What an amazing application of linear algebra. It's almost too good, it makes me wonder why this is not usually given as an example in linear algebra books or linear algebra courses.
This Dr. Wildberger is the most amazing and beautiful generalization I've ever seen. My favorite part of the Algebraic Calculus course - thank you for spreading math awareness through the Internet.
Even though I am still learning many things from this series, a lot of the ways you reintroduce more basic topics makes me really wish I saw this a few years ago.
you have solve my problem. I spent two nights for this and i didn't know that is formula exists. I will generalized the Bernoulli numbers to the K. Thanks man Allah blessed you.
so, it's just change of basis problem. very nice. another way to think about it just from reccursive formula to close form formula using linear algebra. well knowen method yet very enjoy!
Thank you. Rather than perform an inverse directly, you can use an updating scheme. Each row of the matrix of coefficients is calculated by a simple formula of the coefficients in the previous row, except for the Bernoulli number (in column 1), which is chosen so the sum of coefficients in the row is 1. I generate the coefficients in Python using the Fraction class so as to get fractions for the coefficients, rather than float values. The insight I take from this linear algebra approach is I can derive the formulae without using Calculus.
Umbral calculus is the theory behin this exposé ( en.m.wikipedia.org/wiki/Umbral_calculus ) and explains in a more general way how to construct a integration / differentiation on series of powers. Using falling factorials en.m.wikipedia.org/wiki/Falling_and_rising_factorials makes the subject much more clear. The Bernoulli numbers are used to convert from falling factorials to ordinary powers.
Hi, this was a very nice explanation. Thank you. I was thinking about a matrix, a(k,n), where the (k,n)th entry was defined by a(k,n) = Sum x^k for x = 1 to n. Each row is a set of partial sums of a Faulhaber formula. Do you think this matrix contains any interesting properties?
Some feedback on slides 5, 6 and 7. I think the reasoning on slide 6 would be easier for people unfamiliar with this technique if you labelled the telescoping sum on slides 5 and 6 as T_k (where _ indicates sub-script), so that: a) At the top it reads "So *T_k =* yields," and each following line with k=1..4 has a T_k label on it as b1) k=1 T_1 = S_0 = n, and b2) k=2 T_2 = = 2S_1 - S_0 = n^2, and b3) k=3 T_3 = = 3S_2 - 3S_1 + S_0 = n^3, etc. The reason I suggest this is because the S expressions are sums of powers, but without a distinct label for the telescoping sum at the top of the slide, it is easy to mentally confuse the S expressions with the telescoping sum instead of the sum of powers. By labeling the telescoping sums differently, it becomes more clear that they are being expanded into binomial terms and then re-written in terms of sums of powers of a lower order k. T3 re-written in terms of S2, S1, S0, not T2, T1, T0. Then, on slide 7, you could use the T_k labels to label each of the four linear eq'ns (corresponding to the four rows of the matrix).
I made a simple implementation (in the Sidef programming language) of the formula presented at 30:35 for computing Bernoulli numbers, and it works nicely. Source code: github.com/trizen/sidef-scripts/blob/master/Math/bernoulli_numbers_from_pascal_triangle_2.sf
11:40: Prof., this idea of using the Pascal polynomials to 'rewrite' into the sums of powers formulas reminds me of 'changing basis' in lin alg. Is that essentially what you're doing when you perform the matrix inversion? Are there other facts/results about independent or orthogonal bases (I forget the terminology, I hope you know what I'm hinting at) from linear algebra which can apply in interesting ways to this 'power summation' toolbox we're gathering? I'm thinking along the lines of Fourier transforms? (Or again maybe I've got the wrong terminology again; but that thing where functions are re-written using cosines or sines or something, possible due to some weird trick with them always being equal to zero at integer values except at 0 or something.)
Yes you are on a good track here. We will be exploiting in many ways this idea of using linear algebra "change of basis" technology to help us understand a variety of situations.
Prof. (or anyone, actually): For the Pascal array, suppose you added an extra column on the left and a row on the top, perhaps indexed at -1. It seems to me that if the numbers in the array are called P(i, j), then you'd have P(-1,-1) = 1, and P(-1, j) = 0 and P(i, -1) = 0 for i, j >= 0. So, in a sense, you can start your array just with a 1 in the upper-left corner, 0s to the right and down from it, and begin filling in all the rest of the array using P(i+1, j+1) = P(i, j) + P(i, j+1). This kind of seems like a more 'natural' or at least 'simpler' starting seed for the generation of the Pascal array. Is there any useful/practical significance to this? Do these 'marginal' columns and rows ever get used in any sort of math anywhere, ever? More generally, can the Pascal's array be extended even further to the left and above than the -1 index? What would P(-3, -5) equal, if so? It kinda feels reminiscent of the Gamma function extending the concept of factorials from the domain n >= 0, to the complex domain and thus to the real line like Gamma(x) where x (a Real) can be < 0 (except for poles at negative integers, or whatever). I have a gut feeling that the Pascal array can be extended to the left and up also, but I don't have a sense of how that would work beyond the -1 index. Any clues or pointers? [Bonus question: Is there any sense by which Pascal's array could be extended to Real or Complex inputs?]
