Mathematics - 2018-08-28 (page 3 of 4)

But I don't think you can do the original question without the Stone version of the theorem. If you guys didn't do it, presumably it's not on your exam.

Oskar Tegby

Why didn't my previous method work for this case? Where I found some $g(x)=h(x)f(x)$ and showed that $g^2(x)=0$ so that $f(x)=0$.

Ted Shifrin

Because you can't get combinations of $x^{1/(2n+1)}$ to be a polynomial.

Oskar Tegby

6:07 PM

We did do the Stone-Weierstrass theorem. I though that it was the same as the Weierstrass approximation theorem.

Ted Shifrin

Ah, well, so you're going to consider $\{x^{1/(2n+1)}\}$ instead of $\{x^n: n\ge 0\}$. Oh wait. We don't have constant functions.

Hmm ...

Oskar Tegby

Okay. When I had $\int_0^\infty f(x)e^{-nx}dx$ I used performed a change of variables $u=e^x$. So I guess that I have to use a change of variables, or just try to manipulate it some other way to make it a polynomial, but why does it have to be a polynomial to begin with? Doesn't the Weierstrass approximation theorem say that we can approximate any function in $C([a,b])$ with polynomials? That feels backwards.

Ted Shifrin

Yeah, I don't think you can do any clever substitutions here.

Oskar Tegby

Okay.

Is there a difference between the Weierstrass approximation theorem and the Stone-Weierstrass theorem, or are they synonyms?

Ted Shifrin

Approximating a continuous function with polynomials won't help you with fractional powers if you're trying to get $f^2$.

No, Stone is far more general than polynomials. But you need a collection of functions which (1) contains a constant function and (2) separates points.

Oskar Tegby

6:11 PM

Why does it have to be a polynomial? Aren't we approximating a continuous functions with polynomials, not a polynomials with continuous functions?

Ted Shifrin

We're fine on (2), not fine on (1) here.

Think about how your proof is going to finish using the hypothesis.

Oskar Tegby

Hm...

Ted Shifrin

I'll think about it a bit later.

Oskar Tegby

Sure!

2 mins ago, by Oskar Tegby

Why does it have to be a polynomial? Aren't we approximating a continuous functions with polynomials, not a polynomials with continuous functions?

Oskar Tegby

6:29 PM

Never mind! I figured it out. I'm so thankful for all of your help, @TedShifrin. Tomorrow morning will feel so much better thanks to how much better prepared I am thanks to your help, professor.

Leaky Nun

Weather Channel Music

►

man this music is good

Soham Chowdhury

6:50 PM

Havona By Weather Report

►

I see that and raise etc etc

Nick

7:30 PM

-3

Q: Given straws of length $\pi$ and $\\e$, measure $3$

I give this question to my 12 year old brother ever so often and we find he always forgets how it's done. And yes, it can be done. Does it require many prior assumptions? Be my guest and find out for yourself. Take it as a simple challenge to prove. You are given two straws. One measures $3.1415...

I don't want this deleted.

How do I save it?

I think it's a really neat question.

Like if you had a 2cm length and a 4 cm length, you can so easily measure 3cm, right?

mercio

7:55 PM

that guy has a strange value for $e$

Abdullah UYU

@mercio Why don't you say directly to him?

mercio

I am lazy

Abdullah UYU

No. You're coward.

mercio

the first comment is almost as good at pointing out that his value for $e$ is very strange

it also wouldn't improve the question at all to fix it

Robert Frost

Anyone know how to plot a set in Geogebra or Wolfram Alpha?

mercio

8:02 PM

I don't know. with wolfram alpha I would try maybe giving a list of coordinates

Robert Frost

merci @mercio

mercio

de rien

Leaky Nun

merc @merci

mer @merc

me @mer

mercio

aaagh

Robert Frost

$x_{n+1}=trunc(x_n): x_0="mercio"$

leakynu @LeakyNun

Leaky Nun

8:17 PM

> I have no formal education in pure maths.

