Mathematics - 2012-04-13 (page 2 of 2)

Mariano Suárez-Alvarez

06:00

or the variable under a quantifier?

Rob

Fourier used it for his summations.

Mariano Suárez-Alvarez

IMHO the best way to deal with confusions brought by an unfamiliar term is to familiarise the students with it, not pretend it is not there :D

because The World Out There will use the term, and then the confusion is only postponed!

'index variable' is pretty much restricted to sums

but the idea of a dummy variable is more general than that

when you write f(x)=sin x, you have one—that's another example

Rob

@MarianoSuárezAlvarez Have you seen my post on the definition of a variable here?

Mariano Suárez-Alvarez

I 'd say that it is better simply not to define what a variable is

for the purpose of hgh schoolproblems, better wording of the statements is the solution

"find a real number x such that Ax^2+B=0"

"find the real numbers x such that cos x=1"

most mathematicians would not be able to produce a non-absurd definition of what a variable is, and it has never been a problem to any of them! :)

why would high school students need one?

Rob

06:19

Hmm.."The World Out There will use the term" variable, as you said.

Mariano Suárez-Alvarez

but mathematicians use it, and they cannot define them

Rob

"and then the confusion is only postponed "

Mariano Suárez-Alvarez

high school students cannot understand any correct definition of what a real number is

my point is, a definition of what a variable is is useless

specially for the purposes of a high school student

it is better to avoid confusions, like in your comment «The variable, x, in the quadratic equation represents a parabola when it "varies" over a given domain.» to Scott

the quadratic equation does not represent anything

«the graph of the quadratic function is a parabola» rather

and «the quadratic equation is the problem of finding the value x which will satisfy the equality»

and so on

I have troubl coming up with a sentence which uses the word variable which cannot be rewritten without it gaining correctness and clarity

an average English speaking student is able to distinguish a couple of dozens of meanings attached to the verb «to get» (we non-native speakers have all suffered that...)

why would they have trouble with a couple of meanings of «variable»?

Rob

06:35

@MarianoSuárezAlvarez Why students have trouble learning variables? reminds me of the absent-minded math Professor who writes "x" on the board, but says "y" when he means "z" but it should have been "t." ;-)

Benjamin Lim

@Rob are you in high school?

Mariano Suárez-Alvarez

I don't know if they have trouble learning variables

in all likelyhood, they have trouble because someone is trying to teach them "variables"

Benjamin Lim

what if they get to indeterminates then?

Mariano Suárez-Alvarez

they should be being taught how to deal with functions, graph them, solve equations and so on

Benjamin Lim

yeah

Mariano Suárez-Alvarez

06:38

"learning variables" is problematic in itself

Benjamin Lim

@MarianoSuárezAlvarez Have you read La tía Julia y el escribidor por Mario Vargas Llosa?

Mariano Suárez-Alvarez

many many years ago

Benjamin Lim

sorry he is peruvian not argentinian

but I remember in the book he took a dig at argentinians

Mariano Suárez-Alvarez

well, I am not sorry

Benjamin Lim

I don't know why

Mariano Suárez-Alvarez

06:39

the guy's an idiot :D

Benjamin Lim

hahaahahahahahah!!

Mariano Suárez-Alvarez

to be fair, at the time he wrote that he was not

(the guy's son, on the other hand., has probably been an idiot during his whole life)

Benjamin Lim

don't know much about his son

but that novel has some interesting themes

user19161

07:27

@TylerBailey Thanks for reminding me it is Friday the thirteenth! Is Freddy going to make a comeback?

Matt N.

08:05

Good morning everyone.

@BenjaminLim I saw your challenge. I'll think about it later, tonight maybe, m'kay? : )

Rob

@MattN Good morning.

Happy Friday the 13th.

Benjamin Lim

@MattN ok

user19161

08:34

So yesterday I entered the wrong chat room and realized it only half an hour later...

