help-26

[exotic lance]

thank you

see you later

.close

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[pulsar girder]

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[pulsar girder]

im given the problem: for $an \geq 0$ if $\sum_{n=1}^{\infty} an$ diverges, does the $\sum_{n=1}^{\infty} \frac{a_n}{1+na_n}$ converge. i tried some tests but all were inconclusive. any hints appreciated

[thorny flame]

atif

[pulsar girder]

do i need to consider cases if a_n is monotonic or not?

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[pulsar girder]

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[runic pivot]

is a line of invariant points found separately to an invariant line? if a question asks for you to find both, would the answers to either relate to each other?

[runic pivot]

example here, would one answer tell you anything about the other answer in any way or are they kind of separate things?

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@runic pivot Has your question been resolved?

[runic pivot]

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[pine flame]

I need to solve this cubic equation

[pine flame]

Using Cardona's method

2x^3 -x+1=0

Depressed cubic actually

[ruby tree]

Cardano

[ruby tree]

You need to get rid of that 2 in front first

[pine flame]

Mb m

[pine flame]

Ok after that

[ruby tree]

You have x^3 + px + q = 0,
calculate (q^2)/4 + (p^3)/27

[pine flame]

Uhhh yeah

Let's just say

Idk what p and q are

[ruby tree]

I thought you got rid of the 2

[pine flame]

Yeah I did

[ruby tree]

What's your equation now?

[pine flame]

Just divid by 2 right

[ruby tree]

Yes

[pine flame]

Ignore me using t

[ruby tree]

That's fine

t^3 -1/2t + 1/2 = 0
t^3 + pt + q = 0

Do you not see what p and q are?

[pine flame]

Now what:D

[ruby tree]

... tell me the values of p and q

[pine flame]

-1/2

And 1/2

[ruby tree]

Right

Now, define
delta = (q^2)/4 + (p^3)/27

Calculate that

[pine flame]

A minute

13/108

Is it?

[pine flame]

Ping: 13/108

[ruby tree]

No

[pine flame]

Dang

1/8

• 1/8*27

Right?

[ruby tree]

1/4 * (1/2)^2 = 1/16

[pine flame]

Bruhhh

Mb

25/432

[ruby tree]

1/27 * (-1/2)^3 = -1/(27*8)

,calc 1/16 -1/(27*8)

[thorny flame]

Result:

0.05787037037037

[ruby tree]

,calc 25/432

[thorny flame]

Result:

0.05787037037037

[ruby tree]

Right

[pine flame]

Alright now what?

By the way

Can I show you

A pdf?

About the method

[ruby tree]

Just screenshot the part you want to talk about

[pine flame]

Sure

[ruby tree]

Since this is positive, you have a real root x_0 = cbrt(u_1) + cbrt(u_2), where u_1 = -q/2 + sqrt(delta) and u_2 = -q/2 - sqrt(delta)

[pine flame]

The net is hella bad

Taking a bit too long

Here

Dang there

This method

Here is a question solved using the method

Here

The u^2

Wouldn't it comes negative

Or like do we take the difference

[pine flame]

In 6th lines

[ruby tree]

I don't see a u^2

[pine flame]

6th

[pine flame]

Here

[ruby tree]

u^3 = 1/2(35+19) ?

[rigid cloak]

Can I hel

P

I can find roots

Without cardano

[pine flame]

Ofcourse

[rigid cloak]

Here

[pine flame]

Ey yo

Nuh uh

[rigid cloak]

Fr

[pine flame]

Need to learn it

[rigid cloak]

Okok

[pine flame]

Comes in clutch

[rigid cloak]

When u learn

Ping me

[pine flame]

YEAH MB

[violet raven]

what?

[rigid cloak]

So we will cross check

[pine flame]

It comes negative doesn't it

[pine flame]

Bet

[rigid cloak]

Okok

[pine flame]

Wait idk whether negative or not

[rigid cloak]

Imma find it now

[violet raven]

can you show that part in the image

[pine flame]

But shouldn't there be minus

[ruby tree]

I don't know what you mean

[pine flame]

6th lines, u^3

According to formula

[violet raven]

why would it come negative

[pine flame]

It is-4H^3

Over there

Did they just over simplify and write +

[ruby tree]

This?

[pine flame]

Yeah

See negative

No i mean

h is negative

Like the value

Another doubt

[ruby tree]

You mean H = -1/3 in your case?

[pine flame]

Yeah

So that becomes negative

[ruby tree]

-1/6 actually but anyway

G = 1

[pine flame]

Oh wait

It has to be

3H

That is a doubt actually

[ruby tree]

G^2 + 4H^3 = 1 + 4(-1/6)^3

That's positive

[pine flame]

No no

Wait a minute

[pine flame]

Here in general form the gave 3H

[pine flame]

So is H = -6

In our case

Or -18

[ruby tree]

You had -1/2 in place of 3H

3H = -1/2, H = -1/6

[pine flame]

I'm talking about the sa

Ss

First let's get that

Then the problem

[pine flame]

So it is -18

So H becomes -6?

