Doing calculus with a matrix!
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- Опубліковано 23 бер 2022
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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
I think examples like this make a strong case for teaching Calculus and Linear Algebra simultaneously.
@@MichaelPennMath I agree. Do you find it is true that many of your undergraduates arrive with a great eagerness to use calculus but with hardly any linear algebra background? When teaching high school (in FL) I was pressured to leave out all linear algebra topics, and they pressured me to spend 8 weeks at the end of my PreCalculus course teaching limits and derivatives.
Maybe time for an AP Linear Algebra test?
@@barbietripping Please no. High schoolers don't need any more AP classes. Moreover, AP curriculum often discourages learning "nonessential" concepts, so these kind of methods would not be taught in most places because "it won't be on the test". While the ideas behind it (i.e. linear operators, vector spaces, etc.) was taught in my linear algebra class last year, so many concepts were already left out.
@@cuerti8327 good point. I recall taking the AP Calculus test and receiving the maximum score, but my teacher had not gone over proofs or derivations of hardly anything. Perhaps collegeboard is not the right party to trust for this.
But should there not be some attempt to put linear algebra on a similar playing field to calculus in the pre-college curriculum?
@@MichaelPennMath And category theory! (Though in practice, you want to be using techniques from category theory 'in the background' for a while, to build useful intuitions, and then introduce fully generalised abstract nonsense in a later course. Basically, exactly like you are doing it here.)
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 Im{e^((1+i)x)/(1+i)}=1/2*e^x*Im{(1-i)*(cos(x)+i*sin(x))}=1/2*e^x*(sin(x)-cos(x)).
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.
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
This shows how the integral can be solved with complexification.
e^x*sin(x)=Im{e^((1+i)x)}, so it's integral is Im{e^((1+i)x)/(1+i)}=1/2*e^x*Im{(1-i)*(cos(x)+i*sin(x))}=1/2*e^x*(sin(x)-cos(x)).
@@cuerti8327 yep yep
I love the problems where several math fields come together. Great video, Michael!
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
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
Ah that seems very clean. I think for people being introduced (viewers of this video) it helps to see these small examples where the basis doesn’t include the reciprocals of factorials.
@@barbietripping Yeah, for sure it helps if you aren't experienced enough with higher maths, it was just a note for those comfortable enough with the material
You can drop the finite dimensional requirement on the Polynomial space (and just require finite support).
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`
the anti-derivative in the second example was really cool
This is by far my most favorite video of yours: crystal clear and so insipiring! Thank you so mich for uploading this.
This has been one my most favorite videos in all the time I’ve been following the channel.
Thank you, professor.
Great video, very insightful! I especially loved the last part when you calculate the integral of e^x sinx
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
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!
13:18
Videos like these make me love math more and more every day
Thank you for this awesome video!
You often make these kind of videos with totally understandable math (fwiw: some undergraduate but not a math major), but used in a way I've never seen before, that blows my mind. It always feels like a round of applause is needed at the end! So inspiring. Thank you.
Awesome video! Thank you!
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
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!
very eye opening, thank you sir
Good job with this video!
Quite instructive. I enjoy seeing a mixture of disciplines. IMO linear algebra leads well into some concepts in differential equations.
Extremely interesting, thanks!
Kind of like magic at the end!
That was so fricking cool!
Great idea
An interesting finite dimensional vector space to do these calculus operations on the Walsh functions. The subsets of size of powers of 2 are closed under integration and most other arithmetic operations.
woooooooooooooow mind=blown daaamn, felt what i was taught could have been explained this way
This is actually kind of mindblowing, and now I need to pocket your other channel so I can start to really dig into linear algebra like I've wanted to for years. :D
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.
Beautiful. I really like it.
wow this was the best math video I have seen in a while
exitingly clear
This is a very awesome idea
Nice video, good work!!!!
Can you make a video about smooth manifolds and Hopf-Rinow theorem and some examples (i Did a thesis(Tesi triennale italiana)about that marvelous theorem)
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.
