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1:12 AM
smacks Thor for his tautological answer
the zariski topology is perversion and self-abuse. the strong boy will 'cut it out'
or girl
or anything in between
@TedShifrin what are your opinions on the aversion to choosing a basis unless absolutely necessary?
Same with metric, I guess.
i think a point is a pretty great example of a discrete space
when i put 'cut it out' in quotes i'm referencing a social hygiene pamphlet from the 1920s. specifically the 'outdoor life' page of thesocietypages.org/socimages/2011/10/15/sex-and-social-hygiene
a point is a wonderful example of a discrete space.
it may be prehistoric to say it out loud, but the zariski topology may hinder people's progress toward a vigorous manhood.
1:19 AM
@leslietownes what beef do you have with the second best topology?
Are there any algebraic geometers in this chat?
it's just a goofy topology. i saw a friend of mine introduced to it, and suddenly it was all he was doing. it's like heroin. not even once.
why not
what people ought to be paying attention to are weak and weak-star toplogies on function spaces. everything else is goofing around.
1:33 AM
@anakhro Depends on the context. As someone who likes to use moving frames in geometry, I obviously cannot be too strongly opposed to working with (moving) bases. In algebro-geometric things, we obviously don't want to use a hermitian metric to stay in the holomorphic/algebraic category. As I said, depends on context.
Does an algebraic geometer have an opinion on this:
I could never care about those, @leslie. You lose.
This isn't homework, but it's been bothering me a lot recently
the best topology obviously being the quotient topology
my zen master said that sometimes it is necessary to fight a losing battle. that's what i'm doing, here and everywhere.
1:35 AM
@user2154420 I do not think about algebraic geometry in terms of algebra, but I would say that geometrically Daniel Hast's comments are on the money. We're just lifting the projective variety to the affine cone and geometry is the same up and down.
he also said that the buddha taught us to joyfully participate in the sorrows of life. i do not include the zariski toplogy in that. i refuse to participate in it, joyfully or otherwise.
i do agree that it might be one of the sorrows of life.
Why are three- and four-year old posts all of a sudden becoming active? It's nuts.
i'm here for it. most of my posts are about 8 years old. give me those upvotes. when i get to 10k reputation, i'll become the mayor of long beach.
I just added a pedagogical comment to someone from 3+ years ago.
that's how it works, right?
1:39 AM
Upvoting doesn't make posts active, no.
Whether Long Beach is insane enough to elect you is a totally different matter.
Why did I just get invited to a chat in Theme Decisions?
our mayor is actually pretty good. i think i could be elected in my hometown, but they kicked me out long ago. i do know the DA there. i could kill people and not be prosecuted. but i don't because i'm a good person.
Oh, you're right. You do emulate Trompolini.
i'm having my bedroom done with gold leaf on some of the fixtures. it's great. saddam hussein would have loved it.
Only if you put a golden toilet in it.
we're starting with brass. i need more money from the russians
1:43 AM
Maybe brushed nickel is more your speed.
my wedding ring is something like that. it's brushed something. it was cheaper than the other metals.
it's a chromium alloy. i looked it up. it was definitely brushed.
This is why you're so fond of brush-offs.
it's been massively damaged by me using metal door handles. it scratches at a moment's provocation.
Not remotely cheap man's diamond.
it cost something like $120. i think anything relating to marriage is a scam so i got off easy.
my wife's ring was over my credit limit at the time, so she ended up buying it and i wrote her a check. true story.
1:54 AM
We won't ask her if you were worth the credit risk.
definitely not.
2 hours later…
3:57 AM
Q: $\sqrt{\ \ \ } $ hypercomplex ??

mickIn mathematics, hypercomplex number is a traditional term for (an element of) a finite-dimensional unital algebra over the field of real numbers. Examples are complex numbers, quaternions, octonions etc. Complex numbers and quaternions are associative, however non-associative systems like octonio...

4:38 AM
good evening.
hello @TedShifrin
mick, re your question, division algebras and related ideas freak me out. that the dimensions of known examples were the only examples was not proved until, like, the 1950s? and only using algebraic topology of spheres? frightening.
It's morning here!
