Doing calculus with a matrix!

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[navidave]

i could tell where this was going with having an integral be represented with the inverse matrix but seeing it in action and actually working is honestly so cool

[MooImABunny]

about the second vector space you mentioned, V = span{e^x sinx, e^x cosx}, if you go ahead and diagonalize the derivative matrix, you'll find that its eigenvalues are (1±i)
from the point of view of complex analysis, a more natural basis for this space would be to write sinx = (e^ix - e^-ix)/2i, cosx = (e^ix + e^-ix)/2, and find the basis {e^[(1+i)x], e^[(1-i)x]}, which just so happens to be the basis that diagonalizes the derivative D
just another fun little part of it

[mathflipped]

I love the problems where several math fields come together. Great video, Michael!

[cuerti]

Another fun way to solve the integral at 11:22 is complexification! e^x*sin(x)=Im{e^((1+i)x)}, so it's integral is

The "normal way" to solve such integrals in a first year calculus class is integration by parts, and it is such a pain.

In fact, all of these methods are related.

[JonathanMandrake]

About the basis: In our Linear Algebra Lecture, we chose x^n /n! to be the basis, such that D is a matrix with 0 almost everywhere and 1 on the spaces 1 below the diagonal. The Integral could be defined as D^T as long as you can actually integrate the function and stay in P_n, so it has to be a polynomial in P_(n-1) for the Integral to be in P_n

[manucitomx]

This has been one my most favorite videos in all the time I’ve been following the channel.
Thank you, professor.

[camrouxbg]

This is so gorgeous, and so much fun! Having these kind of examples to show that matrices can do more than "just" systems of equations, and really dig in to the abstraction (but in an oddly concrete way) is so fantastic and valuable. Thanks!

[MCLooyverse]

On the final comment about a missing (or not) constant, would it be right to say that it's because we've limited the domain of our derivative, so we can fairly limit the image of our anti-derivative as well? When you say "The anti-derivative of `1` is `x + c`", it's because *any* function of the form `x + c` will have a derivative of `1`, but here, we have functions of the form `(a ⋅ cos x + b ⋅ sin x) ⋅ e^x` only, and so there is only one function of the form `(a ⋅ cos x + b ⋅ sin x) ⋅ e^x + C`, namely `(a ⋅ cos x + b ⋅ sin x) ⋅ e^x + 0`

[alessandrovistocco]

This is by far my most favorite video of yours: crystal clear and so insipiring! Thank you so mich for uploading this.

[mundocanibaloficil]

the anti-derivative in the second example was really cool

[pwmiles]

Another application is to functions

P= (a*x^2 + b*x + c)*exp(x)

In this case the differentiation operator is
D =
1 0 0
2 1 0
0 1 1

The inverse operation i.e. antidifferentiation is
I = D^(-1) =
1 0 0
-2 1 0
2 -1 1

E.g. to find the indefinite integral of

Q = x^2*exp(x)

the result is
exp(x)*[x^2 x 1]'*I*[1]
[0]
[0]
= exp(x)*(x^2 - 2*x + 2)

Of course, this would extend to higher powers i.e. x^3 exp(x) etc

[aristo]

I have to say Micheal Penn yt channel is by far the best in the math category.
Now there’s a seperate channel going in depth??!
We are truely so blessed

[jantarantowicz]

Great video, very insightful! I especially loved the last part when you calculate the integral of e^x sinx <3

[THEDeathWizard]

That is such a cool way to take an integral. If I had more background in linear algebra I could see this technique being extremely useful

[akirakato]

i wish you would go deeper into this idea of defining derivative operator as matrix, specifically explore properties of the square matrix and see what that saids about derivatives of functions and see if that generalizes to any family of functions.

[peterhall]

When I did linear algebra in 1970 in Australia it was mixed with calculus like this. Nice video Michael. You really do package these ideas very well and your students should develop good ways of looking at problems from the high level down.

[Chalisque]

Perhaps a better way of explaining the +C situation is that your function space is factored by the equivalence relation f~g iff f=g+c for some constant c.

[kodirovsshik]

Videos like these make me love math more and more every day
Thank you for this awesome video!

[-sv]

This is a really fantastic video, it’s the first one of yours I’ve really understood as a first year undergrad. I would love more!

[ongzz]

my mind is blown! didn't think linear algebra could be used with calculus like this!