Would that have been at Sydney Uni? If so, just wondering if Terry Gagen was teaching back then?
my mind is blown! didn't think linear algebra could be used with calculus like this!
As usual great video, though if you wanted to change the integral we get from the new context to our old context would we just add +C to the integral we got?
Honestly the best channel on yt
Fun. Thanks
I enjoyed the video. I guess my only methodological objection for the second example would be that if we start with just the integral, how do we define the original vector space that we began with? It seems like that requires some intuition of what goes into the basis, right?
It's possible the inverse situation: f.i e^A, where A is a matrix for a lot of applications in rational mechanic and other...
Great explanation of these mappings! You kind of touch on it near the end, but you can make things a little less messy by using the fact that a linear map can be completely described by its actions on the basis vectors. So instead of differentiating a e^x cos x + b e^x sin x, you could just check what d/dx e^x cos x and d/dx e^x sin x are and that gives you the columns of the matrix.
For sure, and in a course setting, this type of representation theorem would be proven just after these examples!
So cool thankyou :)
Every linear transformation with a basis can be represented as a matrix.
This is so cool
oh! a defined integral is a linear functional which can be represented as < · , v > where v is fixed. fun!
(I thought this video is going to teach matrix derivative or matrix calculus.
en.m.wikipedia.org/wiki/Matrix_calculus )
Beautiful.
Extremely cool!
Interesting.
What happens if the matrix has no inverse?
Is multiplying by the matrix square root related to the half derivative? There can be multiple matrices that square to the same thing.
functional analysis!!!
Can we have more of these videos please. Let's see if we can apply this for any arbitrary smooth function, since that has a nice Taylor expansion we know there is an obvious map between \sum_{n=0}^{\infty} a_n x^n and {a_1,a_2,a_3,...} (infinite dimensional vector space) this means the corresponding linear transformations are infinite dimensional matrices, how can we deal with those?
I have been wondering if this was possible for the longest time
So, could we extend this by exponentiating the matrix to computer the nth derivative/antiderivative? (Exponentiation done via diagonalization)
linear algebra never fails to be cool
How does a dot/inner product work for the polynomial example? Separately, does this mean the {x^2,x,1} basis act like covectors for polynomials?
In your first example with the polynomials (I think you also do this with the exp-trig example), when you draw the arrows down from ℙ2 space to ℝ3, what symbol are you writing next each arrow? It looks like a 2 or a dotless question mark
it is a ~. an arrow with a ~ can be used to mean "isomorphic"
@@MichaelPennMath Great, thanks! I'm guessing you probably state this in your Linear Algebra series on your new channel. I'll go check that out
Sir actually i too love to do math .iam now studying in clss 12th and want to get in to your field .can u pls guide me about from where and how should i go forward.
Good afternoon, Professor. I have absolutely no idea what you are talking about, but it sure does look interesting. What are the prerequisites for this course?
For the stuff in this video the pre-reqs would be a passing familiarity with basic calculus and some fundamental linear algebra. For the expanded course stuff I don't know, but for topics in linear algebra high school math through algebra 2 and some familiarity with mathematical proof is probably sufficient.
That would have been nice (and in my opinion more interesting) to interpret the column of the matrix as the images of a base of the vector space by the linear application.
Yeah agreed. The concept is really neat but I found the video made it overly complicated since the simple fact that this works already relies on the linearity of the derivate. Why not just differentiate each basis vector and see where they are in the image and you get your matrix column vectors for free! This relies on a piece of linear algebra intuition rather than cranking the dreary algebra machine by differentiating the general case of [a,b]ᵀ in function space
That was amazing wtf
this is beautiful
Wait when you related it to R3 why do they have to be the basis vectors, why not any variable non orthonormal vectors?
can we do this for triple integrals as well?
Very cool
I think by trying to do the entire derivative in one (a exp(x)cos(x) + b exp(x)sin(x)) you're maybe more general, but the entire argument that this is possible hinges on the fact that the derivate is linear anyway. But this means that in order to construct the matrix we just need to look "what happens" to each basis vector (with [1,0]ᵀ≡exp(x)sin(x), [0,1]ᵀ≡exp(x)cos(x))when we apply transformation D on it (differentiate).