Hi @Shobhit
4:43 AM
I need help with $\lim_{n \to \infty} n.\sin(2\pi e .n!)$. Since $-1 \le \sin(x) \le 1$, I thought this would diverge, but it's given that it's a finite limit, and the proof is very ugly. Can someone help with an "easy proof" or a helpful tip? Thanks
@Shobhit use stirling for factorial
math.stackexchange.com/questions/76097/… purports to solve this. the question relates to the proximity of 2pi e n! to multiples of pi.
which does implicate stirling's formula, as snapdragon notes.
i don't think you need too precise a version of the formula to evaluate the limit, or at least that solver doesn't seem to need one.
Good evening.
good evening, morning, or afternoon to all.
or brunch time. if someone's having brunch, happy brunch.
I'm heading straight to my glass of white.
4:50 AM
my favorite white varietal is gruner veltliner. the best ones are from austria. sometimes it tastes like lentils, other times it tastes like licking wet rocks. don't ask me how i know that. it's wonderful.
i'll add to my to taste list.
i have a preference for reds, but find they completely inhibit any mathematical thought and have the opposite effect on my vocal cords.
i like light, dry if white, pinos, etc.
got it, finally. thanks, @Snapdragon-X @leslietownes
my dad is similarly a red partisan. i grew up in napa, which is mostly known for its reds. i like wine that tastes like mineral water.
germany has so many great white wines, but they're mostly too sweet for me. and france goes all in on oaky whites. austria is the place to be.
to each their own. many years ago the was a wine glut in norcal and the spillover was taken and made its way into tjs two buck chuck.
chateau de I-5. i'm a fan of it.
value for money is what that is.
4:55 AM
i used to go, buy a bottle to taste, and then a few boxes if it was ok.
currently waiting on a case of 2016 Chateau Lynch-Moussas Pauillac from france.
i think they lifted the tariffs again.
no more of the EU strangulations? i get a commission every time i say EU strangulations on social media. i think this counts.
do not get the notion that i 'know' wines from the above. i find something i like and just buy it again and again
i want ireland to separate slightly from the eu.
but i don't live there so not my call...
i've only visited england, scotland, denmark, and sweden. all of whom i think opted out of the euro. they adopted the euro while i was in sweden and i still paid for things in kronor.
maybe it's different now. i wouldn't mind some euros in my bank account, frankly
yeah, i have a complicated euro setup. the whole money laundering thing really made life difficult for people who have lives in two countries.
ireland was a tax haven for a short while. i think they cut that off. there are still other jurisdictions.
5:01 AM
ireland still has some autonomy in that regard, which they should, i believe.
france & germany feel free to ignore rules when it suits.
i'm an attorney in real life and i've had moments where it's like, "i won't even ask why you're incorporated in the isle of man. i can guess. but anyway, if you put this money in the US the tax authority will know about it." that is usually the end of the conversation.
well, i am a good boy and do my fasb or whatever it is
that's the big somewhat legitimate knock against the EU. it's france and germany. we have the same problem in the states in our national politics, CA vs. TX and everyone else is along for the ride.
but i do feel that money i received from sources that have nothing whateosver to do with the usa should remain out of sight of the us authorities.
just on the lines that it is none of their business. taxes paid, etc.
yeah, i think the US is one of the only countries on earth that still taxes income from whatever source (perhaps subject to deductions). we want jurisdiction over everything.
5:04 AM
i think it hails from a more colonial perspective
eritrea does it, or did it.
it is not like i have lots of money stashed away
the philippines gave it up a couple of years ago.
colonies, all of them, and we were a colony. we should give it up.
5:06 AM
i have an issue with an irish account at the moment, but it is hard to resolve as a non resident.
such a pain.
there is a section of the code where you can exclude a fixed sum of money from income. but by excluding it you also can't get credit for foreign tax you pay on that income. which is, not great.
taxing software is challenging.
my daughter is wondering if she gets paid as an intern in the uk if she has to pay us taxes. its stupid that a 20 yo has to even consider these things.
certainly i did not at that age.
it's quite dumb. we advised a band once for touring the united states. they needed to set up a US shell to get their US-sourced income or it would be taxed twice. nobody is thinking about this. nobody should be thinking about this. least of all people who can do guitar solos.
it takes some time to realise that the usa is really a great place
if you are wealthy
and can afford to have people take care of all the details.
i'm trying to get to a point where i can have people take care of everything. that's definitely the rule. otherwise, good luck
5:12 AM
even little things like putting assets in trusts are complicated.
been a while, i have regressed for a variety of reasons. mostly i am an expert at finding financial potholes :-).
