What is a magnet ? Im confused

@Alessandro: Good! :) Even easier, take an orthonormal basis $w_{k+1},\dots,w_n$ for the orthogonal complement of the span of the $v_i$'s. Apply the determinant formula you knew in the first place to $v_1,\dots,v_k,w_{k+1},\dots,w_n$.

Srr off-topic maybe

$A {-1}$

$A ^{-1}$

I have a linear algebra question (inverses): if A takes vector x to b (i.e. A x = b) ; then does it mean the $A ^{-1}$ takes b back to x ? Is this a good way of looking at it ?

Ah a magnet is $A^{-1}$ :)

Better to discuss that with Semiclassical when he's here, @mick. Go to sleep!

No :)

why ?

Not in general, @analyst.

so what is the general way of looking at it i.e. one without determinants

Only if you know $A$ is nonsingular (invertible). But if $A^{-1}$ actually exists, that's exactly what it does.

yeah sure

given A is non-singular

is it a good way to understand it ?

If you think of the original $A$ as representing a linear function $\Bbb R^n\to\Bbb R^n$, then the inverse matrix is the inverse of that function.

Yes, it's exactly right.

Just like exp and ln.

I wish a bounty was a garantee for a good answer

wanted to confirm this understanding as well

Well, singular would mean that for some $b$ there is no solution at all, and for other $b$ there are infinitely many solutions (a line, probably not through the origin).

@analyst by 1. You mean in geometry or what part of math ??

linear algebra, @mick

@TedShifrin neat, that's much simpler

@analyst: If you're still doing Strang (or looking at my book), you'll get to constraint equations on $b$ for $Ax=b$ to be consistent (i.e., to have a solution).

@TedShifrin i thought the matrix was linear algebra ? ( so 2) )

@Alessandro: That actually was related to your original question about a measure on a manifold. This is in effect what you do.

I am just trying to ask:

where did the word singular come from ?

like what does it mean

invertible is mpretty intituive i.e. you can "invert" the operation

ah ... it means "singular" in the sense of "unusual" ... most square matrices, it turns out, are non-singular (i.e., invertible).

That "most" can be made precise (but I won't do it now).

I think it comes from singularity @analyst

@mick, what is singularity mean ?

@mick: That doesn't help. I gave the explanation.

@Ted

A singularity is an unusual occurrence.

@TedShifrin epxlanation makes "sense" for a novice like my self

K , i was just guessing

@TedShifrin, sounds like a sci fi movie word, like black holes are a "singular" event in the universe

My explanation should make sense, @analyst. I haven't proved it to you here, but it should make sense.

Yes, that's true, too ... something unexpected or unusual.

and we just listed a set of unusual events: like parallel lines, etc etc... right ?

No, your parallel lines is not quite making sense.

I think most means dense in the continuum with uncountable cardinality

You're talking about the row vectors or column vectors being parallel?

I'm talking only about solving the equation $Ax=b$.

@mick: By "most" I do mean that the exceptional ones (the singular ones) form a thin set, a set of dimension less than the dimension we're in.

That makes sense @TedShifrin

Yippee @mick :)

Now, seriously, go get some sleep ...

No

shrug

@TedShifrin, I am talking about in 2X2 matrix that both column vectors are parallel like $\begin{bmatrix}1 & 2\\ 3 & 6 \end{bmatrix}$

@TedShifrin I think I get the main idea but I'll probably ask you a couple more questions on this topic next time we're both online if it's not a problem, I'm going to sleep now though, thanks for your time and the interesting problem!

You are not my mom

Right, I said that up there ^^^^, @analyst. :) Not parallel lines, but parallel vectors.

Night, @Alessandro. :)

@Te

@TedShifrin, what is the difference gemetrically

Maybe Ted is my dad Trololol

The column vectors of a matrix are vectors, @analyst. All multiples of those vectors do give you lines (through the origin). In the case we're discussing, you do not get parallel lines. You get the same line both times.

so what is an example 2x2 matrix of parallel lines ?

That doesn't make sense :)

You can't do it.

