Mathematics - 2015-05-26 (page 2 of 4)

I was having trouble with understanding how $\text{Hom}_{\sf Ab}(G,H)$ is an (abelian) group (what if there's only one homomorphism between the two groups?) until I realised that a group can have one element.

:(

In my defence, the proof used two, and I never noticed that they're free to be the same.

bjb568

Dammit, fluxions sounds so much more kewl than calculus. We should bring back the fluxes. We need flux capacitor.

Soham Chowdhury

@bjb568 Newton, on seeing Gabriel's Horn: "What the fluxion?"

bjb568

:p

Soham Chowdhury

The starboard is impressively oligosyllabic today.

(word?)

Paul Plummer

3:05 AM

Wrong problems are probably the most annoying. I as recently looking at 2 and 3 dimensional topology by Moise, and he mentions that many propostions are stated as if they were true, and part of the problem is to figure that out. Maybe all textbooks should do that so there is no need for errata in the problem section

bjb568

What's oligosyllabacism?

Soham Chowdhury

Having few syllables, I guess.

Oh.

I made it up.

I see.

3:06 AM

It exists!

Woohoo!

bjb568

∃ teh wordz

user147690

$\exists $ oligosyllabic

Soham Chowdhury

"Of three or fewer syllables, as a word; trisyllabic, disyllabic, or monosyllabic: opposed to polysyllabic."

user147690

Hey @Rem

bjb568

Hm… should I bother to install the latex?

Remember me

3:07 AM

Hello@AlexClark

Clarinetist

**Theorem**: let $\{o_i\}_{i \in \{1, 2, \dots, r\}}$ is an orthonormal basis for the column space of a matrix $X$ and let $O = \begin{bmatrix}o_1 & o_2 & \cdots & o_r\end{bmatrix}$. Then $OO^{\prime} = \sum\limits_{i=1}^{r}o_io^{\prime}_i$ is the perpendicular projection operator onto $C(X)$.

In the proof, they say $OO^{\prime}OO^{\prime} = OIO^{\prime}$, $I$ being the identity matrix. How do I know that $O^{\prime}O = I$?

user147690

@Soham I have to do some assignment work now. More Aluffi later.

Soham Chowdhury

@bjb568 Yes, you must.

@AlexClark Okay, later then.

bjb568

Agh, is there a userscript?

Soham Chowdhury

Indeed.

user147690

3:08 AM

@bjb568 It's a bookmarklet on the top of the starboard on the right. $\LaTeX$ in chat

Paul Plummer

Read the star board

bjb568

bookmarklet ≠ userscript

user147690

Oh Paul's got picked up :'(

Remember me

I cant see the 🌟 board coz I am on my phone:(

user147690

14 hours ago, by Alex Clark

> Chat guidelines | $\LaTeX$ in chat

user147690

3:09 AM

12 hours ago, by Paul Plummer

Chat guidelines | $\LaTeX$ in chat

user147690

I should be famous( a star so to speak )

Paul Plummer

Lol

I did not notice you had posted, I was wondering why was no chat guidlines up when I woke up

bjb568

yay, script bound to command-1

Soham Chowdhury

@AlexClark, about how he distinguishes products and coproducts in categories like $\sf{Vect}$: It's sort of similar to how you're supposed to distinguish C# and Db, if you're familiar with a little bit of those things.

pjs36

Sadly, @AlexClark, I somehow got my star badge without posting that message... just by tarting around :)

user147690

3:14 AM

@pjs36 Oh I have had it on three accounts(including this) without needing that haha

pjs36

Oh yeah, I forgot you were the account master!

user147690

@pjs36 Hahahaha

Paul Plummer

$\square \!\!\!\! \checkmark$ Make @AlexClark realize everybody ignores him, by letting him do something, which everybody ignores, then I do the same exact thing, which everybody thinks is the best thing since sliced bread.

6

user147690

@PaulPlummer Hahahahaha

bjb568

☑ this is a better checkbox

Soham Chowdhury

3:16 AM

LaTeX > emoji brah

bjb568

unicode > LaTeX

dammit… less than and greater than are basically the same thing, right?

Soham Chowdhury

:(

user147690

@SohamChowdhury <---this

Soham Chowdhury

@bjb568 LaTeX $\checkmark \left(\frac{10}{10}\right)$

user147690

@bjb568 What? Oh haha

bjb568

3:18 AM

(this)

@SohamChowdhury 11/10 with rice?

Soham Chowdhury

Redditor!

bjb568

:D

Soham Chowdhury

faux-romantic music

bjb568

elevator music, just because

user147690

I remember that thread

user147690

3:19 AM

But are you then following the button?

bjb568

Nah, already red and satisfied.