Cuz 3H right?

[rigid cloak]

Got the roots

[ruby tree]

In the solved problem, yes

[rigid cloak]

Without cardano

[pine flame]

Freaky

[pine flame]

Wait so

Do I take H as minus six

Or eighteen

[pine flame]

-1 right?

[ruby tree]

-6

[pine flame]

Ok thank you

Also now

[rigid cloak]

-1

[pine flame]

6th line

[rigid cloak]

And 1/2+-i/2

[pine flame]

Yeah

[rigid cloak]

Fr

[pine flame]

Is that calculation correct

[pine flame]

What method?

Synthetic division?

[pine flame]

Is it 35+19

Cuz I think there would be a minus between them cuz h is minus six

Or maybe they over simplified

[ruby tree]

H = -6, G = -35, G^2 + 4H^3 = 35^2 - 4*6^3

,calc 35^2 - 4*6^3

[thorny flame]

Result:

361

[pine flame]

Ey yo

So it correct

[ruby tree]

Of course it is

[pine flame]

Good to know lol

Ok now we go back?

😄

[ruby tree]

To what

[rigid cloak]

No

Susgusinabus

[pine flame]

Now can you please solve this in that method

[rigid cloak]

I used simple maths

[pine flame]

Yeah

[rigid cloak]

from seeing the equation

I can say x=-1 is a root

So it satisfies

[pine flame]

Yeah

[ruby tree]

Why don't you try?

[rigid cloak]

so x+1 is a factor

So I wrote

[pine flame]

Sure

Can you correct me?🙏

[rigid cloak]

2x^3-x+1 as (x+1)*Q(x)

Where Q(x) is some quadratic

[pine flame]

Yuh uh

[ruby tree]

I mean if you get the correct roots (that you already know), you should be good...

[pine flame]

Oh long division

Same thing actually

[rigid cloak]

Now I substituted values

And got the quadratic

But did not divide

[pine flame]

Ah yes got it

Thanks

For informing

[rigid cloak]

So I got 2x^3-x+1 as (x+1)(2x^2-2x+1)

[pine flame]

Indeed that has imaginary roots

[rigid cloak]

Yes

[pine flame]

Ok so

[rigid cloak]

This method indeed good

[pine flame]

Now we calculate u cube right?

And find uv

From that derive v

And x = u+v

From that find x

Is this correct

@ruby tree awaken, kind sire

Yeah lemme do it

Dang

This calculation is not sane

Nvm, cardona is stoopid

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[tall swan]

Hi im not sure how to answer questions where theres a radical with a variable(u) attached to it

[acoustic carbon]

whats the ques exactly?

[tall swan]

How do i solve this question where there is a radical with a variable under it

[acoustic carbon]

simplify?

[tall swan]

I dont understand how to do that with a variable under the radical

I can simplify it without the u

But im not sure how the steps are changed or what the rules are for when u have a variable under the radical

For instance sqrt12u x sqrt3

[acoustic carbon]

i am getting 30+root12u^3

[tall swan]

Im not sure how to get there tho

I need the root to be simplified in my answer too

The first step is distributing, no?

[echo tulip]

Yes

[tall swan]

So then

[tall swan]

How does this work

[echo tulip]

But see if you can simplify the 2sqrt(12))u

try that first

[tall swan]

4sqrt3u…?

[echo tulip]

yep

[tall swan]

Then

When u multiply sqrt3u with sqrt3 do u just get sqrt9u…?

[echo tulip]

yes

which can simplify more

[tall swan]

Then

Uhh

Wait how do u do 4sqrt3u x 1u?

4usqrt3u….?

Or-

Nah thats all ive got

[echo tulip]

yes

[tall swan]

Oh

[echo tulip]

but we can combine the u terms

so $u \dotproduct u = u^2$

[tall swan]

Oh u can?

[thorny flame]

Adam

[tall swan]

Then itd be 4sqrt3 u^2?

[echo tulip]

yep

nice

[tall swan]

And then

[echo tulip]

here

[tall swan]

When u simplify 20sqrt9u to 60u…..?

[echo tulip]

we can simplify this

yes

nice

[tall swan]

ohhhhh

So would it be at (60)(4sqrt3u^2) now?

[echo tulip]

wait

I think you are missing a u in that expression

[tall swan]

Oh! Right thx

And then just divide?

This is what ive got so far but apparently the 30 shouldnt have a u

[topaz sinew]

@tall swan Has your question been resolved?

[unreal quest]

What did you start with?