trust stuff is weird. in my own experience, attorneys who are involved in that area either serve the hyper-wealthy, or grift the non-hyper-wealthy. there's no middle ground where you can just pay normal amounts of money to get normal advice for a normal person.
immigration law is the same.
i'll just say it, the USA sucks.
yep, i can attest to that.
i had to call in a politcal favor to get my citizenship moving.
and they still lost my paperwork the day of my citizenship.
criminal law is also the same, which has all sorts of social justice implications. you get the world's best service, or you get some insomniac who can't remember what your last name is.
just bring a big bag of money if you come to america. things will go well for you.
that is true more broadly than the usa...
i mean the criminal law...
i do think plea bargaining is insane.
then again, jury trials...
i have never made it to a jury
it's the cheap way of clearing a case. there's basically nothing resembling a jury trial for most stuff. i mean, juries suck and i've seen that too. the USA is also weird in following english common law as a model, yet persisting with using juries in civil trials. most countries have dispensed with that.
5:18 AM
partly it is because it is such a huge country with many jurisdictions...
we did a trial once where our jury analysis firm had found one of the members selected to be a deranged lunatic. he had a website that was very anti semitic, and we and our clients were very semitic. but we liked him for other reasons. he was like a joker card. everyone else was going to be against us.
it was so divorced from anything resembling the merits of the proceeding. that's what i remember.
if i have an issue in ireland there re two things (1) i can probably find someone who knows & can help and (2) even the government people are human and tend to work with you if you are reasonable.
i am still trying to figure out how to get my daughter's passport renewed without calling a favor. they replied to her today (i asked her to email them), but nothing useful.
i think i would go mad in a jury.
my sister has had good experiences with state job assistance in california. it was as good as therapy, which she definitely needs but cannot afford. the people in the job assistance office were unbelievably helpful. and followed up.
that is good to hear.
whats with the bob hope thing?
i like his films. i realize he was a toxic person.
5:22 AM
:-). everyone has good & bad.
I'm purely good!
i love all of the 'road' movies, even the shitty ones. my true idol is WC Fields but he doesn't have as rich as a pseudonym.
i know a number of decent folks with decidedly questionable pasts.
when i take the DND test it says i am true neutral. neutral neutral everything.
@TedShifrin We know you are good :-).
5:23 AM
Ha ha.
my tastes, like my wine, are simpler.
he's guzzling thunderbird. that's what he's trying to say.
there's no shame in it
Even if it kills me?
Hellloooo @TedShifrin, so...going through some of the previously done chapters and I'm running into an issue expressing the linear transformation. So for direct reference it is example f, on pg 28.

I was trying to derive the matrix $B$. The one representing the reflection.
don't do it if it kills you. that's my motto.
5:25 AM
@TedShifrin My Artin arrived, will take a look shortly.
Oh great!
Go on, @dc3rd
i also like deriving matrices. i do not have example f in front of me.
It's geometric !
it is essentially a two dimensional sort of thing (not really).
So given that I do have the vector $(1,2)$, I was trying to get the expression for the points. Since I couldn't see what to do right away, I referred back to the basic expression used to represent a reflection in the line $y=x$, which is $(y,x)$.

Using that idea, I worked out the equation of the line for this set: $y = 2x$
5:28 AM
it is not quite a swap of coordinates in that case. although this is icky geometry. i feel befouled by it.
This being the case, I had determined that the "reflected corrdinates should be $(x,y) \to (y/2, 2x)$........
Where do the standard basis vectors go?
Reflect $e_1,e_2$ and see what you get.
but now writing this out again it doesn't seem right.
where the standard basis vectors go is absolutely the question.
the entire problem is just two problems. where do (1,0) and (0,1) go.
5:30 AM
I did use the standard basis vectors after I "derived" my expression. They ended up being: $(1,0) \to (0,2)$ and $(0,1) \to (1/2,0)$
Draw pictures. What part is along the line and what part is orthogonal?
You're not thinking,
I am screaming in agreement with Ted.
i limit my approval of all of this to the extent it does not implicate geometry or pictures. but a picture would probably very much help here.
Later we learn better ways to do this, but now you have to use projection and draw pictures.
Should I muzzle leslie yet?
i said it would help. you win. OK? you just win on this.
5:32 AM
i can't believe what i'm even saying, but yes, draw a picture.
its the only way really...
I have evil powers in here.
it might be. i think i need an appointment with my therapist.