I was gifted at maths at school (a long time ago), virtually all of which I have now forgotten. Then I read mathematical economics, econometric methods and the theory of statistics under Professor Pesaran and others at Cambridge University. Most of which I have also forgotten.

then what do you r e m e m b e r?

Robert Frost

@LeakyNun bits of my school maths come back to me as I use it

But the theory of statistics I didn't enjoy and I have never used since

and I have no desire to remember it

Just equations about 50 characters long of strings of matrices

@LeakyNun school was 30 years ago!

I see

What's up, chat?

Yo

8:33 PM

hi @Daminark

Daminark

How's it going?

Leaky Nun

it's going well

Robert Frost

@Daminark any ideas how to plot a set of complex numbers say in Wolfram alpha or Geogebra?

@Fargle hi

Fargle

Heya @Daminark, @Robert, @Leaky

Robert Frost

@Fargle hi

you know how to plot a set of complex numbers say in Wolfram alpha or Geogebra?

Leaky Nun

8:39 PM

hi @Gargle

Robert Frost

Fargoyle

Leaky Nun

@RobertFrost what complex numbers?

Fargle

I'm not sure, actually, other than just treating them as ordered pairs and asking those programs to plot the ordered pairs.

Granted, I'm not very familiar with Geogebra.

Robert Frost

I have the function $f(x)=x^{(2\pi\cdot i/\log(2))-1}$ and I want to plot for a range of odd positive integers $x$

@LeakyNun ^^

Daminark

I'm not terribly familiar with graphing, but try putting in Desmos, I've used it to graph functions online

Robert Frost

8:43 PM

@Daminark ok will try. I know what the plot $x\in\Bbb R$ looks like but want to see the spacing of $x\in\Bbb N$

Leaky Nun

@RobertFrost desmos.com/calculator/6rjnhh8rpx

Robert Frost

@LeakyNun nice job. you have to zoom in to see it!

Leaky Nun

just change 59 to 29

hi @TedShifrin

Daminark

Rehi @Ted!

Ted Shifrin

@Oskar: It seems you need to use the general Stone result. When you have all the functions vanishing at $0$, then the theorem proves that the closure is that set of all continuous functions vanishing at $0$. ...

hi Leaky, Demonark

Leaky Nun

8:45 PM

@Daminark go read my LCFT stuff in garbology :P

Daminark

I will at some point soon but right now Princeton Review owns my soul

Leaky Nun

is that related to the calculus thing?

Robert Frost

@LeakyNun can teh points easily be labelled with their integer?

Daminark

Yup, it's a source specifically designed for the GRE subject test

Ted Shifrin

We like a soul-less Demonark.

Not just calculus. All sorts of stuff.

Leaky Nun

8:48 PM

@RobertFrost I don't think so, but I hope this makes things easier

Daminark

Yeah it's got a bunch, true, though I'm mostly focusing on the calc/multi/ODEs chapters

Robert Frost

@LeakyNun thanks

Ted Shifrin

Yeah, the algebra/analysis/topology/probability stuff you should be good with.

Robert Frost

much

Leaky Nun

@JasperLoy you know what, Loy sounds very Chinese to me

Jasper Loy

8:50 PM

@LeakyNun Yes, that is correct.

Leaky Nun

really? :o

Daminark

If only that were 50% of the test instead... shakes fist at ETS

Leaky Nun

well I can't connect it to any Chinese surname

Jasper Loy

@LeakyNun I am by ethnicity. It is Li in Mandarin, the Li in dawn or LiMing.

Ted Shifrin

The issue, Demonark, is that people take the GREs having had all sorts of different sorts of math backgrounds. So they have to spread the stuff beyond calculus over all the various topics, including applied things that some people might not know and pure things that some people definitely don't know.

Leaky Nun

8:51 PM

then why is it Loy?

Jasper Loy

@LeakyNun It is so in the Hainan dialect.

Leaky Nun

:o

Ted Shifrin

awards points to Jasper

Jasper Loy

But as everyone knows, I am from Antarctica.