As in, the initial expression

[tall swan]

It asks for the area of a triangle when the base is 2root12u and the height is 5root3+1u so i put it into the area of a triangle formula on the top line here, (2root12u)(5root3+1u) over 2

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[twin warren]

can anyone help explain what i do here

[lean pebble]

You can do some angle chasing using inscribed angle theorem

What’s DEB?

[twin warren]

uhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

i think 41 as wel

[lean pebble]

Yeah

[lean pebble]

So what’s FBE

[twin warren]

26

[lean pebble]

Ok, does that give you enough information to finish

Remember what you want to find

[twin warren]

EJF is gonna be supplementary to C right

wait hold on

[lean pebble]

Err why

[twin warren]

i was thinking about the wrong thing

[lean pebble]

You want to find DJB right

That’s supplementary to BJE

Can you find BJE

[twin warren]

ohhhhhhhhhhhhhhhhh

i was overcomplicating everything

ok everyting is way simpler ow

now*

thanks

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[quiet lava]

I understand how to do this what with

(x^2-9)^2 = (x-3)(x+3) and then simplify the fraction.

However why does the question mention the answer cannot be -3? Is that a red herring?

[wheat kestrel]

x cannot be -3 because then the denominator would be 0. as you know, dividing by zero is not something defined in mathematics

[quiet lava]

Oh, I get it. It's just something that makes the question valid and isn't necessarily related with the active arithmetic.

Thanks for the help!

.close

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[stark dragon]

hello i am just confused on why the 4k-6 is before the 16

[lyric veldt]

do you think that it should be reversed?

[stark dragon]

yeah

[lyric veldt]

and what's your reasoning behind that

[stark dragon]

i know the formula is delta y over delta x for slope

[lyric veldt]

that's true

[stark dragon]

but what makes 4k-6 y1?

[lyric veldt]

are you going off the assumption that $\m = \frac{y_2-y_1}{x_2-x_1}$

[thorny flame]

AwesomeRat

[stark dragon]

yeap

[lyric veldt]

well that's true

but

you can assign either point to be the first or the second

since, you're dividing, the signs will be the same anyway

as long as the order of x and y is the same, you're fine

so you can just as well put 16 before 4k-6, but that would mean you would have to put -2 behind 2k+3 as well

[stark dragon]

one sec

[lyric veldt]

as you can see: $\\frac{4k - 6 - 16}{2k + 3 + 2} = \frac{16 - (4k-6)}{-2 - (2k+3)}$

[thorny flame]

AwesomeRat

[stark dragon]

ok i see the issue

i didnt distrubute the negative to the 4k-6 function

the bottom doesnt matter cus its set to 0 anyways for this problem

thx 4 da hlep

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[sharp dew]

It is -1

[sharp dew]

It is -1 bro

[alpine mist]

how so?

[sharp dew]

-1 is the solution

[stark dragon]

its not

[alpine mist]

show your work.

[opal vault]

it isn't

[stark dragon]

did you factor that john

?

[sharp dew]

Im not John

[stark dragon]

lol

i ddint mean that

i meant the equation lmao

did you factor it at all?

or whats your work look like

i could tell u the answer but it doesnt really help

[sharp dew]

I actually got it

[stark dragon]

word?

[sharp dew]

Word bred

[sharp dew]

Your avatar is such an artifact

[stark dragon]

wdym

[sharp dew]

It is a compliment

[stark dragon]

oh thanks

do you know where its from >'

?

[sharp dew]

I do not know. I assume you drawn it yourself

[stark dragon]

alr well thanks and gl on your math endevors

[sharp dew]

Thank you

[rigid cloak]

@sharp dew

.close

[ivory wave]

you can do !done

[topaz sinew]

@sharp dew Has your question been resolved?

[rigid cloak]

!done

[topaz sinew]

If you are done with this channel, please mark your problem as solved by typing .close

[rigid cloak]

I don't have the helpful role

[hollow drum]

You don't have to make people close their channel

[rigid cloak]

@hollow drum ok but his doubt was resolved

[hollow drum]

Okay and?

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[vapid moth]

i need help

[thorn violet]

What have you tried so far?

[vapid moth]

nothing really, i know how to solve it but i just can't figure out that specific point

[thorn violet]

Yeah it's not clear what exactly the radius at that point is

[vapid moth]

yeah

[thorn violet]

I would just guess like approximately 2 and 2/3rds

Maybe 2.5

And say that's what you're doing, then roll with it

[vapid moth]

ok sounds good then yeah i'll prolly just do that

thx

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[fossil harness]

Im confused on how this happens. I try but I keep getting something else.

[fossil harness]

how does it go from the top eq to the bottom eq?

[hollow drum]

Get all the terms with y on one side, everything else on the other

Then factor out a y

[fossil harness]

like this: xy-y=7x+6, then: y(x-1)=7x+6, then: y= (7x+6)/x-1

[hollow drum]

Exactly that

[fossil harness]

ok! Thank you!