I thot about going grey, then I stopped thinking about going grey.

user147690

@bjb568 What number though? Better not be higher than 1

Soham Chowdhury

@AlexClark Non-presser FTW

Paul Plummer

You just have a bad font

user147690

@SohamChowdhury Me 2, I want a 0 or a 1

Soham Chowdhury

3:21 AM

Meh

bjb568

user147690

I will get a 0 or a 1 tonight, since we are down to 280ish users at 8pm where I am from

Soham Chowdhury

I haven't reddited (or been on FB or HN or wherever) in a week

Paul Plummer

bjb568

@PaulPlummer What kind of sorcery is this?

Soham Chowdhury

3:22 AM

Discipline level = pro, @AlexClark

Hah!

user147690

This is what I see imgur.com/LDFiiWa

Paul Plummer

The people who work on $\LaTeX$ are doing the lords work, you shame them by using unicode when you can use $\LaTeX$. @bjb568

2

Soham Chowdhury

WWKD: What Would Knuth Do?

user147690

@bjb568 Filthy presser

Soham Chowdhury

@bjb568 Filthy presser

user147690

3:25 AM

It literally goes lower than 10 most minutes these days

Paul Plummer

That looks really good on your computer @AlexClark

Soham Chowdhury

He probably has a Retina display.

Not sure.

Nope, he doesn't.

bjb568

Paul Plummer

I don't think the display effects screen shots

bjb568

All is Unicode.

Soham Chowdhury

3:26 AM

@PaulPlummer Resolution does.

Paul Plummer

Sure

Soham Chowdhury

That's what I meant

user147690

I have a really nice resolution

Soham Chowdhury

Anyhow, I'll get back to the page that I've been stuck on for half an hour.

user147690

Soham Chowdhury

3:27 AM

@AlexClark OMFG lol

user147690

Haha

N3buchadnezzar

sigh

bjb568

user147690

@SohamChowdhury It's just the uni computers haha

bjb568

I need a 5k display.

Soham Chowdhury

3:31 AM

^ Classic Redditor

Paul Plummer

A 5k display to look at all your precious unicode? @bjb568

2

bjb568

Hey, when you're coding, you need the best equipment! Wait what, silence you, I'm not just using a text editor, it has colors.

Soham Chowdhury

Vim is love, vim is life

bjb568

sublime > vim

sublime is just sublime

Soham Chowdhury

lol

Aaron Hall

3:32 AM

Is there a good intro to real analysis?

Soham Chowdhury

vim is text surgery

bjb568

I realized recently sublime is an english word. English is weird.

@SohamChowdhury and I do surgery like a kid on surgeon simulator…

Paul Plummer

I am sure there is a question on stack exchange asking exactly that @AaronHall

user147690

@PaulPlummer So is 'field norm' common notation to refer to taking any $a+b\sqrt{D}$ to $a^2-Db^2$ via the standard $(a+b\sqrt{D})(a-b\sqrt{D})$?

Paul Plummer

Not sure @AlexClark

Aaron Hall

3:35 AM

well why didn't I think of that... on SO it's considered off topic to ask for resources, but I've been seeing a lot of those types on the other SE sites...

bjb568

Goodnight! I'm going to go sleep with my unicodes…

user147690

@bjb568 Night :)

user147690

@AaronHall Yes, we are nicer here

Paul Plummer

@AlexClark Should we be though...

Aaron Hall

top 3 google hits for my question are SE, top 2 are Math, 3rd is MathOverflow

Soham Chowdhury

3:36 AM

I gave up on SO years ago. It's near-impossible to ask there nowadays.

user147690

Oh I finished a chapter of D&F hmmph

Soham Chowdhury

@AlexClark Now? Assignment purposes?

user147690

Yep

user147690

Sorry a subchapter

Paul Plummer

@AaronHall Here is a good place to start math.stackexchange.com/questions/62212/…

user147690

3:38 AM

431
Blocked malicious login attempts

user147690

Holy crap

Soham Chowdhury

?

user147690

Better strengthen the password

user147690

For my website

Soham Chowdhury

Someone's into you, @Alex.

Aaron Hall

3:39 AM

that's the same book Math Overflow recommends

Paul Plummer

Stephen Abbot?

user147690

@PaulPlummer Oh I have that book(in pdf

user147690

Seems really nice

user147690

Probably run through that and then hit rudin's principles of mathematical analysis

Mike Miller

@SamuelYusim: enjoy Kevin's answer

Paul Plummer

3:40 AM

It is a nice book, I remember for my first course we had a book I didn't care for, then future classes were using that book and I thought it was pretty good (although I never really looked at it in detail) @AlexClark

user147690

My password is now somewhere between 16-32 characters long(won't specify since it makes bruteforcing really easy[in comparison] :P)

Samuel Yusim

I will, it looks interesting

Aaron Hall

ok, I ordered it.

thanks!