N3buchadnezzar

09:14

"Tumbleweed" badge get

Rob

@N3buchadnezzar Good job.

Jonas Teuwen

Hmm.

robjohn

09:33

@JasperLoy which room?

Matt N.

It's this time of the week again : )

robjohn

@Matt hey there

Matt N.

Gotta make my mind up: which seat can I take?

@robjohn Hail to king robmean the first! : )

Jonas Teuwen

@MattN Could you elaborate? 8-).

Matt N.

I think after clicking on the link there's nothing left to elaborate : )

Ok, bbl : )

robjohn

09:39

@JonasTeuwen listen to the video; it becomes evident :-)

Jonas Teuwen

@robjohn I prefer not, 28 million views can't denote anything good 8-).

robjohn

@JonasTeuwen I have a friend like that. He likes something until it becomes popular, and then he thinks it is trash.

Jonas Teuwen

@robjohn Well, I didn't like it before either.

And I have listened to it for about 10 seconds!

I don't mind popular things, but I have never heard something that was this popular which I liked.

robjohn

@JonasTeuwen I agree; the song (which I've only just heard) is a rehash of almost every weekend celebration song made.

Jonas Teuwen

Good 8-).

robjohn

09:48

The thing about "which seat to choose" is that one of her big decisions is whether to sit in the front seat or back seat of her friend's car.

Jonas Teuwen

Oh, does that matter?

Matt N.

(look at the number of dislikes at the bottom right of the video : D)

Rob

141,169 likes, 592,430 dislikes

N3buchadnezzar

Hey

Jonas Teuwen

Nice copy-paste bro.

N3buchadnezzar

09:54

Mind anyone look over some simple calculations (improved euler)

I keep messing up, and I do not have a good way to do this

Hmmm

robjohn

@N3buchadnezzar where?

N3buchadnezzar

I have y' = x + y and given y(1)=0 with stepsize h = 0.5

$$ x_{n+1} = x_n + h $$
$$ u_{n+1} = y_n + h f(x,y) $$
$$ y_{n+1} = y_n + h/2 \left[ f(x_n , y_n) + f(x_{n+1} , u_{n+1})\right] $$

I need to use this to approximate $y(2)$, and I get what is listed in the table. But this is incorrect. The answer should be $y(2) \approx 2.436502$

robjohn

@N3buchadnezzar and $f(x,y)=x+y$?

N3buchadnezzar

@robjohn Yeah

robjohn

10:27

@N3buchadnezzar sorry, I got distracted answering a question :-)

I get $y_1=.22$

$f(x_0,u_1)=1.2$ so that $\frac12(f(x_0,y_0)+f(x_0,u_1))=1.1$

Sb Sangpi

12:28

how can I do to ask about the graph?

on stackexchange mathematics question

Jonas Teuwen

I'll write my own window manager. 8-).

All these things suck.

Hmm, dwm is quite cool.

robjohn

12:57

@JonasTeuwen good idea :-p

Jonas Teuwen

dwm proves that 2000 lines should do it.

robjohn

@JonasTeuwen I assume that dwm is a window manager?

Jonas Teuwen

@robjohn It is! dwm.suckless.org

robjohn

beat you to a link :-)

Jonas Teuwen

8-).

Sb Sangpi

13:02

@robjohn how can I do to ask questions about graph on stackexchange? thx

user19161

@robjohn I wanted to go to the TeX room but ended up in one of the interview rooms there instead and sort of interrupted an interview for their blog!

robjohn

@JasperLoy oops :-)

@SbSangpi have you visited the main site?

@SbSangpi there is a link for asking questions there.

N3buchadnezzar

Bah

I really really hate latex now

Sb Sangpi

@robjohn no,I mean I need to ask question about the graph but how can I draw a graph on question?

N3buchadnezzar

@SbSangpi Make it in paint, then upload image

Sb Sangpi

13:07

aww.ok is there any other way that can draw the graph? thx

user19161

@N3buchadnezzar I love my latex pillow...