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[feral palm]

if i have a function $f(t,y(t))$ and need to calculate $f(t+h,y(t)+k)$ is that approximated by taylor series

$f(t,y(t))+hf_t(t,y(t))+kf_y(t,y(t))$

am i missing any chain rules here? if not, why would i not need to multiply any of the taylor series terms by $y'(t)$

[thorny flame]

xt1984kd

[feral palm]

.close

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[devout bridge]

If f(t)=x(at) then f(t-k)=x(a(t-k)) am i correct? where a is some non zero real constant

[cinder sequoia]

yes

[topaz sinew]

@devout bridge Has your question been resolved?

[outer portal]

that yes is because this works in general for anything by substituting t - k in for t

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[desert dagger]

hi need help with this. rn im getting $\sin(2x) = -\sin(b)
$

[desert dagger]

$\sin(2x) = -\sin(b)$

[thorny flame]

∮

[desert dagger]

then 2x = pi + b and 2x = 2pi - b

then x = pi + b / 2 and x = 2pi - b / 2

is this correct?

[outer portal]

,align\sin2x&=-a
\2\sin x\cos x&=-a
\2\sin x\sqrt{1-\sin^2x}&=-a
\2b\sqrt{1-b^2}&=-a

[thorny flame]

mtt

[outer portal]

wait nvm theyre not asking for you to do this

[desert dagger]

yeah thats what

i mean i think concept we have to apply is of trig equation

[outer portal]

you didnt divide both sides by 2 correctly

,,2x=\pi+b
\x=\frac\pi2+\frac b2

[thorny flame]

mtt

[outer portal]

once you fix that, the answers you got are correct

[desert dagger]

i meant whole devided by 2

so my answers are correct?

ill get 5/5 marks then?

[outer portal]

why are you saying it like that

I dont know how they mark your work

so I dont know if youll get 5/5 marks

[desert dagger]

actually my latex is not very good

then x = (pi + b) / 2 and x = (2pi - b) / 2

[outer portal]

thats just using parentheses to type math with text

this isnt latex

if you want to type the math as latex, you have to use \frac{ ... }{ ... } to tell latex to draw a fraction bar

[desert dagger]

fair enough thanks anyway

[outer portal]

np

[desert dagger]

.close

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[wraith iron]

I don't understand the highlighted part. Please explain it to me in detail ty

[simple orchid]

RIP discrete mathematics

[wraith iron]

this is set theory

[simple orchid]

Set theory is in discrete math

[wraith iron]

oh

i love set theory

but sometimes I just don't get things

[simple orchid]

You and I are not gonna be friends

[wraith iron]

it takes like half an hour to read a page

[simple orchid]

Real

[wraith iron]

😂

[wraith iron]

i finally understood the argument haha
what maths do you like? I am doing set theory only because it requires no maths knowledge. i'd want to learn other maths but i cant. im only 15 now and my classmates are learning quadratic equations

well i know a bit of calculus, but its just full of remembering stuff

[wet harbor]

You didn't wanna learn math so you decided to learn set theory.

Uhmmm okay 🙄

[simple orchid]

Uhhhhh

[wraith iron]

no, other maths are too advanced

i dont have the knowledge requried

i love maths

[wet harbor]

I have studied analysis and abstract algebra and didn't get to set theory yet 😂

[simple orchid]

I am being forced to learn calc 3, discrete math, and set theory, linear algebra, boolean algebra in my first year of university

[wraith iron]

i dont think i have done calculus 1 even

none of my classmates know any calculus

maths is taught very slowly here

like 2 years behind

[wet harbor]

look, trust me whatever is lying ahead of you is easier than set theory

you will run through them like crazy

[wraith iron]

oh wait what about number theory

its also one of those that require no knowledge

i liked it too

[wet harbor]

you can start learning that on your own actually, I think you are smart and you can do it

[wraith iron]

of course, i never had a teacher

[simple orchid]

Do you live in Canada

[wet harbor]

it kinda does yes but even the stuff that require knowledge you can just study the knowledge

[wraith iron]

in terms of set theory, so on

[wet harbor]

that's great

[wraith iron]

UK

[wraith iron]

why?

i sometimes have questions

quite often

[simple orchid]

@wraith iron

[wet harbor]

yes and instead of getting handed the answers for those questions on a plate

[wraith iron]

UK

[wet harbor]

you will tire yourself out and try yourself

[simple orchid]

Sad

[wet harbor]

so you will learn more

you will be much more experienced than anyone who had a teacher

[wraith iron]

i've heard that too. what if you are stuck in a problem for like an hour

[wet harbor]

dear, I spent 4 years on a proof cuz i didn't wanna look it up 😂

[wraith iron]

what is that proof?