Paul Plummer

Hmm, I just got a good idea for passwords, I can have the same password but I hash it in different ways every time, so I can just copy and paste, or figure it out by hashing the pass I already know, instead of going through this buisness of having a million different passwords @AlexClark

user147690

Damnit I can't wait until I TA for that textbook money

user147690

3:43 AM

@PaulPlummer I just have passwords that are based on nice hand motions

Aaron Hall

Did you guys download that 10M username/password set?

Mike Miller

@SamuelYusim he facebooked me with a link to it saying 'oh god what have I done'

it's a good answer though

Paul Plummer

@AlexClark I have some nsfc guesses for your passwords :D

Clarinetist

Wow a lot of stars in the last 25 minutes

Mike Miller

people star too much

10

user147690

3:49 AM

@PaulPlummer not safe for copying?

Paul Plummer

chat

user147690

Oh

Samuel Yusim

wow that answer was hype

user147690

Shoot away I suppose, no way you could guess

Paul Plummer

Oh got into wordpress

user147690

3:50 AM

LOL

user147690

It has 7 symbols for the record

Mike Miller

ugh

Aaron Hall

ok guys, gotta hit the hay, or my wife will fume

user147690

@AaronHall Okay, night

Aaron Hall

thanks again!

Paul Plummer

3:52 AM

Why would she be mad if we don't hit the hay?

:P

user147690

@PaulPlummer Are you actually trying to log in?

Paul Plummer

No

It is sort of funny, it is more convenient for me to go to my blog, then the blogroll in my cite, to find yours, than it is to find yours in my bookmarks @AlexClark

user147690

@PaulPlummer Hahaha, I just go through my blogroll to all my blogrollees also

user147690

Is that a picture of you in your header btw @Mike

Mike Miller

haha

no

Paul Plummer

4:02 AM

It is a picture of me...

:)

user147690

@MikeMiller Was it with the template?

Mike Miller

no

user147690

Is it a musician from the 80s?

Mike Miller

closer

user147690

"The Man Who Fell to Earth"

user147690

4:04 AM

An actor from a 1976 movie who is acting as david bowie?

Soham Chowdhury

@AlexClark god, I'm going to be a poor student in two years, too :(

Mike Miller

no, that's david bowie acting, as actors do, in the film "the man who fell to earth"

fantastic movie

Soham Chowdhury

Paul, what's your blog?

Paul Plummer

mathsalad.com

user147690

@MikeMiller Oops, I thought he was solely a musician

Mike Miller

4:05 AM

mostly yeah

Clarinetist

Is anyone willing to guide me through the proof that $C(XX^{\prime}) = C(X)$, $C(X)$ being the column space of a matrix $X$?

Mike Miller

he's been in a couple of movies; that one's the only one where he did a fantastic job

he was coked out of his mind which made him rather alien to the rest of humanity during the period; as this fit the character, he luckily didn't need to do much acting

user147690

@MikeMiller Oh, very interesting

Samuel Yusim

4:37 AM

@Clarinetist isn't that not true if $X' = 0$?

by which I mean a matrix of all 0's

Clarinetist

@SamuelYusim My book says this is true for any matrix. Hmm

0

Q: Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$. This answer suggests using singular value decomposition. So sure, $XX^{\prime}$ is symmetric and thus $$XX^{\prime} = PD\left(\lambda_i\right)P^{-1}$$ for some orthogonal $P$. Since $P$ is ...

Samuel Yusim

oh, you use $X'$ to mean the transpose?

Clarinetist

Yes, sorry

Samuel Yusim

that's fine, I just hadn't seen the notation before

Clarinetist

Yeah, it seems prevalent in econometrics and statistics

Soham Chowdhury

4:44 AM

Hey @Clarinetist

Clarinetist

Hey @SohamChowdhury

Soham Chowdhury

Does anyone know why free groups are called free?

Samuel Yusim

because there are no rules imposed on products I suppose

Mike Miller

free to do whatever they want

Samuel Yusim

I like that there's an isomorphic copy of the free group on 3 letters inside the free group on 2 letters, that's sort of cute

Clarinetist

4:48 AM

What does $X^{\prime}$ usually mean, if not the transpose?

Soham Chowdhury

I've been stuck on the last page of the section on free groups for a while now.

@Clarinetist Just another variable.