N3buchadnezzar

I still can not seem to find the mistake. Jumps angry up and down

Jonas Teuwen

@N3buchadnezzar What error does it give you?

Just start from scratch. Make a simple table, expand it and keep adding things (and check if it works).

N3buchadnezzar

Well, it compiles. It just looks like poo

Jonas Teuwen

I'm sorry to hear that.

Instead of hating LaTeX you better look at yourself and start learning it properly 8-). Or use Word.

N3buchadnezzar

13:34

@JonasTeuwen crawls into a corner and starts using word

Jonas Teuwen

I'd suggest you follow my advice and start from a very simple table that you expand.

This is almost impossible to debug.

N3buchadnezzar

You actually tried?

Jonas Teuwen

Make a table?

Or debug yours?

I've actually tried that and then I thought bloody monkey, he'd better just start all over again.

N3buchadnezzar

Sigh

Jonas Teuwen

Yeah, it sucks to be you at this moment.

It also sucked to be my colleague yesterday. He went to Düsseldorf to the British Embassy but there were train delays, so when he arrived he immediately could turn back. An eight hour trip. He's trying again today 8-).

N3buchadnezzar

13:46

poo this poo

I am asking on the latex site

I am able to fix some of the problems, but it breaks others.

Jonas Teuwen

They will ask you for a minimal example.

So they basically ask you to strip anything until up to the point where the error occurs. They will not debug your complete code 8-).

You lazy bastard.

N3buchadnezzar

@JonasTeuwen Sneaky me, I just added the tag "Best practices"

And my code runs just fine, so there is no need for any bug fixes. I am just asking if there is a better way to make such a table.

robjohn

14:22

@SbSangpi there is no built in way to draw graphs with MathJax, as far as I know.

@SbSangpi You can check here for what is supported

Jordan

14:48

What is the difference between finding the definite integral and the area under a curve using approximating rectangles?

As far as I can tell approximating with rectangels gives the right answer and is far easier to use

user19161

@Jordan What do you mean by rectangle approximation? The definite integral is the limit of the area of approximating rectangles in a way.

Jordan

well in my book I either have to use some insane sigma wizardry for definite integrals or I can just add up the points and multiply then by the sub interval for the rectangle approximation

I think I was suppose to learn something from this chapter but I don't know why I use that stuff and I don't understand how so I am not using it and I am getting the right answers

user19161

@Jordan Using rectangles we can only approximate the area. Using the integral calculus we get the exact value. That is one immediate difference.

Jordan

but the math appears to be the same

"A table of values of an insteasing function f is shown. Use the table to find the lower and upper estimates for the intregeal from 10 to 30 f(x) dx

I am then given x and f(x) values, no equation just numbers

this would be the same if I add then up and multiply by the delta x

user19161

@Jordan So you see these are overestimates and underestimates of the area under the curve using rectangle approximation. Using integration we get the exact value if we can compute the integral easily.

Jordan

14:55

But I dont see how that is possible with a table of values

user19161

@Jordan By integration I mean finding the antiderivative in this case. Maybe you have not done that.

Jordan

no I am not suppose to do that

I am doing definite integrals and I have to find the limit as x approaches infinity of the summation of some crazy formula

user19161

@Jordan You will learn it later on then.

Tyler Bailey

Howdy

I just got your ping from last night, Jasper. How's your Friday so far? :)

Howdy Jordan

Jordan

hi

Tyler Bailey

15:01

Riemann sums are really tough because of all the notation that's going on inside the sum

it's difficult to keep track of at first

but that's really the hard part, is keeping track of that business

since adding up rectangles isn't too bad :p

Jordan

yeah I just cant figure out the sigma stuff

Tyler Bailey

Well you're really just adding up a bunch of points of the (height of the function)times (some length)

the rest is just the gory details

but it sounds like you have the right idea, judging from the problem you quoted above

Jordan

ok

I wasnt sure if I was suppose to be doing it different but I guess he really just wants me to learn the idea of the rectangle stuff

Tyler Bailey

yeah

Jordan

thanks

Tyler Bailey

15:26

Have you had to draw pictures to see what's happening when you're doing this?

like, the left endpoint vs midpoint vs right endpoint method?

if that's even what you're doing, I can't recall how stewart does it.

