This is the English translation of a Japanese video posted in March 2024.
Таймкоды
00:00:00 Введение в проблему
- Обсуждение понятия дифференцирования.
- Представление проблемы: попытка связать дифференцирование с матрицами.
00:01:03 Основы умножения матриц
- Объяснение работы умножения матриц на двумерный вектор.
- Идея использования матрицы для преобразования двумерного вектора.
00:02:00 Преобразование многочленов в векторы
- Матрицы могут преобразовывать только векторы, а не многочлены.
- Сопоставление коэффициентов многочлена с вектором.
- Дифференцирование многочлена и попытка представить его с помощью матрицы.
00:02:56 Создание матрицы дифференцирования
- Определение матрицы D для дифференцирования многочленов.
- Проверка соответствия матричного расчёта результату дифференцирования.
00:03:52 Расширение матрицы дифференцирования
- Ограничения текущей матрицы: возможность дифференцирования только многочленов не выше второй степени.
- Расширение матрицы D для дифференцирования многочленов любой степени.
00:04:46 Применение расширенной матрицы
- Необходимость бесконечных векторов для умножения на расширенную матрицу.
- Пример дифференцирования многочлена с помощью расширенной матрицы.
00:05:40 Проверка результата
- Детальный расчёт компонентов вектора и проверка соответствия результату дифференцирования.
- Подтверждение совпадения матричного вычисления с результатом дифференцирования.
00:06:38 Применение к другим функциям
- Обсуждение возможности дифференцирования тригонометрических и экспоненциальных функций с помощью расширенной матрицы.
- Представление синуса и косинуса как бесконечных многочленов и их дифференцирование с помощью матрицы.
00:07:37 Дифференцирование и матрицы
- Дифференцирование можно представить как матрицу, независимо от классического определения через предел.
- Матрица дифференцирования d не зависит от исходного определения дифференцирования.
00:08:36 Векторы и функции
- Вектор f представляет функцию f от x, а матрица d — дифференцирование.
- Существуют разные способы связать функции с векторами, что влияет на форму матрицы дифференцирования.
- Коэффициенты бесконечных полиномов, таких как ряды Тейлора и Фурье, используются для представления функций.
00:09:32 Возведение в квадрат и вторая производная
- Возведение матрицы d в квадрат соответствует взятию второй производной.
- Результат возведения d в квадрат можно использовать для вычисления второй производной без повторного расчёта.
00:10:28 Обратная матрица и интегрирование
- Обратная матрица d не существует, так как дифференцирование устраняет постоянный член.
- Интегрирование означает антидифференцирование, но постоянная интегрирования не учитывается.
00:11:27 Производные от производных
- Производная от производной не является обратной матрицей из-за нулевого верхнего левого элемента.
- Первый столбец матрицы дифференцирования d состоит из нулей, что делает невозможным получение единичной матрицы при умножении слева.
00:12:20 Заключение
- Обсуждение представления дифференцирования с помощью матрицы изменило представление о дифференцировании.
- Подчёркивается важность начальных условий в дифференциальных уравнениях для восстановления постоянного члена.
Расшифровка видео
0:00
what what is this this is what is called
0:02
differentiation who is it just who are
0:08
you oh my I feel like I had a bad dream
0:12
good morning zindan M good morning let’s
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get started today’s problem is here I
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just woke up
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though well um represent G by GX as a
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matrix uh what does it mean this is
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quite a tricky problem I have no idea
0:28
what we are supposed to do
0:30
differentiation in a matrix are
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completely different things right that’s
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right they’re completely different
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things but the fact that it came up as a
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problem like this means there might be a
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way to mathematically interpret and
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represent differentiation with a matrix
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let’s give it a try understood try to
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recall how matrix multiplication works
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as an example let’s consider a 2×2
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square Matrix like this then multiply
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this Square matrix by a two-dimensional
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Vector a matrix with a single column can
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be identified as a vector you know
1:02
that’s how it works this matrix
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multiplication is first focus on this
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part and the first component becomes
1:09
like this next focus on this part and
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the second component becomes like this
1:16
now that you mention it I think it was
1:18
something like that note that the result
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is also a two-dimensional Vector see in
1:24
other words this Matrix can be thought
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of as transforming a two-dimensional
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vector indeed that’s correct based on
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this idea let’s think about whether
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differentiation can be represented with
1:36
a matrix however handling a general
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function right away is difficult so
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let’s start by considering a polinomial
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in X for instance something like this
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this is a quadratic polinomial hold on a
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second I’m starting to lose track of
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what we’re doing but what we’re trying
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to do now is create a matrix that
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differentiates this polinomial right oh
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so you do understand menad unfortunately
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matrices can only transform vectors and
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they cannot transform polinomial that’s
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a sharp observation what it’s true that
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as it is it can’t be transformed using a
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matrix that’s why we need to identify
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polinomial with vectors so we’ll extract
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the coefficients of the polinomial and
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create a vector let’s identify this
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Vector with the original polinomial
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that’s quite an unusual thing to do
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you’ll understand what we’re doing soon
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enough now then let’s differentiate the
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polinomial let me do it for this
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polinomial when you differentiate it
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with respect to X it becomes like this
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that’s exactly right now let’s assume we
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can represent this operation with a
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matrix so there exists some Matrix and
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if you multiply this matrix by the
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vector of polinomial
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coefficients the result should be a
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vector corresponding to the
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differentiated polinomial H in other
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words the calculation in this part