Clarinetist

Oh I see

Samuel Yusim

the notation I've seen is $X^T$ or $X^t$, not that it's a big deal

Clarinetist

I think what baffles me most is how much I'm struggling to finish some of these proofs

It's been a welcome change compared to actuarial work, but dang, quite a struggle for me

Samuel Yusim

welcome to math

Clarinetist

4:53 AM

More like welcome back :)

Samuel Yusim

this would be a good time for a thumbs up if the chat could to that

so pretend I gave you one

Soham Chowdhury

$\checkmark$

is close enough

Clarinetist

Hmm, I wonder if there is a way to prove $C(XX^{\prime}) = C(X)$ without using SVDs

and without using ranks

I suppose I could try containment

Samuel Yusim

my first thought, not that I've worked it out, is to expand out the product into a big mess and from there show that every column of $X$ is a linear combo of columns of $XX'$ and vice versa

Clarinetist

$C(XX^{\prime}) \subset C(X)$ because... you're basically taking a linear combination of the columns of $X$ by multiplying it by something after you look at the column space of the resulting matrix

Samuel Yusim

4:57 AM

sounds too fuzzy

Soham Chowdhury

@Chris'ssis if you can, try $\int_0^1 x^{-\Gamma(x)} \text{d}x$ when you come on

Clarinetist

@SamuelYusim I considered that at first... might want to try it again :P

If $Z = [a_{ij}]$, $$ZZ^{\prime} = \begin{bmatrix}
\sum\limits_{k=1}^{p}a_{1k}^{2} & \sum\limits_{k=1}^{p}a_{1k}a_{2k}& \cdots & \sum\limits_{k=1}^{p}a_{1k}a_{nk} \\
\sum\limits_{k=1}^{p}a_{2k}a_{1k}&\sum\limits_{k=1}^{p}a_{2k}^{2} & \ddots & \vdots \\
\vdots & \ddots & \ddots & \sum\limits_{k=1}^{p}a_{(n-1)k}a_{nk} \\
\sum\limits_{k=1}^{p}a_{nk}a_{1k} & \cdots & \sum\limits_{k=1}^{p}a_{nk}a_{(n-1)k}& \sum\limits_{k=1}^{p}a_{nk}^{2}
\end{bmatrix}$$

So now I gotta look at the column space of this thing

Soham Chowdhury

Woo, what latex-fu.

Samuel Yusim

I get the feeling this won't be nice

Clarinetist

@SamuelYusim Yeah, hence why I gave up on that approach :P

@SamuelYusim This book I have basically says that $C(XX^{\prime}) \subset C(X)$ is obvious. Any reason why it would seem that way?

Hey @KarimMansour

Samuel Yusim

5:05 AM

obvious? I'm having to stop and think about it, at least.

is it something to do with the decomposition you mentioned earlier?

Soham Chowdhury

@Chris'ssis sorry, I meant $$\int_1^\infty x^{-\Gamma(x)} \text{d}x$$.
Apparently converges to around 1.04078

Clarinetist

No idea @SamuelYusim

Karim Mansour

hi @Clarinetist

just had busy day with gf lol

tommrow doing physics all day and some algebra

Clarinetist

Nice. I've been doing linear algebra all day

Karim Mansour

cool

Clarinetist

5:10 AM

@KarimMansour Are you familiar with column spaces and SVDs?

Karim Mansour

yeah

what is SVDs I am familiar with column spaces

is it short hand for something

Clarinetist

Singular Value Decomposition

Karim Mansour

yeah

Clarinetist

Maybe you could help me with this one :P

Karim Mansour

I did that in my matrix theory class

Clarinetist

5:11 AM

1

Q: Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$, $X^{\prime}$ denoting the transpose of $X$. This answer suggests using singular value decomposition. So sure, $XX^{\prime}$ is symmetric and thus $$XX^{\prime} = PD\left(\lambda_i\right)P^{...

linear-algebra proof-strategy vector-spaces

Karim Mansour

tommrow I will look at it oke because its my bed time :)

Clarinetist

K 'night! :)

Karim Mansour

nights !

Paul Plummer

5:38 AM

It seems like (and this was one of my two plausible theories) that it is because the group is generated by an independent system of generators @SohamChowdhury

Soham Chowdhury

Do you have a copy of Aluffi with you right now? I figure I need a bit of help with page 77. I've been stuck on it for a long time.

If you're free, that is. :P

Paul Plummer

:) You can ask, (I am guessing it has something to do with free product and coproducts of $\mathbb Z$

Soham Chowdhury

I basically got lost when he says "Now for the general case."

How does he know to define $H^{\oplus A}$ as that subset?

That bit about $H^{\oplus A}$ having an operation induced on it and being a group needs a little bit of explanation.

Paul Plummer

So, like why not have it just every function $\alpha: A \to H$?

Soham Chowdhury

Yes, that's one.

Wait. Each $j_a$ corresponds to an $a \in A$, right?

I think I'm getting a sort of handle on this now.

Paul Plummer

5:55 AM

The idea is that every element is a combination of the generators, and from that perspective it does not really make sense to have infinite combinations. That is all that is seen from the structures perspective is finite combinations, so there is never a need to have infinite combinations (which is sort of what direct products are)

Although was that what was confusing you?

Remember me

Hello@Paul

Paul Plummer

@Rememberme Hello

Transcript for

$Mathematics$ Mathematics