Jordan

yes

but now he is moving away from pictures and into the weird summation stuff

Tyler Bailey

did you label the pictures and everything? It might help to take one of the sums and draw the picture and label the things, so you can see what they actually correspond to

rather than just like "What the heck is that $x^*_i$ doing??

because that's what happened to me

Jordan

I understand the idea of the rectangles I just have trouble applying that to a formula that uses summation

Tyler Bailey

heck, i just went to a Prof's office hours yesterday and he just drew me a picture with rectangles ;p

i felt like an idiot

i had been thinking about the problem for a few days

hmm

Rob

Jordan

15:36

I understand the pictures, just not the sigma notation that uses the n(n+1)(2n) or whatever

I am going to watch some khan academy later today

Tyler Bailey

The sigma notation is just a (very) compact way of writing sums.

that might be a good plan :)

good bye

good luck Jordan

N3buchadnezzar

Sigma is just a simple way to add similar things for example

$$ \sum_{i=1}^{100} i $$

Is a really compact way of writing 1 + 2 + 3 + ... + 99 + 100

Rob

=5050 ;-)

$user image$

Daniil

17:02

Hi

Rob

17:16

@Daniil Hi.

robjohn

17:31

@N3buchadnezzar: did you see my reply about your numerical integration?

7 hours ago, by robjohn

I get $y_1=.22$

Ah, he is already gone.

Rob

@robjohn Did you notice Jordan was back?

robjohn

17:50

@Rob I saw.

Rob

@robjohn Any word from Kannappan?

robjohn

@Rob not since yesterday. I was supposed to go over a paper for him, but I ran out of time.

Rob

@robjohn Do you want to hear a riddle?

robjohn

@Rob why not?

Rob

@robjohn Two individuals are born from the same biological mother and father at the same time but they are not twins. How is that possible?

robjohn

18:02

I guess it depends on the definition of twin and whether surrogacy was involved.

Rob

@robjohn No surrogacy was involved. One biological mother at one time.

@robjohn Yes, the definition of a twin is very restrictive...

robjohn

Twin

A twin is one of two offspring produced in the same pregnancy. Twins can either be monozygotic ("identical"), meaning that they develop from one zygote that splits and forms two embryos, or dizygotic ("fraternal") because they develop from two separate eggs that are fertilized by two separate sperm. In contrast, a fetus which develops alone in the womb is called a singleton, and the general term for one offspring of a multiple birth is multiple. Statistics The twin birth rate in the United States rose 76 percent from 1980 through 2009, from 18.9 to 33.3 per 1,000 births. The Yoruba hav...

it doesn't look that restrictive.

Rob

@robjohn That is the first place I went to :-)

@robjohn "A twin is one of two offspring produced in the same pregnancy."

robjohn

so you are talking about triplets or more

Rob

@robjohn They are two members of triplets or quadruplets or etc...

@robjohn The phrase " Two individuals are born" has a built-in natural assumption that only two are born.

robjohn

19:15

Finally a question about continued fractions :-) the proofs in my paper are pretty similar.

robjohn

20:25

@Rob yep

N3buchadnezzar

@robjohn Yeah, I saw it. I just did not have time to look at it closer. It seems I need to look over my code again. In the last day or so I have been trying to make a nice report for my physics class. But thanks for looking into it, I really appreciate it =)

robjohn

@N3buchadnezzar sure. it looks as if you were not using $u_k$ in your computation

N3buchadnezzar

@robjohn You have been working on that paper for 35 years?

robjohn

@N3buchadnezzar On and off. I extend it when I discover something new.