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becomes a a sub 1 so the first row
3:02
should look like this next this part
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becomes 2 a sub 2 so the second row
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should be like
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this finally since the result becomes
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zero the elements of this row can also
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be zero looks good with this we have
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represented the differentiation of
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polinomial with a matrix let’s denote
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this Matrix as d h is this the Matrix
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that represents differentiation I don’t
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quite get it all right let’s try using
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this to differentiate we’ll consider a
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polinomial as a function of X this can
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be expressed like
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this and the corresponding Vector looks
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like
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this now let’s differentiate this which
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means multiply the differentiation
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Matrix d by the vector F I’ll skip the
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calculation steps but the result is here
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this corresponds to the result of
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differentiating f ofx if we actually
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differentiate F ofx the result looks
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like like this so you can see that the
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Matrix calculation matches the result
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well it’s true that multiplying by The
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Matrix D gives the same result as
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differentiation but you can only
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differentiate polinomial of degree 2 or
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less with this that’s not good you’re
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right then to handle any polinomial
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let’s extend the differentiation Matrix
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D when we do that D takes this form ah
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the blank parts are considered to be
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zero I feel like this showed up in my my
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dream maybe it’s just my imagination by
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the way how far does this Matrix go it
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continues infinitely what does such a
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matrix even exist that’s a short
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question from here on we’ll be dealing
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with infinite dimensions and infinite
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sums in reality we need to rigorously
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Define their meaning in calculation
4:50
rules but for now we’ll prioritize
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intuitive understanding so please keep
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that in mind I see now let’s
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differentiate a polinomial using this
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Matrix this time we’ll consider a
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polinomial like this extracting the
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coefficients just like before the
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corresponding Vector takes this form uh
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that makes sense so we also write the
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vector as infinite dimensional yes the
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vector also has to be infinite
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dimensional otherwise you can’t multiply
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the matrix by the vector I see all right
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zon try differentiating f using the
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Matrix D okay if I think about it the
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same way as before when you multiply G
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by F you should get the result of
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differentiating f if we write G and F
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explicitly it looks like this H the
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first component is we should focus here
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in the first row the only nonzero value
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is one and this one corresponds to that
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one so the first component can be
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considered one that’s true next let’s
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calculate the second component in the
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second row the only nonzero value is
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this two and the two corresponds to zero
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so the second component becomes zero it
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ended up being zero now what about the
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next one uh the third component is this
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three corresponds to the two here which
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means the result is six that’s right
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since the four in the fourth row
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corresponds to zero the fourth component
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is zero from here on the vector
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components remain zero so the
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calculation results remain zero as well
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that’s how it is all right let’s confirm
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that this calcul corresponds to
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differentiation the derivative of f ofx
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is like this and the result turns into
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this The Matrix calculation result
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matches the result of differentiation
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that’s true with this Matrix you can
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differentiate polinomial of any degree
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but somehow I feel a bit tricked what do
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you mean you can only differentiate
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polinomial with this there are many
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other functions like trigonometric
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functions or exponential functions it’s
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true that not every differentiable
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function can be differentiated using
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this Matrix but if it’s a trigonometric
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or an exponential function you can
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differentiate it with this what really
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for example s of X and cosine of x can
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be expressed like this simply put they
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can be represented as infinite