These are great questions! But the answers will have to wait till we start working with "formal power series" i.e. "on-going polynomials". That is one of the features of moving to negative exponents.
I wonder if there a recursive method to reuse the solution of the Sn matrix in order to generate a solution for the higher order Sn+1 matrix. If so, this would be a more computationally efficient method than solving large matrices directly and thus reinventing the Bernoulli number wheel each time as the k power index increases.
I believe the mathematical journey is perhaps more important than the mathematical destination. As my high school mathematics teacher once said " There is more than one way to skin the cat'.
There is simple way to do that without using matrix. All you have to do is to get the so-called dimensional constants and directly substitute into the generalized equation. If k is the exponent, then there are k dimensional constants. Dimensional constant is a function of k for these sequences. The generalized equation is applicable not only integers raised to common power but also to other integers sequences based on uniform differences. The generalized equation is so simple even without simplifying to the known common form of the nth series. You can readily substitute any value of n since the generalized equation is in factored form.
Sir can you do derivation bernoulli number method with covering whole stuff of Bernoulli numbers In a simple way including try to explain hard types too I am now entered in collage so I am not yet so much at it.. But i like this stuff But i have understand your matrix Mathed but i couldn't able to digest your Bernoulli method So please derivation... By the way i am Ankit
I don't think the derivation is complete. You have not derived the relation with the Bernoulli numbers and the binomial coefficients. You can only prove in special case that the Faulhaber's formula is true.
I think my question nails down to: why is the Bernoulli numbers obtained in each sum the same, e.g. why is there only one B_2? and not one B_2 for each sum?
What an amazing application of linear algebra. It's almost too good, it makes me wonder why this is not usually given as an example in linear algebra books or linear algebra courses.
This Dr. Wildberger is the most amazing and beautiful generalization I've ever seen. My favorite part of the Algebraic Calculus course - thank you for spreading math awareness through the Internet.
Very good teaching style. Easy to understand. Very helpful. Thanks
My mind has been blown about 60 times in the last few days
This may be the best video of yours that I've yet seen, and I've watched a LOT of your videos! Fascinating stuff!
best explanation for bernoulli numbers on Internet so far
Omg thank you sir! I love you! You saved my life!
Even though I am still learning many things from this series, a lot of the ways you reintroduce more basic topics makes me really wish I saw this a few years ago.
you are genius sir📈
you have solve my problem. I spent two nights for this and i didn't know that is formula exists. I will generalized the Bernoulli numbers to the K. Thanks man Allah blessed you.
Thank you for your precious time.
so, it's just change of basis problem. very nice. another way to think about it just from reccursive formula to close form formula using linear algebra. well knowen method yet very enjoy!
Great explanation.. thank you
You are welcome!
Very, very nice!
Thank you. Rather than perform an inverse directly, you can use an updating scheme. Each row of the matrix of coefficients is calculated by a simple formula of the coefficients in the previous row, except for the Bernoulli number (in column 1), which is chosen so the sum of coefficients in the row is 1. I generate the coefficients in Python using the Fraction class so as to get fractions for the coefficients, rather than float values. The insight I take from this linear algebra approach is I can derive the formulae without using Calculus.
Clever idea. Matrix inversion is quite expensive.
Amazing video
Umbral calculus is the theory behin this exposé ( en.m.wikipedia.org/wiki/Umbral_calculus ) and explains in a more general way how to construct a integration / differentiation on series of powers. Using falling factorials en.m.wikipedia.org/wiki/Falling_and_rising_factorials makes the subject much more clear. The Bernoulli numbers are used to convert from falling factorials to ordinary powers.
I was searching for this kind of content, I hope umbral calculus was more widespread It seems like a quite deep and usefull theory
Hi, this was a very nice explanation. Thank you.
I was thinking about a matrix, a(k,n), where the (k,n)th entry was defined by a(k,n) = Sum x^k for x = 1 to n. Each row is a set of partial sums of a Faulhaber formula. Do you think this matrix contains any interesting properties?
Some feedback on slides 5, 6 and 7. I think the reasoning on slide 6 would be easier for people unfamiliar with this technique if you labelled the telescoping sum on slides 5 and 6 as T_k (where _ indicates sub-script), so that:
a) At the top it reads "So *T_k =* yields," and each following line with k=1..4 has a T_k label on it as
b1) k=1 T_1 = S_0 = n, and
b2) k=2 T_2 = = 2S_1 - S_0 = n^2, and
b3) k=3 T_3 = = 3S_2 - 3S_1 + S_0 = n^3, etc.