dont talk about it if its something i cant understand

[wet harbor]

probably not yea but you will get there, it's regarding something called tensor calculus

[wraith iron]

never heard of that

sad

[wet harbor]

don't be sad you shouldn't know that now

[wraith iron]

by the way, one day some undergraduate came to my school

[wet harbor]

I actually think you are in a great position

[wraith iron]

and taught me 'lagrangian mechanics'

he said it was some adfvanced physics

[wet harbor]

that's beautiful, lagrangian mechanics is at the core of physics

every new piece of physics is formulated in terms of a lagrangian nowadays

[wraith iron]

i derived the equations of motions of the n-pendulum after he taught me

on my own

however i dont understand where the euler lagrange equation comes from

it makes no sense to talk about T-V

[wet harbor]

you will later learn to prove that lagrangian mechanics support conservation of energy

[wet harbor]

I proved it

it requires calculus

[wraith iron]

is it elegant

like in a page?

[wet harbor]

Yes i think it doesn't get more than a page

[wraith iron]

thats so cool

[wet harbor]

you will also learn than lagrangian formulation is merely a method in something called the calculus of variations

it's a way to optimize over entire functions rather than just values

so physics is basically an optimization problem

[wraith iron]

yes, i heard that it is analogus (is that the right word?) to newtons second law

[wet harbor]

it's way more cool

because it's not limited to physics

you can use lagrangian to prove that the shortest distance between two points is a straight line

or prove that the surface with the smallest area for a given volume is a sphere

[wraith iron]

how?

[wet harbor]

you have to understand the broader concept

[wraith iron]

i mean they are intuively obvious

[wet harbor]

yes they are but then you can start proving other non obvious stuff

[wraith iron]

what I mean is, such simple concepts should be derived by simple axioms and theorems, rather not by a complex mechanical system

[wet harbor]

like how if you hang a chain from two points it gives you the shape of a hyperbolic cosine

[wraith iron]

i havent learnt cosh yet

i just know its like e^x +e^-x

[wet harbor]

math is all axioms and theorems, you develop the complexity from scratch, you can't learn all that in one day

[wet harbor]

it's half of that

[wraith iron]

oh okay cool

wait i actually have a problem

do you want to help?

[wet harbor]

If it's set theory i may not be good enough to help xD

I didn't do set theory with depth

[wraith iron]

noo its not

its recurrence relation

[wet harbor]

alright show me

[wraith iron]

okay one second

so i basically solved a second order (general) recurrence relation

im wonering how I can do it again for third order

[wet harbor]

it's constant coefficient right ?

[wraith iron]

(i know it is based off the characteristic equation, but i dont want to use it)

[wraith iron]

yes

fixed coefficeints

[wet harbor]

I actually like that solution

you can do it with matrices

[wraith iron]

also I have heard that if the roots are equal we have a different solution

[wraith iron]

can you show me?

im not very good with matrices tho

[wet harbor]

you know about eigen values and eigen vectors ?

[wraith iron]

no

[wet harbor]

it's really simple I think you are really smart you just need to get the knowledge

[wraith iron]

can you teach me them now?

[wet harbor]

okay so you know about maps

[wraith iron]

on wiki's page it is directly related to the char. eq

[wet harbor]

and functions

yes it is

[wraith iron]

yes, function is a one-to-one relation

what is a map again?

what is the difference

[wet harbor]

no difference just different names xD

[wraith iron]

oh ok

[wet harbor]

I like the word map more because it describes what it does

it maps values to other values

[wraith iron]

it does

[wet harbor]

but it doesn't have to be one to one

it's just that one value cannot map to more than one output

[wraith iron]

oh yea many to one

i forgot the defn

[wet harbor]

it's okay

a function can map anything to anything you say f:S->M
meaning a function that maps from the set S to the set M

and those sets can have anything

[wraith iron]

yea

S = dom F

M subset of ran F

i think

[wet harbor]

ran F is subset of M

[wraith iron]

i just read this a few hours ago

[wet harbor]

M called co-domain

[wraith iron]

oh yea that makes more sense

yes

youre right

[wet harbor]

so one set in particular that is very interesting

is the set of all vectors in maybe the plane for example

which you would call R^2

[wraith iron]

by vectors you mean points?

or they are the same

[wet harbor]

yes

[wraith iron]

okay

[wet harbor]

for our discussion they are the same

defintions are made whenever convenient

there is no laws on math only what is most efficient

[wraith iron]

true

[wet harbor]

so you can map from a set of vectors to another set of vectors

turns out there is a special type of map called a linear map which can be defined on sets of vectors

it's a map which is compatible with addition and constant multiplication

[wraith iron]

im not sure, is it kinda like a transformation

[wet harbor]

you know what

I know you are young but i will give you the offer and you can make the choice

[wraith iron]

how old are you?