Leaky Nun

I think that's related to Hokkien

micsthepick

8:52 PM

@AbdullahUYU I was able to show that if x = p/q for coprime p, q, then pq|(p+q)^3

Ted Shifrin

@JasperLoy now deducts points

Leaky Nun

@JasperLoy indeed lôi is a reading of 黎 in Teochew that is used for surname

Ted Shifrin

@micsthepick I don't understand. How does $x$ figure in here at all? Are you suggesting that $pq|(p+q)^3$ when $p,q$ are coprime?

Jasper Loy

@TedShifrin Haha. Oh after doing more research I think there is no graphical way to typr equations in pages. One needs to key in the latex himself unless he installs something called mathtype.

Leaky Nun

@TedShifrin I learnt Gram-Schmidt today :P

Ted Shifrin

8:54 PM

@Jasper: Yes, I typed LaTeX into the equation editor yesterday. What do you mean by a graphical way to type?

Jasper Loy

@LeakyNun Yes, that is the correct character.

Ted Shifrin

Good @Leaky. Now you can prove some interesting things about the topology of $GL(n)$.

Jasper Loy

@TedShifrin Point and click, like in Microsoft Word.

Leaky Nun

@TedShifrin also, how many equivalent statements does your version of invertible matrix theorem have? (yes, this is a competition)

Abdullah UYU

@TedShifrin There is a question in the background.

Ted Shifrin

8:55 PM

OH UGH @Jasper ... you mean like off a menu of selected symbols? I HATE that.

Leaky Nun

@TedShifrin the one in my notes has 18

Ted Shifrin

@Leaky: Totally absurd waste of time.

Jasper Loy

@TedShifrin I mean, if one doesn't do that, one might as well just use latex all the way.

Ted Shifrin

I'm not playing this game.

Leaky Nun

:(

lighten up

@TedShifrin such as?

Jasper Loy

8:56 PM

@LeakyNun I think I have seen only five or so, not 18.

micsthepick

supposedly

Ted Shifrin

@Leaky: It deformation retracts to $O(n)$.

Leaky Nun

ah!

that is very interesting indeed

Ted Shifrin

@Jasper: I'm fine with using LaTeX all the way.

micsthepick

does that not mean that p and q share the same unique prime factors?

Fargle

8:57 PM

I remember the textbook I had in my linear algebra class a long time ago had a similarly long list of statements that are equivalent to invertibility. I don't necessarily like that style of presentation.

Ted Shifrin

The linear algebra version is the $QR$ decomposition of nonsingular matrices, Leaky.

Leaky Nun

so QR is continuous?

Ted Shifrin

Heya @Fargle !!! :)

Jasper Loy

@TedShifrin Yeah, you use it to type letters even. By the way, in case you didn't know, latex does have a letter document class, but it is seldom used.

Ted Shifrin

indeed, Leaky.

Leaky Nun

8:58 PM

that is very interesting!

Fargle

Howdy @Ted, how goes it?

Ted Shifrin

I have my own, @Jasper. But I did originally start using the letter document class.

Hope you weren't too upset by my email, Fargle ...

Jasper Loy

@TedShifrin Then there is also the minimal document class that is used for troubleshooting.

Fargle

lol, not at all

Daminark

So, my professor didn't state it at the beginning and then go and prove it out or something but one day he did just kinda walk in and to get us talking he was like alright give me a bunch of characterizations of invertibility that we'll list on the board

To get us talking and also because he was about to start talking about eigenvalues

Ted Shifrin

8:59 PM

Sure, I like that for a classroom discussion/game, Demonark. Just not for a textbook.

Leaky Nun

textbook is dead, what can I do

I'm revising things on my own

Robert Frost

@LeakyNun See this; desmos.com/calculator/pwzbwjdf5n I added the even numbers. Now how do I plot a line on top between two numbers?

Leaky Nun

school starts on October

Jasper Loy

@LeakyNun Make the book come alive.