5:34 AM
and my wife's therapist. she wants to meet me for some reason. i think it's just to increase her income and not to figure stuff out.
my therapist is rifle range road in tilden
Oh oh. Three, but not all my fault.
i saw a bobcat there once. i love wild animals.
So I don't know if my reflection drawing is shitty, but I drew a line and reflected it in the $(1,2)$ vector and it "appeared" perpendicular....is that the right idea?
you have my complete understanding if your drawing is "shitty." i have never done a non-shitty drawing.
5:36 AM
i was going to take a pic of my notebook where is try to find a nice geometric explanation for the det=ab-bc formula after a few glasses of wine. but its too embarrassing.
Do some arithmetic. Use projections.
it's less important that it be shitty than that it be accurate. test the picture with algebra.
get some graph paper or whatever its called in the usa.
Still isn't accurate.
projections are involved eh?, well I worked out matrix $A$ using projections so I'll work with this hint. give me 5
5:37 AM
forget about the matrix
find where the basis vectors go.
you might even just draw the point and try to see what it seems to correspond to.
this is profanity from me. it's impossible. i can't believe what i'm saying.
pictures are beautiful.
not even concerned with the matrix @copper.hat, I know how that comes about. I wanted to derive the explicit linear transformation and then verify it with the standard basis vectors
5:38 AM
would it be math porn to redo rudin with pictures?
Rudin After Dark
Rudin Nights
i'm here for it
then you need to think of parallel and perpendicular to the $y=x$ line.
can (or rather, should) we learn something from the documentation?
I sometimes find the documentation hard to understand
@dc3rd let $v_1$ be a unit vector along the line $x=y$ and $v_2$ a unit vector perpendicular to that. Then $v = \langle v_1, x \rangle v_1 + \langle v_2, x \rangle v_2$. What will $Lv$ be, where $L$ is your reflection?
that was the funniest thing about the admonishment to RTFM. which i haven't seen since the 90s but i presume people still do. nobody thinks that TFM might be written by someone who is not shakespeare. putting it mildly.
5:47 AM
it used to be that clive molers matlab (fortran back then) would reply with RTFM if you typed help.
Why $y=x$? We're reflecting across $(1,2)$.
my apologies, i thought i saw $y=x$ somewhere.
this is a problem i deal with all day at work, overlapping threads of conversations, inevitably some stuff gets lost...
i really like my office's IT people. once i asked them something, and it was not documented, and they said "we're asking microsoft about this." like grown ups. RTFM is edgy but doesn't get you very far. and i understand that RTFM is a proper response to 99% of requests.
@copper The $y=x$ example is done in the text.
sadly math.SE seems to mostly be RTFD (definitions) these days.
5:51 AM
@TedShifrin i see, sry about the noise.
but that is what it is, i don't need to stomp on people.
i find, when dealing with customers, it is faster for me to read the manual and tell them.
besides, time is money :-).
echos of brendan behan.
guy was a genius.
behan comes back all of the time, whether you want him to or not.
emerson, mark twain, etc. gems.
behan was not house trained.
that would be oscar wilde.
we have clients who sometimes tell us, "we get that, but it needs to be this way." and it's like, OK, that's the world of make believe. that's where mr. rogers had the puppets. come back to reality, we do not lie in court for our clients.
dylan thomas had some good ones too.
5:54 AM
ah yes
Tromp's lawyers lie in court all the time .
customers always want the impossible.
Sorta like parents of college students.
alternative reality they call it, i think.
trump was just a nasty person.
i know a guy who was appointed to a lifetime judgeship by trump. he is, surprisingly, not an asshole. trump's friends will be repeatedly disappointed.
5:56 AM
Still is.
the problem is that everything he did is tarred by his obnoxious persona.
he turned everything into a vote on how much you loved him. that was his toxic essence.
public health, well, how do you feel about trump?
still, he had a few followers.
one time when i was teaching the parent of one of my students left a nasty message on my answering machine after a midterm grade. implying that it would not go well for me if i didn't change it. i was on the last year of a three-year postdoc. it was a weird and shitty threat.
wow, the parents got involved???
they left a recorded message?
5:59 AM
yeah it was the dumbest thing imaginable. i knew the dean, he was basically the person who hired me.