[wet harbor]

I offer to explain this over a call, you can refuse if it's not comfortable

i'm 27

[wraith iron]

i dont think i am

[wet harbor]

you don't think you are comfortable

it's fine I figured xD

there is just so much to explain

[wraith iron]

i dont like to talk to people

in general

if its not about maths

[wet harbor]

I mean it's gonna be about math, i'm asking because i'm a stranger

[wraith iron]

so i just dont really talk

[wet harbor]

if the reason is because i'm a stranger it's okay i get that

[wraith iron]

im just trying to explain why im not comfortable

no even to people i know

i just dont like to talk

[wet harbor]

sure sure it's okay

[wraith iron]

okay

[wet harbor]

so you can add structure to sets

this gets us to a branch of mathematics called group theory

[wraith iron]

i know a bit

[wet harbor]

for example a group is a set with an operation on it which composes to elements

[wraith iron]

yes

i remember something called an ablidean group

that has a negative

and something else

[wet harbor]

it's closed under composition, and it's associative, has an identity and an inverse for each element

[wraith iron]

what does 'closed under composition' mean

[wet harbor]

abelian groups you mean

[wraith iron]

yes

[wet harbor]

means composing any two elements in the group gets you back in the group

[wraith iron]

oh ok

so like N is closed under +

[wet harbor]

yes

[wraith iron]

alr

[wet harbor]

you are correct

for example the integers aren't a group under mutliplication because inverses don't exist in there

1/2 is not an integer

and it's the inverse of 2

[wraith iron]

yes

it is for the reals

i think

well reals - {0}

[wet harbor]

so adding structures to groups adds structures to the maps applied to the groups

[wet harbor]

yes

so you can now define something called a homomorphism

[wraith iron]

whats an example of adding a structure?

[wet harbor]

i'm explaining right now

[wraith iron]

okay

[wet harbor]

if you have group 1 and group 2 G1 and G2 then a map f:G1->G2 is called a homomorphism if f(ab) = f(a)f(b)

you use the new laws of composition to put restrictions on the maps

you call such a map compatible with the law of composition as it sends products to products

you with me ?

[wraith iron]

okay kind of

what are the new laws of composition?

[wet harbor]

first we had sets only now we added a law of compostion and made it a group so that's our law of composition

so now we had a chance to look at specific types of maps which are compatible with that law of composition

[wraith iron]

oh ok

[wet harbor]

example to clear things out, integers are a group under addition

right ?

[wraith iron]

yes

[wet harbor]

the map which sends the integers to the reals by e^x will have the property e^(m+n) = e^m * e^n

[wraith iron]

yeah

[wet harbor]

so from the group of addition on integers to a specific group of multiplication on the reals, this is a homomorphism

[wraith iron]

ty this was very helpful

[wet harbor]

happy to help, there is more !

[wraith iron]

and because it is a homomoprhism it is compatible

right?

[wet harbor]

yes it's what it means to be a homomorphism

[wraith iron]

okay

[wet harbor]

now you can do more !

for the real numbers

you can add two groups not one

one under addition and one under multiplication excluding the zero

this structure of having both groups + the distributive law creates a new structure we call a field

so a field is two group structures + a way to connect them (distributive law)

[wraith iron]

okay

[wet harbor]

a homomorphism of fields would look like this
f(x+y) = f(x)+f(y)
f(xy) = f(x)f(y)

compatibility with both laws !

[wraith iron]

what is an example of such function?

[wet harbor]

honesly don't have something in my head right now xD

[wraith iron]

oh okay

[wet harbor]

but simply a function which takes each element to itself is always a homomorphism

the identity transformation

[wraith iron]

just to be sure you are talking about f(x) = x

[wet harbor]

yes

[wraith iron]

oh ok

[wet harbor]

you can make another very special structure from all the past ones

it's called a vector space

a set of vectors which is a group under addition, with a field of numbers

the field numbers can be composed with the vectors using what is called "scalar multiplication"

and a homomorphism for that one would look like this
f(v+w) = f(v)+f(w)
f(c w) = c f(w)

compatibility with addition and scalar mutliplication

[wraith iron]

is it not f(c)f(w) for the last one

[wet harbor]

it's not because f operates on a vector space and c is a scalar not a vector

c is in the field

[wraith iron]

oh okay

[wet harbor]

it just so happens this homomorphism has another name

they call it a linear transformation

[wraith iron]

that is more familiar

[wet harbor]

yes but this is the abstract algebra background for it

so now those sets of vectors I spoke of earliers

aren't just sets of vectors

they have structure

they are closed under addition for example

[wraith iron]

yes

[wet harbor]

each vector has an inverse and there is an identity which is the zero vector

this structure when imposed in the map makes it a linear map

those two ways to compose can create something we call a linear combination

looks like this
w = c1 v1 + c2 v2 + c3 v3

the cs are all constants in the field F

and the vs are all vectors in the vector space V

[wraith iron]

then what is w?