Use some spell from Harry Potter.

Leaky Nun

@RobertFrost 1. you haven't saved 2. what do you mean a line on top

Fargle

9:00 PM

@JasperLoy I think that would require the use of the Book of the Dead Books.

Daminark

So he wanted to have the stuff written on the board so that he could immediately talk about char poly, multiplicity, etc. For textbooks... Yeah probably not the best of ideas

Robert Frost

i just mean a line between two points sorry

@LeakyNun I didn't save :(

Ted Shifrin

@Jasper: You have a good answer for this?

Leaky Nun

@JasperLoy li e hiao gong min nam weh?

@Daminark I think the number of statements in the invertible matrix theorem is the hallmark of a successful book

Robert Frost

@LeakyNun I guess I need an account to do that

Leaky Nun

9:01 PM

(don't be triggered Ted, I'm just joking)

@RobertFrost sure

Robert Frost

@LeakyNun Ah here we go: desmos.com/calculator/yj8s3fzgjs

Ted Shifrin

Demonark: Did I tell you my GRE story about getting an integral of a 2-form over the unit sphere on my exam? The dopiest thing I ever did (because I was so used to teaching undergrads the classical way) was translate into $\vec F$ and take divergence ... rather than instantaneously taking $d$ of the 2-form. I laughed hard at myself after that one.

Jasper Loy

@LeakyNun Nope, I mostly only know the Mandarin dialect, not the other dialects.

Leaky Nun

@JasperLoy but you did understand my question lol

Ted Shifrin

@Jasper: Be warned. Leaky gets extremely competitive/obnoxious at times.

Jasper Loy

9:02 PM

@LeakyNun Hence the word mostly, precisely speaking.

Leaky Nun

I think he knows me

Ted Shifrin

Oh, I thought you mostly started living in here after Jasper disappeared the last time.

Jasper Loy

Don't worry I know everyone cos I am crazy.

Daminark

And look at you now, he who bears the flag of moving frames in the face of a lot of people who don't seem to like it as much

Jasper Loy

If you are bored or need someone to talk to, just email me. My email is in my profile.

Leaky Nun

9:04 PM

@JasperLoy be warned. I get extremely competitive/obnoxious at times.

@RobertFrost what do you mean a line on top

Jasper Loy

No problem. I can handle all obnoxiousness.

Leaky Nun

also, the road not taken

Jasper Loy

And I also don't compete with anyone over anything.

Robert Frost

@LeakyNun Just take two points and plot a line from one to the other. E.g. from f(3) to f(10)

Fargle

MSE chat: We Might As Well Call It Levity StackExchange

Leaky Nun

9:06 PM

@RobertFrost still don't undersatnd

ÍgjøgnumMeg

@Jasper bet I compete with fewer people over fewer things

Robert Frost

@LeakyNun pick any two dots on this: desmos.com/calculator/yj8s3fzgjs and draw a stright line from one to the other

Leaky Nun

I can't just pick any two dots

this isn't an interactive graph

Daminark

@ÍgjøgnumMeg lmao

Ted Shifrin

Demonark: That's been true my whole mathematical life. But Chern, Griffiths, Bryant and their students were/are largely in the camp. :)

LOL @Fargle

ÍgjøgnumMeg

9:07 PM

lolol hi @Daminark @Ted @Leaky etc.

Ted Shifrin

hi @ÍgjøgnumMeg

Leaky Nun

Tag

Ted Shifrin

I sure wish you had a short name/nickname we could type.

Daminark

Hmm

@Igjo does this ping you?

ÍgjøgnumMeg

yep

Daminark

9:08 PM

Yes it does okay perfect

ÍgjøgnumMeg

oh no

will I be Igjo now

Robert Frost

@LeakyNun Then draw a straight line from any n to any n+1

*every

@Ig

Did not

@Igj

9:08 PM

Igj works

lol

Daminark

Okay you need at least 3 letters

Fargle

Would I read that as something close to "Iggy"?