[D][1,0]ᵀ = [1,1]ᵀ
[D][0,1]ᵀ = [-1,1]ᵀ
So D= [[1,-1],[1,1]]
Isn't this a more elegant way to find the matrix? Less algebra and more intuition based surely. In order to construct a matrix I always just look at where the basis vectors end up after the transformation.
I like given just for the support represented in the T-Shirt.
as I read once: mathematics is the study of linear algebra
nice!
I deal with Taylor represemtations of Taylor maps (via truncated power series algebra), so I eat Prof. Penn's stuff for breakfast! My work is in accelerator physics where the map around a ring can be approximated by a Taylor series sometimes and then analysed.
Very useful indeed.
Who?
Cool!
yo i am learning about linear algebra now!!!
what if the base of the vector space are the roots of the polynomial instead of its coefficients?
The base chosen in the video isn't made of the coefficients, it's made of monomials: {x^2, x, 1}. The coordinates of the vectors with respect to this base are the coefficients.
So if you wanted to find a base B, such that the roots of the polynomial are its coefficients, then that's impossible because such a polynomial has at most 2 roots (except for the 0 polynomial), but the space is 3 dimensional.
lol
It's cool how much can be done without Mat, Vect_k, or the rest of the heavyweight category-theory machinery, just by studying mappings between Mat and Vect_k without introducing the categories.
So you're saying when I forgot the +C on my calculus exams, i could have just mentioned I was working in a certain function space? Lol
Any of you guys still interested in doing calculus with matrices step by Dr. Peyam too! He has some treats on there.
Sadly it doesn't work for rational functions since it's not stable by integration
Sent the +C to the shadow realm
Would be pretty cool to learn calculus starting from these simple operations, which is actual numbers and data structures you can understand, and then later on applying it to limiting processes of secant lines, which is the baffling way we learn calculus nowadays
That's an interesting take. I would think the "limit of secant lines" appeals to geometry and is easier to digest. Where I'm from we don't touch matrices in highschool and it makes things complicated for sure.
Cool! So would this fall under representation theory?
kind of.... derivative operators on a function space form a Lie algebra so this is towards a representation of the Lie algebra of derivations of the polynomial vector space.
In general representation theory is the study of algebraic structures via linear algebra -- so you look at how they act on vector spaces.
NICE! Great way to show what a definite integral is.
13:08 "because one of the basis vectors is not equal to the number 1" you keep doing this in more and more scenarios now- you use "is not" as the simple negation of an all/every/exists statement, which does NOT logically match what you intend: where your intention matches 'the claim "x is y" is false', that is not how mathematical logic proceeds when you have quantifiers in play. The reason you provided states that if ANY of the basis elements are not 1 (or, generously, a nonzero real numner), that a +C is unnecessary.
11 vedios in math major is all video of linear algebra there no more video
The course is still being constructed...
Hey! Your shirt is the flag of the Jewish Autonomous Oblast!
You impressed me with the rainbow t-shirt (I guess)
Is your shirt based on Jewish Auutonomous Oblast flag?
is the prideshirt a statement?
Do I support LGBTQIA+ causes, yes... Was I thinking about that when wearing this awesome shirt, not really.
@@MichaelPennMath
Its this type of stuff that let's us spot eachother. Thought u may have clued us in on something.
I still like the video and i think the shirt adds
@@nevokrien95 I’m just here for the shirt. (jk)
In all seriousness though: I’d prefer that we don’t out people that aren’t out, especially if they aren’t even “in” to start. And, speaking as a gay man, I hope that we’d all support you and your excellent math explainers no matter what!
And that’s a good place to stop
I started the vid and within 30sec i know that I won't understand a bit
A lot of omeomorphismes...
The tall pansy oceanographically replace because slipper contextually expect per a uptight overcoat. thankful, wrong swimming