does it have any meaning

[wet harbor]

w is a linear combination of all of them

[wraith iron]

is it f(c1, c2, c3)

[wet harbor]

it's a definition of the word

[wraith iron]

oh ok

[wet harbor]

so w is anything

turns out for some vector spaces

you can represent the whole space in terms of a finite number of vectors

so any vector in V would look like v = c1 v1 + c2 v2 + ... + cn vn

those are called finite dimensional vector spaces

the minimum number of vectors required for such representation is called the dimension of the space

for example R^2 can be represented by the vectors (1,0) and (0,1)

since any vector (a,b) = a (1,0) + b (0, 1)

[wraith iron]

yea its essentailly (a, b)

[wet harbor]

yes

[wraith iron]

so every point

whole space

[wet harbor]

and it's two dimensional

yes

[wraith iron]

you have (1, 0, 0), (0, 1, 0), (0, 0, 1) for R^3

then

[wet harbor]

yes

there is a beautiful proof that shows

[wraith iron]

induction?

[wet harbor]

any map from F^m to F^n for any field F

[wraith iron]

i thought you meant a beautiful proof that shows how the unit vectors can cover the whole space

so i said induction

[wet harbor]

if it's linear then it must be an m x n matrix.

[wraith iron]

please continue

[wet harbor]

no that one is trivial

the other one doesn't seem very obvious

[wraith iron]

it doesn't

[wet harbor]

it just has to be a matrix

so you think of a matrix as a map of vector spaces from now on

this is what it means in its best form

[wraith iron]

okay

what does the proof look like?

[wet harbor]

let's keep that for later, you can try it if you decide to go deeper in this

example of a linear transformation is rotation in the plane

[wraith iron]

okay

[wet harbor]

it's a 2x2 matrix which rotates a vector

[wraith iron]

ive seen it, just forgot

i cant memorize things

[wet harbor]

yea it has sines and cosines

[wraith iron]

but it is only 0s and 1s

oh yes if its any given angle

[wet harbor]

oh this one rotates by 90 degrees

yep

now a miracle happens

turns out

each one of those transformations

has specific vectors which never rotate but only get scaled

for example reflection around the y axis is a transformation which fixes any vector on the y axis

it scales it by one and never rotates it

these are called eigen vectors !

[wraith iron]

can i have an example?

[wet harbor]

the reflection is one of them

[wraith iron]

ok let me think

[wet harbor]

if you reflect a vector around the y axis

what happens to any vector on the y axis ?

[wraith iron]

x |-> -x

or y

it is negated

[wet harbor]

the x component is negated

the y component stays the same right

[wraith iron]

yea

of course

[wet harbor]

so any vector that only has y

never changes

that is called an eigen vector of the reflection

now there is more

the vector in the x direction is also an eigen vector because it gets scaled by -1

so the line didn't rotate

scaling by -1 we are still in the same line

hence (1,0) and (0,1 are both eigenvectors of the reflection around y)

[wraith iron]

okay

one second

[wet harbor]

if you wanna try that yourself here is the reflection matrix.

sure

[wraith iron]

why is 0, 1 an eigenvector of reflection around y?

i meant 1, 0

[wet harbor]

because it gets scaled

by -1

it stays on the same line but goes negative

the definition of an eigenvector looks like this
Av=cv

[wraith iron]

can one argue that it is also a rotation of 180 degrees?

[wet harbor]

yes but the definition allows that to be included as scaling

as if c = -1

Av = -v

it works

[wraith iron]

both x and y has to be scaled by the same factor right

[wet harbor]

nope

[wraith iron]

for (x, y) to be an eigenvector

[wet harbor]

that's the fun part

[wraith iron]

oh okay

[wet harbor]

those scaling factors

can be different and will be different

they are so important they got their own name

they are the eigenvalues

those are the ones that come out of the characterstic equation

[wraith iron]

okay

why tho?

[wet harbor]

I was about to say xD

[wraith iron]

oh ok

[wet harbor]

how can we prove the characteristic equation ?

check this out

you know about algebraic equations right ?

[wraith iron]

yes

[wet harbor]

what is an equation

[wraith iron]

A = 0 where A is an expression with some number of variables

i dont know how to define

one

[wet harbor]

good

[wraith iron]

is it limited to only some operations

[wet harbor]

they can come in a shape which requires you to solve it

like x+3 = 10 for example

[wraith iron]

okay

[wet harbor]

well it happens that those form of equations can come in many fields of math

you can have a matrix equation

most popular one is Ax = b

You ask yourself what is the vector which if transformed by the matrix, gives you b

[wraith iron]

b x matrix^-1?

[wet harbor]

if the inverse exists!