Leaky Nun

@RobertFrost desmos.com/calculator/zxani8srfj

Daminark

Actually speaking of moving frames, among probably hundreds of things in the "meaning to do at some point" file, I have considered trying DG once more. Let's say that even though I've done some stuff in the bootcamp, it's been a while and I struggled with that stuff even then

ÍgjøgnumMeg

@Fargle however you like!

Robert Frost

9:10 PM

@LeakyNun you beauty

Daminark

What would you say is the best avenue for me to try and reapproach it?

Leaky Nun

@ÍgjøgnumMeg i'll just call you "through me"

Fargle

How would I pronounce the whole thing, @Igj?

Ted Shifrin

Demonark: I'll be happy to discuss it with you (as, I'm sure, will Eric) whenever you're interested.

ÍgjøgnumMeg

@Fargle something like "uy-jugnon meh"

@Leaky go for it

rofl

Ted Shifrin

9:11 PM

Let's save this discussion for after GRE :)

Daminark

That's true

Leaky Nun

ígjøgnum meg

Fargle

Thanks! I find it's easier to remember names if I can pronounce and spell them.

Ted Shifrin

@Igjy, is this an actual name in some language?

Leaky Nun

[ʊiˈdʒœɡnʊn] [meː]

Ted Shifrin

9:11 PM

me too, @Fargle, me too.

ÍgjøgnumMeg

^

Leaky Nun

"through me" in Faroese @TedShifrin

Ted Shifrin

whoa ... what's Faroese?

Fargle

Spoken in the Faroe Islands, I would imagine.

Daminark

I pronounce it ÍgjøgnumMeg

ÍgjøgnumMeg

9:12 PM

@Ted no it's just a phrase I heard in a song that I liked so I put it into Faroese (which is my dream language) lol

Leaky Nun

it's a North Germanic language related to Norwegian

spoken in the Faroe Islands

Fargle

Was at one point Norwegian land, I think?

ÍgjøgnumMeg

@Daminark nise

Ted Shifrin

oh wow ... we should really become the linguistics chat.

ÍgjøgnumMeg

hahah

Faroese and Icelandic tend to be referred to as "Insular Scandinavian"

lol

Leaky Nun

9:14 PM

@TedShifrin I'm in total support of that idea

Fargle

@Daminark I've been meaning to give DG another shot as well, now that I have free time.

...Then again, I, too, have to take the GRE.

Daminark

Wait are you taking it in September?

Fargle

Either Sept or Oct, I have to check

October

Daminark

Ah, good luck

I'm doing it in September. I'll retake in October if September doesn't go as hoped but I'm sorta hoping it doesn't come to that since I won't really have much time to study then

Ted Shifrin

I always encouraged my advisees to take it earlier so that they could retake once, if necessary, before graduate applications are due in December/January.

The problem, Demonark, is this: Do you find out the score on the advanced exam soon enough to sign up for the second test? I think the answer is no. The deadline for October comes very soon.

Leaky Nun

9:21 PM

@TedShifrin I just proved that $\operatorname{proj}_{U_1+U_2}(\vec v) = \operatorname{proj}_{U_1}(\vec v) + \operatorname{proj}_{U_2}(\vec v)$ when $U_1 \perp U_2$ as subspaces of $\Bbb R^n$ lol

Daminark

Yeah that's the problem, I think I only have a week after taking the first test to sign up for the second

Ted Shifrin

That's why I tried to get my kidlets to take it in the preceding spring for the first time.

Daminark

So I'll either have to be confident that I did well enough in September (what does that usually entail anyway? 75th percentile?) or pre-emptively do October

Ted Shifrin

Leaky, are you using the characterization of projection in terms of the subspace and its orthogonal complement, or are you using some matrix formula for projection?

I actually don't know about percentiles. That's very dependent on which schools you're aiming at. Harvard/MIT/Princeton want higher GRE scores than Georgia.

And it's very competitive because so many foreigners do better than US students, to be honest.