N3buchadnezzar

That means you are 35+ wow =)

It means you have been working on a paper for a tad shorter than twice my lifespan, that really is something.

robjohn

20:32

@N3buchadnezzar it might :-)

you are 17 I guess and I misread your comment

N3buchadnezzar

Norway is about 3 years behind when it comes so mathematics, so I am slightly older.

And me being childish I suppose do not increase my credibility either.

anon

20:48

@robjohn: Cool, you have a note on Soddy's theorem. Where were you when I got my Tumbleweed badge? :P

N3buchadnezzar

:p

@anon I just obtained it too, but I got it fair and square.

anon

Hey, I also got it fair and square. (cough Srivatsan cough) I wasn't trying to get it.

N3buchadnezzar

Oh, my bad!

anon

What I was trying to do was soak in upvotes for pretty pictures, but that didn't work :P

N3buchadnezzar

I really do like your question though

upvotes because of pretty pictures.

anon

20:54

Damn MathWorld and its "personal communication" citations.

Matt N.

Kannappan hasn't shown up here for quite a while.

anon

He said something about not coming back to chat.

Matt N.

Heh? What happened?

anon

Rustled feathers between Gigili, FaceDavid and Jordan.

hhh

Could someone look here about simplex? I cannot still get that...

Matt N.

21:01

People should take other people less seriously.

hhh

"not to revert the signs of the inequalities but d,e can be arbitrary because the identities survive multiplication by any constants. "

?

anon

I don't know about linear programming really, or what a dual is in that context.

Henning Makholm

I used not to understand how people were willing to quit the entire site in anger over one single user's actions. Now I'm an experience richer, thanks to Aryabhata. :-(

Matt N.

What happened?

Henning Makholm

@MattN This meta thread.

Matt N.

21:11

I see.

Henning Makholm

Really, I've been this close to start deleting all my answers. Only didn't because many of them are accepted and the rest would just be restored by moderators anyway.

Matt N.

I think you're overreacting.

Henning Makholm

Of course it's overreacting -- doing anything of that sort in response to a single user is automatically overreacting.

Matt N.

Exactly.

Henning Makholm

But I'm rather disappointed in the lack of support from the community here.

Matt N.

21:15

I have no opinion on this either way but if you don't mind my asking a curious question: why is it so important whether those questions stay here or are asked / moved to CS.theory?

(but I did upvote your meta question)

Henning Makholm

I want to be able to answer them.

Matt N.

And you can't have a CS.theory (or whatever it is) account?

Or is that too inconvenient?

Henning Makholm

(It's sort of like asking a rabbi in the 16th century why it is so important whether his people can settle anywhere in town when there's a perfectly fine ghetto just over there).

Matt N.

: D

anon

What relation does the unexpected hanging paradox bear to Godel's theorems?

Henning Makholm

21:19

@MattN Using more than one site would be such a stupendous timekiller because one has to check separate question lists, reputation pages and so on. Once you have checked all of them to see there's nothing new, enough time has passed that something interesting may have happened on the other site, so you need just to quickly check that before going back to work. Repeat ad infinitum.

Matt N.

Yes, I can see that it's inconvenient.

Henning Makholm

@anon Not much, I think. But if we're thinking explicitly about the incompleteness theorem, one might attempt an argument that both are somehow about the difference between what is true and what can be inferred rationally to be true.

@MattN So I'm not going to do that.

anon

I see. A post on the mainsite insinuated they were related.

Henning Makholm

There's probably some formal sense in which all paradoxes are isomorphic, by virtue of being paradoxes. (And a Gödel sentence is close enough to being a paradox -- without actually being one -- that it could receive a honorary inclusion).

robjohn

@anon as far as I was able to research (and I talked to some people active in the field) the result in that paper is new. One of my few new results.

Matt N.

21:23

Wow. Cool. : )

robjohn

@anon however, I guess I need to come up with a poem for it ;-)

Henning Makholm

@anon I was about to write an answer along the same lines as your comment.

