7:00 AM
you have thought about it for less than 5 minutes!
if by tomorrow at this time you have not gotten it, ask me
and I'll tell you :-)
I really doubt your life depends on knowing how to prove right now

Also, as a general thing: do not accept answers which do not solve your problem! («solving your problem» includes your understanding the solution)

I tried doing an elementary proof
but I could only get it to work with vector spaces

You may have, but you have not been thinking for a few days about how to prove my two maps are mutually inverse, have you?

Thanks, I'll try that then. Also - I accepted that answer because it looked really damn neat. :)
(To someone who doesn't know category theory)

7:02 AM
that's one reason why people end up talking mumbo jumbo
things one does not understand are never neat
in fact, to really understand what is going on in Qiaochu's answer I posit that it is necessary to have done not one but several similar arguments to the one I sketched
understandable should always, always beat cool
2

If $f:F \rightarrow L$ and $g:L \rightarrow F$, is it at least OK to say that $fg=id_L$ since they agree on $B$?

you can say it if you can explain why that is enough :-)

Is it because $B$ is a basis for $L$ and the uniqueness thing?

then sure, you can

I'm still thinking about $gf$ though. Grrrr

7:17 AM
well, $B$ as a subset of $F$ has a property... use it

Oh god
OK thanks a lot
I don't know how the hell I didn't see that

now go write an answer to your question which you can understand
and accept it
(youwill have to wait a day or so, to accept your own answer, tho)

That would be too embarrassing

no, it is not
if you must, tell the world that I forced you

So would you say that using the Yoneda lemma for this result is going a bit too far?

7:22 AM
not really
but if you do not understand the lemma or the way it is used here, then it is useless
the Yoneda lemma is the abstraction of this result (and many others like it)

Interesting
makes me want to learn category theory

?

the best moment to learn category theory is when you already know it
4
and you still don't :-)

Another question - what's with the quotes on the right hand side of the screen?
3

7:25 AM
I should go to bed... I am speaking in apothegms...:-/
what do you mean?

well
who puts the sentences up there

people can "star" things we say here
and then they get put there
each of out lines here is in a box: put the pointer to the right of that box
a star will show up

ah thanks

if youclick on it, then you have starred the line
for some reason, people in stackexchange seem to looooove the :hover property

Good morning all

7:29 AM
hello sir

hi

@MarianoSuárez-Alvarez Hi

Hello Ben

@MarianoSuárez-Alvarez I have a lie theory exam tomorrow :D

cool :-)

7:32 AM
Just proved that if $G$ is connected
then $Z(G) = \ker Ad$
@MarianoSuárez-Alvarez How come you don't hang out so much in chat anymore?

well, I do have a job :-)

@MarianoSuárez-Alvarez You met Peter Tamaroff?

these last few days, for example, I am reading a thesis

Ah ok fair enough
@MarianoSuárez-Alvarez I find Lie algebras exciting

they are very nice

7:34 AM
You learn that if $G$ is connected

I did meet him
and he came to my office last week too

then you have a galois correspondence thingy happening between normal subgroups and ideals of the lie algebra
@MarianoSuárez-Alvarez So lucky he is
Thankfully I can invade Jarod's office at any time
@MarianoSuárez-Alvarez I asked a question on linear algebraic groups
have you seen the book by malle ?

after you are done with your test and all, you should browse a bit Serre's book on lie groups and lie algebras
you'll probably enjoy it
not familiar with the book, but I've seen it

@MarianoSuárez-Alvarez I am going through questions from here
@MarianoSuárez-Alvarez But I do say that without a knowledge of differential geometry, it is hard to see what is going on behind

nimbus roman font, ugly! :-P

7:37 AM
Algebraic topology is coming in
Differential geometry is coming in
Like today we proved that if G is simply connected then a map of lie algebras induces a unique lie group homomorphism

@MarianoSuárez-Alvarez Can they really be summed?

lie groups is a nice subject because it mixes up everything

we used the fact that G simply connected implies if you have $f : K \to G$ a covering space map then $K$ is isomorphic to G
Yeah
Like I keep hearing about flows, submanifolds and smooth immersions :D

$f : A --> R$, with $A \subset R$. What do we call $f(a)$ where $a \in A$?