Damn yes. The new era. Heard of helicopter parents?
we giggled about it a little and hit the delete button.
i am one, but i am certainly not getting into my kid;s college grades.
here's a sonnet i like as we leave winter. romantic-circles.org/editions/poets/texts/emmonsail.html
if you read it out loud, it's simply wonderful.
its ornithologist porn.
6:02 AM
yeah i'm a birder. you spotted that.
There was a young man
From Cork who go limericks
And Haikus confused
Spotted wren.
i was attacked by an ostrich once.
my daughter's middle name is wren.
an ostrich kicked me once, too.
they can really kick.
ostriches are athletic XD
6:04 AM
well, to be fair, i was holding on to its neck.
Question: Should I be projecting onto my vector$(1,2)$ like in matrix $A$ @TedShifrin?
two of my brothers wanted the middle name of their children to be Danger.
i was just trying to vibe with the ostrich. i was doing a report on ostriches for my fourth grade class. my dad knew a guy who had an ostrich farm, we went there, obtained pieces of ostrich eggshells, and met an ostrich. who was anxious and kicked me.
i still have a pic somewhere of me with the ostrich, post-kick. we made friends.
i got whacked in the head by its beak. it hurt.
scanning back to make amends for my earlier misleading comments
i got marked off a little bit because in my ostrich report i drew an ostrich in a way that suggested it had four, and not two, legs. i think my fourth grade teacher was a little persnickety about that.
i later met his brother at a math conference, and we had no idea who each other were, and then narrowed it down, "wait, is your brother the guy who teaches fourth grade?" "yes." we're all friends now.
6:08 AM
i can't find the original question so i can't make amends.
ostrich is tasty.
anyway it was weird that my fourth grade teacher turned out to have a brother who was a functional analyst that i separately met. functional analysis is not a large world.
eating an ostrich burger was part of my report. it was delicious.
at some point my daughter and myself (elementary school time) wanted to make a one egg omelet for her friends (but not tell them it was an ostrich egg, of course).
how did you get into mathematics from law or whichever came first?
the nearest ostrich egg was gilroy, but that was too far for a single egg.
andronicos used to carry them at some stage.
i started in math. some of my friends from undergrad had gone from math BA to law and seemed OK with it. i needed a way to make money. my family kind of fell apart while i was doing a postdoc and i couldn't do anything for them on postdoc wages. and it was just, how do i get some money.
it worked. it might not have been best for my mental health, but it worked.
$ > mental health
one you need, the other is optional
i basically had to quit academia because my mother might have been dying and i couldn't afford a live-in nurse. the cash grab did work. we're all happy now.
6:12 AM
what is an example of a ring wu
@dc3rd Are you reflecting in the line through $(1,2)$?
whose zariski topology is trivial?
the strong boy will "cut it out"
Yes @copper.hat, that's the line I'm supposed to be reflecting across
medical is tough here.
@dc3rd let $v_1 = {1 \over \sqrt{5}}(1,2)$ and $v_2 = {1 \over \sqrt{5}}(-2,1)$. Write $v = \langle v_1, x \rangle v_1 + \langle v_2, x \rangle v_2$. What does $Lv$ look like?
6:16 AM
my mom was a nurse when she was able to work. she dealt with a lot of death and maybe was too direct with me about what she was seeing. in the late 80s and early 90s a lot of people were dying of AIDS in our area and she had to deal with extremely negative family situations. it's probably some form of trauma
the important thing is she has my amazon account for groceries and we're all just waiting out this pandemic now
$Lv = <v_{1},x>Lv_{1} + <v_{2},x>Lv_{2}$.
@dc3rd Yes, but how does $L$ operate on vectors parallel & perpendicular to the line in question?
yeah, my mom passed last november. luckily i was at her bedside at the time. no mean feat given that she was in a home and the whole quarantine/covid thing going on.
i can't imagine not seeing people anymore, through this pandemic stuff. i want everybody to make it through to where i can see them again.
sorry, i made some sort of connection there and for the life of me cannot imagine what it was now!
oh, yea the amazon account.
@copper.hat I think that is what I'm trying to "see" either algebraically or geometrically. I know it will be similar to the regular $y=x$ case, but not completely the same.....
6:21 AM
my 90 year old god mother just got her vaccination today, i am very happy to say.
@dc3rd How would $L$ map the vector $v_1$?
that's wonderful. 90 years. i wouldn't wish that on anybody but i like hearing about it.
she's in great shape (for 90)
still drives.
it's similar to the y=x case in the sense that the picture looks the same. i hate saying this, but have you drawn the picture?