@robjohn Sounds ambitious. Wouldn't a Latin anagram suffice?

robjohn

@HenningMakholm You probably haven't read The Kiss Precise

Matt N.

I'll end my day here. Good night everyone!

robjohn

21:39

@anon you have cool images and animations for that post.

@MattN good night!

Matt N.

@HenningMakholm It's not worth being so annoyed about something so insignificant.

user19161

21:50

@TylerBailey Many analysis problems can be solved by drawing the appropriate pictures in your mind. Even if the problem is not in the Euclidean plane drawing pictures in the plane may help motivate the actual proof.

user19161

@Rob Nice pics there!

robjohn

@anon: do you know a proof of Kollros' Theorem, but looking for a simpler one?

anon

IIRC I was looking for help visualizing an inversive geometry argument for why KT-with-extra-turns was true. I'd have to remember what I was talking about back then a little.

anon

22:10

If $\displaystyle W\subseteq\bigoplus_{i\in I} V_i$ and $\lambda_i\in\mathrm{GL}(V_i)$, if $(\bigoplus\lambda_i)(W)\subseteq W$ is $W$ necessarily a graded subspace?

The original problem I have is with $V_i$ defined over an infinite field $k$ and $\lambda_i=\alpha^i$ with $\alpha\in k^\times$ specifically (also $I=\Bbb N$), but I think it might generalize.

robjohn

@anon: I think I see the 3 of the 3,6 case. What is the 4,4 case of Kollros?

I'm not sure I understand the terminology.

Henning Makholm

22:25

I'm seeing broken MathJax output on the main site, producing, e.g., "i&ltk-1" for $i<k-1$. (There's a semicolon missing after &lt). Anyone else?

E.g. here

robjohn

@anon: I think I see now. Is this only for $\mathbb{R}^3$ or does it extend to $\mathbb{R}^n$? MathWorld does not make that clear.

anon

Pretty sure it's R^3, but I'm having trouble remembering the 4,4 case :P

@Henning: I'm not seeing issues.

Ironically, I think I can see the 4,4 case prior to inversion...

Oh, nevermind, I was visualizing it incorrectly. After inversion it is easier.

After inverting the 4,4 case you get two planes and two spheres in between them (like an hourglass). Just put four new spheres around the hourglass big enough to touch the bounding planes and you're done.

Anybody understand the graded vector space question? :D

Henning Makholm

@anon Here's what I see:

robjohn

forgot a ;

Henning Makholm

Yes, but the source of the question has TeX there. It's MathJax that goofs somehow.

robjohn

22:52

There is the source. It looks as if the error is before MathJax gets the data

However, it renders fine for me

t.b.

Hi all.

robjohn

Hey there

t.b.

Am I being to harsh on the 'gician here?

Henning Makholm

"The most famous [nonassociative algebras] are probably ..." and then not a word about Lie algebras?

t.b.

well, that was already pointed out by zyx

leo

23:05

just to say hi

Henning Makholm

@tb I don't think you're too harsh. Both incomplete and not very enlightening.

leo

so HI

t.b.

HIIIII

Thanks, Henning.

leo

coffee time! :-)

t.b.

enjoy!

Henning Makholm

23:08

Hmm, you write "You also have the action of the additive group" -- except unless I'm missing something that's not even a group action. (The point that it must mesh with scalar multiplication in a particular way is still valid, of course).

leo

@tb what question should (and already is)be deleted?

t.b.

@HenningMakholm right. Thanks for catching that goof.

@leo there were two questions having somewhat lengthy answers that were deleted by the OP. We undeleted both questions, but one of the answerers deleted his contribution afterwards (because it somewhat missed the point), so it was fine to delete the question again.

leo

i see

well, I must go

see you

t.b.

see you

Benjamin Lim

23:28

hey

t.b.

hey

Benjamin Lim

@tb I can't believe the problem of proving that if you have $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ exact

and $M'$, $M''$ finitely generated

then $M$ is finitely generated

can be proven using some simple knowledge of linear algebra!!

t.b.

why not?

