitively instead of going through the math.
I searched for "What is determinant of a matrix". Now i am left with more questions.
The last determinant where he got a 9 right? It was all inside one matrix so what was the original dimension and what are the new dimensions of the cube?
i think its relative to the identity matrix
Excellent Video!
Thanks, I'm glad you liked it!
Ur vdeo speaks volume 😄 thanks alot
very cool stuff, thanks for sharing!
Glad you liked it! =)
brilliant sir
What a POV changing video!!!"❤
does it mean that the determinant of a matrix (in dimension 3x3) tell us how much can we magnify another matrix (also 3x3 representing a cube) if we multiply the first one by the second??? If this is it, it´s astounding awesome!!!
Wasn't what I was looking for but mind blown anyways
I understand the criticism. This video is probably one of my more controversial ones because it is trying to give an intuitive description of the mathematics instead of showing the math, itself.
An amazing video. Thank you.
Glad you enjoyed it!
Now I have a intuitive sense of determent only 'cause of you thank you and God bless you
Hey, I'm glad this was helpful! I actually took this video down for a while because people were saying it was too complicated. I'm glad to hear other people find this discussion useful!
very good video!
Truly a genius !
As someone who has suffered from matices for years,and I mean yeeeeeeeears,,Thank you
but why do we compute the determinant of 3x3 matrix like that? is there any reason of hiding rows and columns and alternative + and -?
even the number of elements are 9 in that matrix xd...awesome vid i was curious about the reason behind determinants and what they are used for...this video made it so much easier to understand cuz my teacher just plays around with properties what i lacked is the reason to use them...but i am still curious those elements in the matrix what do they represent in terms of the cube?
very helpful video for matrix
What happens when you apply a Matrix Transformation whose det=0 to a unit cube? What will be the resultant cube look like? Is it that there will be infinite possible resultant cubes with infinite shapes?
Great explanation! While I get the idea the determinate is the factor we scale the original one, but I'm still wondering how can a square matrix and its transpose have the same determinant intuitively? I can check the formula det(A) = det(A^T) by induction for a square matrix A, but how to understand the intuition behind it? Thank you!
how could u align a cube in the direction of eigen vectors ? Are eigen vectors of any matrix are mutually orthogonal to each other ?
aravind gopal yes eigenvectors are basis spanning the eigenspace, they are linearly independent of one another thus orthogonal too.
@@budasfeet Eigen vectors are not orthogonal to each other unless the A matrix is symmetric, which is the case here in this example. Second, linear independence of of two vectors (a) doesn't depend on them being orthogonal, (b) and can still span the entire 2D space without being orthogonal.
This may have some VERY limited use in finding how an area (2 dimensional matrix) or a volume (3 dimensional matrix) has been stretched from its "unity" base.
But what about higher dimensional matrices..................better still.........what if you're not dealing with areas in the 2 dimensional domain (or 3 dimensional)?
What does knowing the determinant do for you, in those cases?...................if anything?
Wow thank you so much
The person who found determinant must be a ridiculously intelligent guy.
I love ur videos
I, m a bit confused can you please tell me why did you or how did you multiply the matrix with the cube
This video describes ways in which we can understand the determinant. One way is the area of a transformed cube after application of a transformation matrix to each unit vector.
Excellent!
I'm glad it was helpful!
So you're saying that determinant has a connection with eigenvalue and eigenvector? I might as well learn those 😃
i just knew how to find a determinant but never has someone told me why we find it..
does this generalize? is the determinant of a 2x2 increase in area and whatever it's called for 4 dimensional objects. What if the object you are transforming is not a cube? but some other arbitrary shape. Does it still work?
my friend im in serach of this Q -> Is time the determinant of all events in the enviroment?
hey thanks for this
Hi Leios
I have basic question dont think its silly question. why we need matrix ? what is the application ?
in schooling we were thought have to perform matrix operations but not the application
Machine learning uses Linear Algebra (matrices and vectors) extensively!
and this is only to name one application which is already transforming the world!
Good job, thanx bro.
I'm glad it was useful!
brilliant
Can anyone tell me. What is the acutal meaning of determinent?
if we take a number of vectors in a vector space, they (at least in 2D and 3D) form a shape. Well, we seek an answer to the question: "by how much the area is scaled"? Therefore, it makes sense to associate to every transformation a unique scaler value, denoted by det(T). Noting that Det: a number of vectors ---> scaler field.