One of my neighbors is 94 years old
she is unmarried. at 70, one of her exs proposed again. my sister asked her why, my aunt replied "i'm waiting for the right person".
6:24 AM
@leslietownes, yes. It is very similar to it. So let me work up to this scenario......In the $y=x$ case $L$ would map $v_{1}$ to $\frac{1}{\sqrt5}(2,1)$
And $v_2$?
my grandmother had a very unusual mental lapse. she had alzheimers. it affected her ability to drive. she could still sit in the driving seat and drive around and whatever. but if you put her in the passenger seat she could not attach her seatbelt. sidedness, orientation, handedness, they're nontrivial things.
so here's a heck of a workaround
dementia is painful for everyone.
my collaborator uses keynote, and i'm trying to dig through their stuff to get started on an upcoming presentation
but i have windows, not mac. luckily i have an iphone and thus access to icloud. so i can edit the presentations via keynote for icloud
that's not the workaround
6:26 AM
$v_{2} \to \frac{1}{\sqrt{5}}(1,-2)$
i don't like keynote. a guy i work with does that too. can you just email me what you want? no, it's somewhere in keynote.
the obstacle is that i can't figure out how to copy slides from one presentation to another within icloud
searched around a bunch, nada. it's trivial in keynote for mac but nada for icloud
...but, iphone does have keynote as well :)
my five year old cell phone crashes if i get a text massage while i'm on a phone call or vice versa.
sorry, not appl here. not since tim cook refused to let my company put the aapl logo on our website ;-).
6:28 AM
we're not having keynote.
so i get on my phone, open up keynote, copy/paste the slides within keynote on my phone, and edit them on icloud
and i'll have to do that every time i want to copy/paste from one presentation to another. the shortest distance between two presentations on icloud is through my iphone
productivity tools.
@dc3rd think in terms of $v_1,v_2$ Where does $Lv_2$ end up? Visually.
writing out the diagram right now @copper.hat
Note that $v_2 \bot v_1$.
and I see why it was a "nope for the second one...
6:30 AM
huh, didn't know $\bot$ was upsidedown $\perp$
a drawing is always my first step.
is \bot just \perp
i just can't get myself to type perp
if i'm being entirely honest, a drawing is also my first step.
then what was i trying to remember as upside-down perp
right, \top
...\bot is just \top flipped. of course
6:32 AM
i always struggle to place stuff above & below arrows.
@dc3rd you re thinking or drawing too much.
tex just sucks for that stuff. or mathjax or whatever.
Homework spelt backwards is krowemoh
which means child abuse in latin
i find that hard to believe.
defintiely the thinking......Ted's here to blast me...looking at it $v_{2} \to \frac{1}{\sqrt{5}}(-1,-2)$
@LeonhardEuler I verify the truth of the first of your statements. we may need to discuss about the latin.
6:36 AM
@Semiclassical So is \top.
@dc3rd $v_2$ is perpendicular to the line through $(1,2)$.
Where does your reflection end up in the mirror?
yeah, i got mixed up
@copper.hat the reflection in the mirror is really telling me $\frac{1}{\sqrt{5}}(1.-2)$....
6:45 AM
let me see what your image says..
you don't really need specific number here.
that is why i have $v_1,v_2$.
actually I can't even access your image....I put up productivity blockers so I can't acces imgr at the moment
@dc3rd What you need to realize is that the part of a vector parallel to $(1,2)$ stays the same, and the part orthogonal to that reflects, i.e., turns into its negative. Do you understand that?
That's what reflection in a mirror does.
@TedShifrin yes I understand that explanation.
6:49 AM
So what is $Lv_2$ in terms of $v_1,v_2$?
So the part parallel to it is the projection. What's orthogonal is what you get by subtracting.
@Copper is asking you to rephrase what we just said.
His approach is ultimately the right approach when we get to chapter 9 (although you may have gotten there in other books).
using inner product coefficients to express things. I've seen it and vaguely remember its workings. But for now it is this foundation I'll work with
You know enough to follow that through now; it's a fancier way of doing what I said a second ago.
I really think the picture will help.
Just use projection and find what's left over (in @copper's $v_2$ direction).
Of course the picture helps.
Even leslie admitted that an hour ago.
6:56 AM
Good night folks!
Night, @copper.
01:00 - 07:0007:00 - 00:00

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