Benjamin Lim

yeah

I was like wait a minute

and then realised the proof was exactly the same in proving that if you have a vector space and a linear map from it to somewhere else

the kernel and image of the linear map finite dimensional implies that the vector space is itself finite dimensional.

anon

either of you know any hints for my $\bigoplus$ question above?

t.b.

23:33

I missed it.

anon

1 hour ago, by anon

If $\displaystyle W\subseteq\bigoplus_{i\in I} V_i$ and $\lambda_i\in\mathrm{GL}(V_i)$, if $(\bigoplus\lambda_i)(W)\subseteq W$ is $W$ necessarily a graded subspace?

Benjamin Lim

@anon Unfortunately my algebra is not so advanced to know about graded rings, etc

anon

1 hour ago, by anon

The original problem I have is with $V_i$ defined over an infinite field $k$ and $\lambda_i=\alpha^i$ with $\alpha\in k^\times$ specifically (also $I=\Bbb N$), but I think it might generalize.

@Benjamin: graded VS just means a direct sum of VSs, and a graded subspace is just a direct sum of subspaces (thereof)

Benjamin Lim

VS = vector pace?

anon

yes

The problem I don't think specified anything about the dimensions of the $V_i$ but they might need to be finite or something.

Benjamin Lim

23:38

@anon So a graded vector space is one that is the external direct sum of some vector spaces, and if $W \subset V$ is a subspace, $W$ graded means that $W$ is the internal direct sum of subspaces?

t.b.

@Ben: It means that $W = \bigoplus_{i \in I} W_i$ where $W_i \subset V_i$.

Benjamin Lim

Oh ok.

@tb Hectic hey the proof on $M$ being finitely generated!!

anon

Hmm, I think there might be a simple counterexample now that I'm trying to visualize it.

Benjamin Lim

@anon what is $\bigoplus \lambda_i$?

anon

Yes, consider $$\bigoplus_{\Bbb N}\Bbb R.$$ and $\lambda_{1,2}=\mathrm{Id}$ while the others can be anything, and $W$ is generated by $(1,1,0,0,0,\cdots)$.

@BenjaminLim $\bigoplus \lambda_i : (v_i)_{i\in I}\mapsto (\lambda_i v_i)_{i\in I}$ (I made up that notation, but it's probably standard... maybe..)

t.b.

23:42

Yes, I think you need that condition on the full graded endomorphism ring, not only the invertible group.

(the counterexample seems correct to me)

Benjamin Lim

@tb So now the $\lambda_i's$ need not be invertible linear transformations?

t.b.

yes, but then it is rather clear; just set all entries but one to zero and the other one to the identity.

anon

@tb: I'm not sure I follow. $\oplus\lambda$ is fixed so if the idea fails using GL surely it does with End too?

t.b.

Oh, I misread then: I thought that the condition was supposed to hold for all combinations of $\oplus \lambda_i$ with $\lambda_i \in GL(V_i)$.

anon

My counterexample wouldn't be a counterexample then :D

Benjamin Lim

23:47

So $W$ is a stable subspace...

t.b.

Oh, right. Well, guys, if you need further proof that you shouldn't ask me about algebra... :D

anon

Anyway the original problem is with $\lambda_i:=\alpha^i$ (scalar multiplication with $\alpha\in k$ nonzero and $k$ infinite).

Benjamin Lim

@anon Perhaps we can think of a subspace $W$ of the direct sum

such that

it is not say $\bigoplus_{i \in J} V_i$ for some $J \subset I$.

anon

@BenjaminLim There are graded subspaces of $V$ that can't be written as $\bigoplus_{i\in J}V_i$ for any $J\subseteq I$. Suppose $\dim V_i>1$ for each $i$ and consider $W=\langle (a,0,0,\cdots)\rangle$ for some $a\in V_1$ not zero.

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