@GustavoBandeira, what do you mean?

7:40 AM
You can only talk about the limit of partial sums

@GustavoBandeira You can only talk of limit of partial sums.

@IssacM The image of $a$ under $f$?

You look at $s_N$, and the infinite sum is defined to be $\lim s_N$

@BenjaLim Damn! We have posted exact similar message.

@yunone Ok so it's fine to use image even though image is also defined as $\{f(a) : a \in A\}$?

7:41 AM
@MarianoSuárez-Alvarez I must say that in these more advanced courses

@JayeshBadwaik @BenjaLim He's really stoned then?

They do demand that you have an overall understanding of what is going on in general in maths
@GustavoBandeira He is talking nonsense
@GustavoBandeira Like in Fourier theory
so many times you deal with infinite integrals and things like that
what makes you so sure you can just do a change of variables?

@MarianoSuárez-Alvarez "can you point to any reference where an 'infinite sum' is considered?" He said it's in Rudin's book - but he didn't say in what page lol

well it's because in your infinite integral, you are taking the limit as $x \to \infty$

@IssacM I could call that set the image (or range) of $f$. But I could call $f(a)$ the image of $a$ under $f$. Notice the slight difference in wording.

7:42 AM
and certain operations commute with limits
and then you're fine.

@IssacM Sure thing.

Well I'll be off for lunch.
@GustavoBandeira See you around later.

@MarianoSuárez-Alvarez Maybe I should get back to preparing for my exam
bye guys
see ya

7:44 AM
bye!

Bye!

@GustavoBandeira, you cannot add innfinitely many numbers

@MarianoSuárez-Alvarez Oh wait before I go I wanna tell you guys a joke
I'm like I don't know business/finance/marketing/accounting whatever I don't know
And that person was like

@MarianoSuárez-Alvarez I don't have a strong background in math but this seems so plausible.

"Those are all different subjects you know"
I'm like:
UP TO ISOMORPHISM THEY'RE ALL THE SAME.

7:45 AM
@BenjaLim lol

go study :-)

@GustavoBandeira You know what it means, up to isomorphism ?
hahahahahahahahahahahahaha

@BenjaLim I know what isomorphism means - But I guess I can deduce the rest.

1 hour later…
8:52 AM
Take about the sum of the parts being greater than the whole :-D

Some Random Guy/Gal : Business/Finance/Marketing/Accounting. Those are all different subjects you know!
BenjaLim: UP TO ISOMORPHISM THEY ARE ALL THE SAME.
@JohnJunior Just for capturing one more really great line in the chat. Star my previous message.
@BenjaLim LOL
hahahahahahahahahahahahahaha

9:09 AM

@JohnJunior Perfect.

9:31 AM
hi @JohnSenior

10:00 AM
@JohnJunior Hi - I'm back

user19161
10:51 AM
This John thing is getting confusing: robjohn, John Senior, John Junior, Jonas Meyer, Jonas Teuwen, John Nash...

Jayesh, Jasper, Justavo
The "J" thing is confusing.

user19161

user19161
26 letters, 1 alphabet, remember kids...

@WillHunting Haha thanks 8-).

10:53 AM
How many vowels Sir?

user19161
@JonasTeuwen A lot of J's I know in real life are crazy, like me.

Maybe I should change my name then ...

@JonasTeuwen I was trying to increase the number of "J"'s :P

The craziest of them all is this Junior guy.

@JonasTeuwen
I applied an identity wrongly
I feel happy in knowing my mistake

10:55 AM
Shit happens man.
Yes, that is good.

user19161
@BenjaLim As long as you are not crazy, it's good enough.

@JonasTeuwen @WillHunting Look at my answer here:
@JohnSenior I know happy in knowing where was my mistake

user19161
@BenjaLim Hey Lee's Smooth Manifolds second edition will be out later this month. It also has Lie groups and algebras.

@WillHunting Stop discussing about books and learn math!
5

user19161
@BenjaLim The time has not yet come, patience man...