Multiples of a vector give you a line through the origin.

What do you call the analogue of clickbait in a chatroom ?

So you cannot get parallel lines from what we're talking about. You get parallel lines by considering the solutions of $Ax=b$ for different $b$ when the matrix is singular.

$\begin{bmatrix}1 \\ 3 \end{bmatrix}$ is not through the origin, right ?

Im afraid Ted is right

All multiples of it form a line through the origin.

1x + 3y = 0 -> x,y = (0,0) is one possible solution . Ah I see !

but

No, no. Careful.

?

We're doing columns here, not rows.

This is the same confusion you had because of Strang a few weeks ago.

We don't get a linear equation from the columns. We get it from the rows.

So if I take $A = \begin{bmatrix} 1 & 2 \\ 3 & 6\end{bmatrix}$ and $b=\begin{bmatrix} 1\\3\end{bmatrix}$, the solutions are the line $x+2y=1$.

Hey there folks

Your lineair algebra is good @TedShifrin

But if I take $b=\begin{bmatrix} 6\\18\end{bmatrix}$, what line do I have?

@mick: I've written a few books. Seriously. :)

$\begin{bmatrix}1 \\ 3 \end{bmatrix}$

or 6 * $\begin{bmatrix}1 \\ 3 \end{bmatrix}$

No, it's a line parallel to the first line, @analyst.

Read what I wrote up there ^^^

Also note that if I take $b=\begin{bmatrix} 1\\2\end{bmatrix}$, there are no solutions at all.

Lin algebra books @TedShifrin ?

how is 6 * column vector parallel to orig vector ?

One, yes, @mick. And one mixing linear algebra and multivariable calculus.

amazon.com/Multivariable-Mathematics-Algebra-Calculus-Manifolds/…

How could I go about analysing the nature (convergent or divergent) of the series $$\sum_{n=1}^{+\infty}{\frac{1}{\sqrt{n^3}}\sin(\frac{\pi}{n})}$$

What do you mean, @analyst? When are two vectors parallel?

yes

When one is a scalar multiple of the other.

well before that:

@Bernardo: What do you think if you ignore the $\sin$ completely?

what do 2 col vectors represent ? given row represent lines

I mean geometrically

Rows only represent lines because you have an equation $Ax=$ something.

@TedShifrin Divergent b/c it fails the integral test

Really, @Bernardo? $\sum \dfrac 1{n^{3/2}}$ fails?

No, I'm stupid

:)

so geometrically, what do 2 col vectors represent ?

Wait

Let me see for sure

They represent vectors you will take linear combinations of. Remember, we talked about this, @analyst.

Multiplying your matrix by the vector $\begin{bmatrix} x\\y \end{bmatrix}$ takes $x$ times the first column plus $y$ times the second column.

yeah thats what they represent algebrically, i.e. combination of col vectors

but what about geometrically ?

That is geometric :)

ok

Linear algebra is about vectors as well as being about lines and planes.

@TedShifrin Derp, of course it converges!

@Bernardo: So then you use comparison, since $|\sin|\le 1$, right?

A more interesting question would be $\sum\dfrac1{\sqrt n}\sin(\pi/n)$. @Bernardo

what are they ?

@TedShifrin I guess it doesn't matter, since the sine is a limited function it will not alter the characteristic of the series, no?

Right. That's what comparison test tells you.

What do you mean, @analyst? They are, of course, parallel. But what do you mean?

@TedShifrin If I pass my analysis exam tomorrow I'll come around and give you some love

or some beer, I'm versatile

I wanted you to realize that the second $b$ gives you the line $x+2y=6$, which is parallel to $x+2y=1$.

LOL, sure, @Bernardo :)

Thanks for the hand!

Sure thing.

OK, I need to disappear now. @analyst: Work on this :)

@TedShifrin, I plotted both the lines x+6y = 0 and 3x + 18y = 0 and then it is the same lines

but not parallel lines, right ?

@TedShifrin that converges slowly

Why doesn't different form matter when two statements have an equality?