Daminark

Yeah I was considering that. At the time I didn't feel it was worth it because my classes were real heavy and I thought I'd just burn 150 dollars and likely not do well enough to not retake anyway, though in hindsight I might have to pre-emptively burn $150 anyway in case I'm not sure it was a good idea

Gah

Leaky Nun

9:24 PM

Weather Channel Music

►

ÍgjøgnumMeg

@Ted would you say it's reasonably common for people to take some time out before grad school, and does it seem like it has a negative effect on performance?

Leaky Nun

I've been looping this a day

@TedShifrin the more algebraic one, i.e. the former :P

Ted Shifrin

@Igjy: It happens, but I wouldn't say it's reasonably common (in the US). People forget stuff quickly ....

@Leaky: A matrix formula is more algebraic.

The former is more geometric!

Leaky Nun

I see

ÍgjøgnumMeg

@Ted that's kinda what I'm afraid of lol, I basically have to take 6 months out because I'm poor... hoping I don't end up forgetting too much

Ted Shifrin

9:25 PM

But I have exercises in my book that ask students to use the definition to prove things, so I like that.

@Igjy: 6 months or a year aren't that big a deal at all. Several years is a bigger deal, I imagine.

Leaky Nun

well the latter requires an orthonormal basis right

or at least orthogonal

Ted Shifrin

No, Leaky.

Not at all.

ÍgjøgnumMeg

@Ted well that's the kind of reassurance I was looking for! lol

Ted Shifrin

It's rather a cool idea, actually. If $A$ has columns giving a basis for the subspace, then $A(A^\top A)^{-1}A^\top$ gives projection onto the subspace. Prove it.

Leaky Nun

:o

I learn something new from you every minute

Daminark

9:27 PM

Including while you're asleep! :0

Leaky Nun

yep

Ted Shifrin

smacks Leaky and Demonark

Daminark

ow

Leaky Nun

so I need to prove that $x^\top A (A^\top A)^{-1} A^\top x = x^\top x$

Robert Frost

@LeakyNun Turns out you can put e.g. ${n} in the label and make dynamic labels as here: desmos.com/calculator/76fzd8hewo

Leaky Nun

9:30 PM

I'm also learning something new from you @RobertFrost

ÍgjøgnumMeg

G'Night chat

Robert Frost

@LeakyNun even the one legged man kicks ass sometimes

Ted Shifrin

Whoa. No, @Leaky.

Leaky Nun

@TedShifrin right nvm

Ted Shifrin

You need to show that $A(A^\top A)^{-1}A^\top x$ satisfies your definition for the projection of $x$ onto the subspace.

Leaky Nun

9:32 PM

I need to prove that $\langle Av, x - A(A^\top A)^{-1}A^\top x\rangle = 0$

Robert Frost

@LeakyNun I need to plot the collatz graph on these numbers because then good things will happen

Ted Shifrin

And, ultimately, it's cool to understand where this comes from.

That's half of what you need to prove. Admittedly, the other half is trivial.

Leaky Nun

$v^\top A^\top A (A^\top A)^{-1} A^\top x = v^\top A^\top x$

yeah ok proved

so where does this come from?

Fargle

@ÍgjøgnumMeg Hey, me too, lol

ÍgjøgnumMeg

@Fargle sleeping or taking time out? lol

Ted Shifrin

9:36 PM

@Leaky: It comes the fact that the kernel of $A^\top$ is the orthogonal complement of the column space of $A$.

Fargle

@ÍgjøgnumMeg Taking time out, haha. It is not my bedtime yet.

Sorry to keep you, and goodnight

ÍgjøgnumMeg

@Fargle that's cool! Hope it all works out in the end, down with money!!!

Ted Shifrin

Night, @Fargle :P

ÍgjøgnumMeg

lol

Goodnight

Fargle

@TedShifrin :|

Ted Shifrin

9:37 PM

LOL

night, @Igjy.

Fargle

I can smack back, you know.