10:58 AM
Some of us learn from books still...

user19161
@JohnJunior Some of us are too unwell to even learn now...

@BenjaLim Excellent

Someone has been downvoting my lie theory questions
at least a few of them now.

@WillHunting ...who?

user19161
@JohnJunior Like me.

11:00 AM
@JohnSenior I like the feeling of relief...

@BenjaLim I saw one that got closed and then re-opened - seemed a bit odd that it was ever closed

@JohnSenior It was my question on algebraic group books

user19161
@BenjaLim Why do you think so?

@BenjaLim ah yes

Because each one has a downvote

11:02 AM
@WillHunting If Steven Hawking can do it ...

user19161
@BenjaLim No need for downvote even if they don't like it.

user19161
@JohnJunior But John Nash could not...

user19161
@BenjaLim Maybe the downvoter is sick of your Lie questions, so you must go from Lie to Lee!

I don't know why they downvote, I don't exactly post low quality questions.

11:04 AM
@WillHunting Very few people can.

user19161
@JohnJunior It depends on the exact nature of the condition and its severity, no one size fits all...

Someone upvoted my questions?

@WillHunting Yes that is true, the size of one's heart does not fit all.

One day I must find out what Lie groups are all about

user19161
@JohnSenior It is a manifold with a group structure on it such that the group operations are continuous with respect to the topology of the manifold, duh!

11:09 AM
@WillHunting Yep - just looked them up on W|P :)
Hmmm Chromium just seg-faulted :(

user19161
@JohnSenior This Chrome is really bad at multiple chat tabs, so I am back to FF.

@WillHunting Yeah - might do the same, but I like Chromium's feature of sync-ing my bookmarks when I reinstall

user19161
@JohnSenior The ONLY reason I can think of for using Chrome is the flash it comes with.

Hmm - way too many books - I've spent 2 hours sorting books and papers this morning :(

@JohnSenior Burn them... burn them!
Or trade them for whisky. You don't need books man.
Only makes you look intelligent, that's all 8-).
Erdos seemed to have only one.

user19161
11:24 AM
@jonas I will get Lee's Topological and Smooth books this year. My friend is returning from the US so free shipping for me!

I always have free shipping.
"free"

@JonasTeuwen I ought to start a resolution that everytime a bring a new book into the house, and old one has to go out :)

Fahrenheit 451?

@JohnJunior Yes.
@JohnJunior Give them to somebody that can actually use them.
For example, poor talented students.

user19161
My friend is a genius bro. He just finished his post-doc.

user19161
11:26 AM
He does algebraic statistics which uses algebraic geometry for statistics problems.

So...?
Too many crazy.

user19161
Nothing.

I go out for a walk! It seems to be fine outside.

user19161
A Walk to Remember.

Yep.
Have a walk too bro.
Good for you.

11:29 AM
There was a time in education that there were no lecturers.

user19161
@JonasTeuwen Maybe I will walk in Heaven soon...

@WillHunting First walk outside.
Practice, practice.
And don't you dare to die PhDless!

user19161
@JohnJunior I met some very good and some very bad lecturers in uni.

We ALL have.

user19161
Some of them are so bad I wonder how they got the full professorship.

user19161
11:32 AM
Their papers must be full of errors as well.

Some of them were so good I wondered why they were wasting their time lecturing.

user19161
@JohnJunior I believe that people really good in research will lecture well too. Those who don't lecture well are probably a little good in research only and they got lucky.

I don't think luck makes a man wise.

user19161
@JohnJunior But it is true that luck plays a significant part in life...

user19161
Hard work yes, but luck too.

11:42 AM
There are no royal roads to geometry.
There is no royal road to logic, and really valuable ideas can only be had at the price of close attention.

12:07 PM
hi all

@ajay hi

how does probability relate to luck? continuing on the current discussion :)
luck, fate, destiny, chaos, probability ...............

@ajay Probability measures luck.
Where measurement refers to the value of a quantity when that value has been ascertained by counting, measuring, or estimating.

Hello!