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polinomial wow I see and if you extract
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their coefficients you can represent
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them as vectors h i get it let’s take
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the vector corresponding to sign and
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differentiate it using this Matrix I’ll
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skip the calculation steps but the
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result will be a vector corresponding to
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cosine wow that’s amazing wait but this
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kind of feels obvious doesn’t it why do
7:42
you think so Express s of X as an
7:45
infinite sum and if you differentiate
7:47
each term it matches the form of cosine
7:50
of x expressed as an infinite sum so the
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earlier calculation was just expressing
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that in terms of a matrix you’ve
7:58
realized it’s in the
8:00
but there’s something I’d like you to
8:01
know here what is it it’s that
8:03
differentiation can be defined
8:05
independently as a
8:07
matrix try to recall it differentiation
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was originally defined using a limit
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like this oh yeah that’s right and the
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Matrix the defined this time took this
8:17
form they look completely different
8:19
that’s the important point this Matrix
8:21
is defined to represent differentiation
8:24
but it doesn’t depend on the definition
8:26
of differentiation using a limit in
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other words
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you could Define this Matrix without
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relying on the definition of
8:33
differentiation and it would still work
8:36
well defining such a matrix out of
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nowhere would make you look eccentric uh
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anyway we can think of a vector F that
8:43
represents a function f ofx and consider
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a matrix D that represents
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differentiation but a matrix is simply a
8:51
collection of numbers so it can be
8:53
defined independently a structure
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similar to that of differentiation
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actually exists in the world of matrices
9:00
which seems unrelated this is truly
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fascinating now that you mention it it
9:04
does feel that way ah for those who’ve
9:07
studied linear algebra this
9:09
correspondence might feel natural as a
9:12
side note there are other ways to
9:14
associate functions with vectors and
9:16
depending on that the Matrix
9:18
representing differentiation might
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change wow really in this discussion
9:23
when thinking about vectors
9:24
corresponding to functions we’ve
9:26
generally extracted coefficients of
9:29
inite polinomial so-called power series
9:32
well when we actually calculate the
9:34
coefficients we often consider it a
9:36
tailor series but let’s set it aside for
9:38
now in practice fora series are often
9:41
used instead of power series we won’t go
9:43
into detail about 4A series but remember
9:46
that there are various ways to map
9:48
functions to vectors H understood now as
9:52
a result of what we’ve done so far we’ve
9:54
roughly established that this Matrix key
9:57
represents differentiation here let’s
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try doing something Matrix like what do
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you mean for example how about squaring
10:04
D of course squaring d means multiplying
10:07
G by itself as a matrix then it becomes
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like this what could this be actually
10:12
this corresponds to taking the second
10:14
derivative or differentiating twice what
10:17
this is pretty interesting D feels like
10:20
both a matrix and
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differentiation here if you do this
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calculation once you can reuse it every
10:27
time you calculate a second derivative
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if you you memorize d^2 you don’t have
10:31
to calculate it every time you could
10:33
probably think about General n
10:35
derivatives in the same way that’s true
10:38
let’s look at another example this time
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let’s consider the inverse Matrix of d
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by the way an inverse Matrix is a matrix
10:46
that when multiplied results in the
10:48
identity Matrix I what do you think this
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will be uh if it’s the inverse of
10:54
differentiation is it integration that’s
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almost correct what do you mean by
10:58
almost actually the inverse Matrix of D
11:01
does not exist you tricked me if we try
11:04
defining a matrix corresponding to
11:06
integration as D Prime D Prime looks
11:09
something like this here the integration
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refers to anti- differentiation the
11:13
process of finding a function that
11:15
returns to the original when
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differentiated note that the constant of
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integration is ignored here it has a
11:22
form that is symmetric to the
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differentiation Matrix this is
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interesting in its own way now let’s
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calculate DD Prime will skip the
11:30
calculation process but the result takes
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this form in other words this is the
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infinite dimensional identity Matrix wow
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so D Prime is the inverse Matrix of D
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after all not so fast next let’s
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calculate D Prime D the result comes out
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like this what the top left is zero
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that’s right which means this is not the
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identity Matrix so D Prime is not the
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inverse Matrix of D how strange why does
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this happen look at the First Column of
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the differentiation Matrix D the First
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Column is all zeros this ensures that
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the top left of the result will always
12:08
be zero so no matter what Matrix you
12:11
multiply from the left it cannot become
12:13
the identity Matrix that is D does not
12:16
have an inverse Matrix to begin with so
12:18
that’s how it is if we interpret this
12:21
from a calculus perspective
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differentiating eliminates the constant
12:25
term and integrating can’t restore it if
12:28
you want to store it additional
12:30
information is needed like initial
12:32
conditions and differential equations I
12:34
don’t really get differential equations
12:36
but at least I understand that D Prime
12:38
is an almost inverse Matrix it was just
12:41
one step away it seems in a way this is
12:43
very calculous light and interesting
12:46
that’s true so that was the discussion
12:49
on representing differentiation with the
12:51
Matrix today what did you think we
12:53
talked about infinite dimensions and
12:55
infinite sums in a somewhat intuitive
12:57
way but I feel like my perspective on
13:00
differentiation has changed a bit well
13:03
I’m not entirely sure if we can just
13:05
connect these with an equal sign maybe
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someone felt like writing it this way
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okay let’s rack it up M well then take
13:12
care everyone see you again
13:15
[Music]