[wraith iron]

im not very good working with matrices

yes

[wet harbor]

it may or may not exist

[wraith iron]

when does it not

[wet harbor]

matrix is just a name

it's simply a map of sets that have a structure on them

if you understand those

you will understand matrices

[wraith iron]

that kind of makes sense

matrix puts a restriction on the algebraic equation

[wet harbor]

so when does it not have an inverse

quite funny actually

[wraith iron]

something to do with determinant i think

i just forgot again

well done me

[wet harbor]

it doesn't have an inverse for the same reason a map my not have an inverse

[wet harbor]

it's fine it's too much info

[wraith iron]

something to do with injectivity or something

or osme other word

some*

bijectivity?

[wet harbor]

yes

that is correct

[wraith iron]

what does that mean?

[wet harbor]

i will tell you it makes so much sense

let say you have a map which maps even numbers to 0 and odd numbers to 1

does that map have an inverse ?

[wraith iron]

um

no

[wet harbor]

why

[wraith iron]

because if you do inverse(0) the output is an arbitary even number

[wet harbor]

that's correct

[wraith iron]

we can only say that inverse(0) in 2N

[wet harbor]

in fact it has an inverse

the inverse is all the even numbers or all the odd numbers

but that's not the correct question

every map has an inverse

but the question is ... is the inverse also a map ?

here is the answer is no

because 0 mapping to the entire even function is something a map would do

[wraith iron]

why?

[wet harbor]

because you know the definition

what is a function

it can only have one output right ?

[wraith iron]

many-to-one relation

yes

[wet harbor]

yes

can't be one to many

so when people say does inverse exist ?

what they actually mean is

does an inverse "map" exist

and if you have something that is not one to one it's clear that it wouldn't exist

[wraith iron]

it doesnt make sense to talk about mapping to the 'entire' even numbers though?

of course it is one of the even numbers

but the even numbers are a infinite set

[wet harbor]

you can map anything to anything as long as it's many to one

it could be infinite

you just say any even number maps to 0

the domain is free to be infinite

[wraith iron]

that makes sense

so what kind of functions does not have an inverse

[wet harbor]

so when is a matrix non invertable ?

when it's not one to one

simple as that

bascially when it's not bijective

bijective means one to one and onto

onto means it maps over the entire co domain

because if it doesn't then there will be elements the function can't reach

aka outside the range

and those cannot be inverted

as they never came out of the function to begin with

[wraith iron]

oh okay

[wet harbor]

the answer to the question

[wraith iron]

when it is not bijective

[wet harbor]

is a matrix invertable

yes

it's related to another question

solutions to the equation Ax = 0

why ?

because if Ax=0 has any other solution except zero

say u for example

then Av = A(u+v) hence the matrix isn't one to one

as A(u+v) = Au+Av = Av since Au = 0

[wraith iron]

i need some time to read this

[wet harbor]

sure

[wraith iron]

oh yes, because the Au is gone

alright

[wet harbor]

yes

[wraith iron]

it doesnt make sense to ask that question

i just realized

[wet harbor]

only linear ones because we used linearity here A(u+v) = Au+Av is the property of linear transformations if you remember

[wraith iron]

yea

[wet harbor]

and they have to be defined on a vector space

aka matrices

[wraith iron]

yes

[wet harbor]

so how do we find those eigenvectors

Av=cv -> Av-cv = 0

which becomes (A-cI)v = 0

[wraith iron]

(A-c)v = 0

[wet harbor]

can't subtract a matrix and a constant

so cv becomes c I v where I is the identity transformation

[wraith iron]

oh, I is identity

oh okay

[wet harbor]

but guess what

if v is an eigen vector then so is any constant multiple of it right ?

[wraith iron]

im not sure

[wet harbor]

try it

if Av = cv

then Akv = kAv = kcv = ckv

hence kv is an eigenvector as well

[wraith iron]

okay, what does the phrase 'constant' do?

constant multiple

[wet harbor]

it tells you something very important

[wraith iron]

i mean for any k, kv is also an eigenvector

[wet harbor]

yes

[wraith iron]

why cannot k = g(d)

like another function

variable

[wet harbor]

so when solving for an eigenvector, multiple ones should give you zero

[wraith iron]

yes

[wet harbor]

because a constant is enough

it says that the map cannot have a single solution

in other words it's a non invertable map

the map A-Ic to be specific

a piece if information that requires a somewhat lengthy explaination is that the determinant of a non invertable matrix is zero

hence det(A-cI) = 0

that is called the characteristic equation

[wraith iron]

what is the determinant?

like apart from the formula

what does it give?

[wet harbor]

a map which takes a matrix to a real number

it has a significant meaning geometrically

if you put 3 vectors in a 3 x 3 matrix

the determinant is the volume of the shape formed by the edges of the vectors in like a way i should show, can't phrase it on words

see this