ÍgjøgnumMeg

Night @Ted :)

Ted Shifrin

Such threats, @Fargle.

Leaky Nun

@TedShifrin sorry I still don't understand how that fact leads to that formula

Ted Shifrin

Write $x=Au + w$ where $w\in V^\perp$.

Now use what I said.

Leaky Nun

9:41 PM

The column space of $A$ is $V$, so $w$ is in the kernel of $A^\top$, so $A^\top x = A^\top A u$

Ted Shifrin

OK, go on.

Leaky Nun

$A^\top A$ is invertible (I don't actually know why), so $u = (A^\top A)^{-1} A^\top x$, so the projection is $Au$ which is $A (A^\top A)^{-1} A^\top x$

very interesting

Ted Shifrin

Now what hypothesis did I make relating $A$ and $V$?

Leaky Nun

V = span(A)

Ted Shifrin

GRR

Leaky Nun

9:44 PM

the columns of A form a basis of V

Ted Shifrin

Okey dokey. So why does that make $A^\top A$ invertible? Maybe you can use one of your 18 equivalent reasons :P

Leaky Nun

If $A^\top A x = 0$, then $A x \in \ker (A^\top) = (\operatorname{im} A)^\top$, but $Ax \in \operatorname{im}(A)$ also, so $Ax = 0$.

Since the columns of $A$ are linearly independent, $x = 0$

Ted Shifrin

There you go. Except you have a typo. You mean $\perp$ ...

Leaky Nun

indeed

Ted Shifrin

OK, my lesson is over.

Leaky Nun

9:48 PM

thank you for your lesson :P

Ted Shifrin

This stuff is actually applied to giving least squares solutions of inconsistent equations.

hi @Jake

Leaky Nun

I see

Jake Rose

@TedShifrin hi

just the man I was looking for

Ted Shifrin

hides quickly

Jake Rose

I’ve been trying to understand matrices from the linear transformations. I think I get most things just not why matrix multiplication isn’t commutative. We haven’t done linear transformations or anything further at uni yet so I was wondering if you could give me a hand?

Leaky Nun

9:52 PM

@TedShifrin what is your favourite proof of JNF?

Ted Shifrin

$f\circ g$ is usually not $g\circ f$, @JakeR. But you should be able to write down some very simple $2\times 2$ examples to demonstrate.

Module theory, of course, @Leaky.

Jake Rose

what does the circle mean?

Ted Shifrin

composition

Leaky Nun

@TedShifrin is that also the one you teach?

Jake Rose

what does composition mean?

is that just a way of saying something acts on something else?

Ted Shifrin

9:54 PM

No, I only have taught it a few times, Leaky. Not in my linear algebra courses.

Mancala

Can anybody help me? I suspect I use the equations of structures, but I can not ... math.stackexchange.com/questions/2896776/…

Ted Shifrin

@JakeRose: $\sin(x^2)$ is the composition $(f\circ g)(x)$ where $f(y) = \sin y$ and $g(x)=x^2$.

@JakeR: So notice that $(AB)x = A(Bx)$ whereas $(BA)x = B(Ax)$. So, yes, acting on something else.

@Mancala: I know all about structure equations, but I cannot decipher your question unless you give me the book. You'll have to provide all the notation. Your question is otherwise hopeless.

Leaky Nun

@JakeRose because moving your body forward and then rotating it by 90 degrees isn't the same as rotating it by 90 degrees and then moving your body forward

Ted Shifrin

@Leaky: But moving your body forward is not a linear map.

Leaky Nun

oh nvm

p r o j e c t i v i z e :P

Ted Shifrin

9:57 PM

LOL, ok.

Fargle

@TedShifrin Ah, fine. ;|

Ted Shifrin

@JakeR: Try $A=\begin{bmatrix} 0&1\\1&0\end{bmatrix}$. Try $B=\begin{bmatrix}1&3\\2&4\end{bmatrix}$. Compute $AB$ and $BA$. Notice what multiplying by $A$ first does and what multiplying by $A$ second does. :)

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