HI

Knowing that the side y = 0 of the plate 0 <x <a, 0 <y <b is isolated, that the side x = 0 is maintained at temperature u = 0 and that the sides x = a and y = b are kept at temperature u = Uo, determine the steady-state temperature u (x, y)

Someone know how assembly this equation?

@user32720 Sorry I don't. It might be a good idea to try the physics forums.

The first condition I'm sure

Honestly I'm a bit unsure of your coordinate system. Is the origin (0,0) at the center of the plate or somewhere else?

@user400188 The origin is on one side of the plate

Do any of the axis pass through the center if the plate?

No

@user400188 Maybe, I believe that need of the other condition in zero for the problem make sense

Gonna sleep

@mick Night

Which condition do you believe should be zero? And why would it make more sense?

Because this example:

@user400188 There are 2 conditions = 0 in one same variable

@JessyCat I found some time to answer your question. Let me know if the answer and the level of detail therein are sufficient.

@SteamyRoot I did not understand

@user32720 They were messaging another user.

@user400188 Oh ok. sorry

Sorry user32720. I'm not sure how to answer your question. I do think the physics forums might be able to help though.

@user400188 Right. I sended this problem. I gonna out, but I thank you for your help

Hey there

hey @arctictern around ?

Can anyone help me with notation? I just started studying rings an there's the notation $\mathbb{Z}/n\mathbb{Z}$ but I don't know what the bar ("division") stands for

@LucasHenrique It's notation for a quotient ring :) en.wikipedia.org/wiki/Quotient_ring#Examples

@AliasUser Thank you! :D

No worries :)

Hmm, so let me see. The $n\mathbb{Z}$ set is $\{\ldots,-n, 0, n, \ldots\}$, right?

I believe so.

I just recently became aware of Riemann's Rearrangement Theorem which states you can permute any conditionally convergent sum to yield whatever result you wish. With this in mind, why is it that some conditionally convergent series are written with 'conventional' values they converge to?

I saw on the internet that $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$, where $\mathbb{Z}_n$ is the set of residues from the division of any integer by n

Correct

The easiest way for me to think about it is $\mathbb{Z}_n $ is $\mathbb{Z}$ modulo $n$

Yes, same thing

and you are correct for your interpretation of $n\mathbb{Z}$

notation for an ideal, so if n was 2, you have the even Ideal

multiples of 2

But I don't understand what is the idea of "making a rule" of $n\mathbb{Z}$ on $\mathbb{Z}$ so it becomes $\mathbb{Z}$ modulo $n$

the word quotient is the key

it's like you 'divide out' all the elements of the ideal

and what's left is the quotient ring

Divide out like, remove?

like as in the definition of division

Oh lol

a|b is same as saying there is some c such that bc=a

yeah it took quite a few rereadings for me to get it too lol

So most of the times when we work with the ring $\mathbb{Z}$ we'll probably want to use number theory?

whoops b = ac

and i cannot answer that lol, i dunno

i'd say that number theorists would be interested in the integer quotient rings, yes

but idk if there are other constructs that prove useful for the study of the integers

number theory is definitely one of my weaker subjects in math, along with combinatorics

(yes ikr, remainder and quotient are obvious but... I just can't get the logic of, example, ring $S \subset \mathbb{R}$, find $\mathbb{R}/S$)

start with the equivalence classes

I studied that

check some numbers in R

a, and b, and do a-b

if a-b is in your given S, then they are said to be equivalend

like, $[a] = \{x \in S / x \equiv a\}$?

*equivalent

ouch

something like that lol

i think its like a~b <=> a,b in R, a-b in I

$a\equiv b \iff a,b \in \mathbb{R}, a-b \in S$

@logical123 I need to read that theorem carefully

:p

that's the definition of quotient ring

reading from a text i got, but wiki has same defn. en.wikipedia.org/wiki/Quotient_ring

$R/S$, you dummy

:p

its the set of equivalence classes under that equivalence relation

so for Z/2Z you get 0 and 1 as equivalence classes

So you need to have the context of what means equivalent for that ring

ok so yeah exactly, equivalence relations are kinda like functions

you can define whatever equivalence relations you want

but there is a 'specific' one we agree on to use in the definition of a quotient ring

I'm gonna go crazy haha

I'm thinking of ring quotient on the physical law $PV = nRT$ which can all be made of a certaing function... blah blah blah

oh interesting

That's what you get if you're a highschooler trying to understand academic stuff lol

but we dont tend to plug integers into PV=nRT do we?