12:22 PM
$\sqrt{-49} + \sqrt{-64} = 15i$ ?

yes

Hehe. thanks
$i^2= ?$

right
I wonder SE chat does with FooPlot links.
Nothing. :'(
Still nothing
@JohnJunior Does it only work with wikipedia?

12:43 PM

@JohnJunior Ah ok.

and within stackexchange

Do you know if there is any documentation on this?
Ah!
"Some links will be automatically inlined if posted on a single line by themselves, such as:

Stack Exchange questions,
Chat messages and rooms
Images
Amazon products
Github gist"

Good job :)

12:51 PM

@JohnJunior Haha... Are you that old?

@DantheMan They only took this song away last year from Monday Night Football?

@JohnJunior Hahaha. Ok. I don't watch football. xD

But, yes it is an old song :)

user19161
@JohnJunior Back from dinner.

12:57 PM
@WillHunting What did you have?

user19161
Chicken, egg, vegetables with rice.

Which came first the chicken or the egg :-D

user19161
Maybe the rice. :-)

Lol! This post is insane.
Wrong one. sorry
140

As per Jeff's suggestion here in the comments. You can use this question as a formatting sandbox (if you can edit CW questions) and you can post answers if you want to test out formatting there as well. Beware that since the changes to syntax highlighting in December 2010, and the inline hints ...

1:32 PM
@DantheMan Seriously.

Hey guys, here is my feature-request
2

"Oneboxing" is a great way to share content quickly in chat. It would be awesome if it would work with FooPlot (an online graphing calculator), that way we could share graphs in the Math SE site without having to screen shot, crop, and upload it. It would also be helpful if this could be implemen...

I like the idea concept :)

:) upvote it

I got $\bar n=(6,3,2)$, how can I check whether it is right?
I used $\bar n = (\bar x-\bar y)\times (\bar x -\bar z)$
(then normalization to get $\hat n$)
$\hat n = \frac{1}{7}(6,3,2)$

user19161
1:50 PM
Hello @tom, I see you in two rooms at once.

@WillHunting Forever & always! :D
And not just 2.

user19161
@JohnJunior That is 5^2.

Indeed a power of 5.

Could someone explain normal-vector in n-dimensional hyperspace?
Actually, is it just nabla?

1:54 PM
Would that make it hyper-normal?

So if I can somehow create an implicit form for the plane, actually $\bar F=\hat i +2\hat j+3\hat k$.
so $\nabla F = a (1,2,3)$ where $a\in\mathbb R$.
But it is not the same as $\hat n = \frac{1}{7}(6,3,2)$ so some mistake?!

When was maths.SE opened?
What year?
I know it must be between 2000 when SE started and 2011 because I remember it being around last year.

Moved the normal-vector -question here.

@alan2here The public beta started on August, 27th, 2010, according to Area51

@DantheMan Thanks

2:11 PM
@alan2here No problem

Hi guys

Could someone tell me what this:
$$\frac{f(b)-f(a)}{b-a}$$
Represents? Is it the angular coefficient of a line?

I have no idea, sorry

2:38 PM
It is the formula for the slope of a function f.
If your function happens to be a line then it could be thought of as "the angular coefficient of a line."
@unNaturhal Are you familiar with what "slope" means?

@JohnJunior Slope? Never seen before :p

@unNaturhal How about "rise" over "run"?

@JohnJunior I'm not enough familiar with english terms of mathematic :/

@unNaturhal How do you calculate "the angular coefficient of a line"?

2:51 PM
Mmmmh.. with a derivative of a function (if it is a line), or a specific point of that function (if it is a curve). Or with this:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

@unNaturhal OK, in the numerator, the part above the line, substitute f(b) for "ysub2" and f(a) for "ysub1" and in the denominator, the part below the line, substitute b for "xsub2" and a for "xsub1."

What negative number is equal to its reciprocal?

haha

@JohnJunior Oh.. right xD You're right, I'm sorry :)

2:58 PM
@DantheMan What positive number is equal to its reciprocal?

@JohnJunior 1?

Yes, don't question it. Work it out :)