The reason I suggest this is because the S expressions are sums of powers, but without a distinct label for the telescoping sum at the top of the slide, it is easy to mentally confuse the S expressions with the telescoping sum instead of the sum of powers. By labeling the telescoping sums differently, it becomes more clear that they are being expanded into binomial terms and then re-written in terms of sums of powers of a lower order k. T3 re-written in terms of S2, S1, S0, not T2, T1, T0.
Then, on slide 7, you could use the T_k labels to label each of the four linear eq'ns (corresponding to the four rows of the matrix).
Anyone notice that each row of the inverse matrix sums to 1?
I cannot wait to discover this is going to have an application in probability theory and quantum computing!
That explains the integration property!!
Thankyou
I made a simple implementation (in the Sidef programming language) of the formula presented at 30:35 for computing Bernoulli numbers, and it works nicely.
Source code: github.com/trizen/sidef-scripts/blob/master/Math/bernoulli_numbers_from_pascal_triangle_2.sf
18:00
11:40: Prof., this idea of using the Pascal polynomials to 'rewrite' into the sums of powers formulas reminds me of 'changing basis' in lin alg. Is that essentially what you're doing when you perform the matrix inversion? Are there other facts/results about independent or orthogonal bases (I forget the terminology, I hope you know what I'm hinting at) from linear algebra which can apply in interesting ways to this 'power summation' toolbox we're gathering?
I'm thinking along the lines of Fourier transforms? (Or again maybe I've got the wrong terminology again; but that thing where functions are re-written using cosines or sines or something, possible due to some weird trick with them always being equal to zero at integer values except at 0 or something.)
Yes you are on a good track here. We will be exploiting in many ways this idea of using linear algebra "change of basis" technology to help us understand a variety of situations.
Prof. (or anyone, actually): For the Pascal array, suppose you added an extra column on the left and a row on the top, perhaps indexed at -1. It seems to me that if the numbers in the array are called P(i, j), then you'd have P(-1,-1) = 1, and P(-1, j) = 0 and P(i, -1) = 0 for i, j >= 0. So, in a sense, you can start your array just with a 1 in the upper-left corner, 0s to the right and down from it, and begin filling in all the rest of the array using P(i+1, j+1) = P(i, j) + P(i, j+1). This kind of seems like a more 'natural' or at least 'simpler' starting seed for the generation of the Pascal array. Is there any useful/practical significance to this? Do these 'marginal' columns and rows ever get used in any sort of math anywhere, ever?
More generally, can the Pascal's array be extended even further to the left and above than the -1 index? What would P(-3, -5) equal, if so? It kinda feels reminiscent of the Gamma function extending the concept of factorials from the domain n >= 0, to the complex domain and thus to the real line like Gamma(x) where x (a Real) can be < 0 (except for poles at negative integers, or whatever). I have a gut feeling that the Pascal array can be extended to the left and up also, but I don't have a sense of how that would work beyond the -1 index. Any clues or pointers? [Bonus question: Is there any sense by which Pascal's array could be extended to Real or Complex inputs?]
These are great questions! But the answers will have to wait till we start working with "formal power series" i.e. "on-going polynomials". That is one of the features of moving to negative exponents.
I wonder if there a recursive method to reuse the solution of the Sn matrix in order to generate a solution for the higher order Sn+1 matrix. If so, this would be a more computationally efficient method than solving large matrices directly and thus reinventing the Bernoulli number wheel each time as the k power index increases.
[0 1/4 1/2 1/4] = ( (1)[1 0 0 0] +(-4)[1/2 1/2 0 0] + (6)[1/6 1/2 1/3 0] + [0 0 0 1] ) / 4
Do you see the pattern?
I believe the mathematical journey is perhaps more important than the mathematical destination. As my high school mathematics teacher once said " There is more than one way to skin the cat'.
My approach to those types of sequence do not need to determine the bernoulli numbers, rather a different constants which can be solve independently.
0:46 zero is a natural number 👌
There is simple way to do that without using matrix. All you have to do is to get the so-called dimensional constants and directly substitute into the generalized equation. If k is the exponent, then there are k dimensional constants. Dimensional constant is a function of k for these sequences. The generalized equation is applicable not only integers raised to common power but also to other integers sequences based on uniform differences. The generalized equation is so simple even without simplifying to the known common form of the nth series. You can readily substitute any value of n since the generalized equation is in factored form.
Sir can you do derivation bernoulli number method with covering whole stuff of Bernoulli numbers
In a simple way including try to explain hard types too
I am now entered in collage so
I am not yet so much at it..
But i like this stuff
But i have understand your matrix
Mathed but i couldn't able to digest your Bernoulli method
So please derivation...
By the way i am Ankit
I don't think the derivation is complete. You have not derived the relation with the Bernoulli numbers and the binomial coefficients. You can only prove in special case that the Faulhaber's formula is true.
I think my question nails down to: why is the Bernoulli numbers obtained in each sum the same, e.g. why is there only one B_2? and not one B_2 for each sum?
π
No "irrational number" without quotes on this channel! XD