No, but we can plug reals

nah that's great mate, start young

And make equivalences depending on the situation we're analyzing

sure, but i'll steer you into a couple diff directions

PV=nRT is trash :P

it doesn't work for many gasses other than the simplest of ones

Yes, I'm also reading a thermodynamics book on that.

wonderful

Even tho it's useful for quasi-adiabatic transforms

you should find things like morse potential

fairly interesting, and holy moly impressive, i only learned quasi-adiabatic processes 2 terms ago

I'm going to stop all the physics and chemistry to do math

Just finished my Physics undergrad, studying math on the side and apply to PhDs

in Physics

@logical123 wow, congrats!

well definitely stop chem, that's just applied physics

@logical123 hahahaha

the battle is only begun my friend, the application process is arduous

and i'm so torn between math and physics but i found my niche

obligatory!

@LucasHenrique lol

Gonna ask one more time before i post, but I just recently became aware of Riemann's Rearrangement Theorem which states you can permute any conditionally convergent sum to yield whatever result you wish. With this in mind, why is it that some conditionally convergent series are written with 'conventional' values they converge to?

and should that be posted in real analysis?

@logical123 it's almost like life - it's very, very beautiful but I don't understand a thing :p

basically there's a very wide interdisciplinary field called control ttheory

gets into optimization, maybe youve heard of operations research?

@logical123 is it 4 dimensional?

nope!

that's 3-d

And in theory we could literally do that experiment

But time was also included

we just cant generate that perfect of potential wells as well

oh ok lol if you wanna go there

3 spacial one temporal

lol

those sims probably took ages to run

basic example for control theory is say you got a factory, you can control a variety of factors and want to maximize profit output

usually there is some sort of 'goal' in the problems

and in this bose einstein condensate, we want to get the condensate in a specific shape

[if wondering applications, think fusion reactor cores]

@logical123 I don't get how it can be proven that the permutation function exists

@logical123 beautiful

what do you mean by "exists"?

all functions exist

it's a matter of whether something is or is not a function

Are all subrings of infinite rings also infinite?

how would I know?

All I can think of is $\mathbb{Z}$ ._.

I know what rings are in the sense that I know the term

but I have 0 experience with them other than the obvious examples in math

sorry

Same thing, I've just started reading about them

@LucasHenrique must the multiplicative identity be the same in a subring?

@DHMO I think so

Then it must be infinite

infinite ring has characteristic 0

@LucasHenrique I'm not going to learn them for some time... if ever.

subring has the same characteristic

so the subring is also infinite

@DHMO does every conceivable geometry upon a surface have an alternate surface with the same characteristics such that the distance between any two points on the surface is finite?

i think it might be true

but impossible for me to really say one way or another

I have no idea @TheGreatDuck

I asked it as a question a while ago

pretty sure it was deleted as not mathematical

meh whatever

thought maybe you would know. You seem geometry savvy in particular

Oops, gotta go

1:44am here in Brazil

0.0

Goodnight, everyone, and thank you so much for your time guys!

no way

Bye ;)

a 4 hour difference?

that doesn't make sense. America is a straight shop up pretty much and it's 9:44 PM here.

:/

hmm?

If you follow a meridian from brazil north, you end up in the atlantic ocean

oh

nevermind

I though brazil was central south america

:p

@logical123 Because that value is what you get when you compute the series in the order given. You add up terms to get partial sums, in order, and that's the limit of the sequence of partial sums.

@TedShifrin thank you

that literally is all i needed clarification of

@TedShifrin hi. How's it going?

Sure, @logical123. I figured it was something simple :)

Hi Duck.