help-27

[light pecan]

got it

thanks mate

.close

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[lament schooner]

i did all the calculations and its still wrnog

[urban jungle]

could you show your calculations

[lament schooner]

yes using desmos it shows one triangle has side 3 and 3/4

another triangle has 3 and 13

i even usd a calculator

then added up the area

[urban jungle]

should be 3.25 not 3

[solar goblet]

limit your bounds to 0 and 4

should be 2 triangles instead

[lament schooner]

yes i did that

also how do yo udo that in desmos

[lament schooner]

oh that might be why

[lament schooner]

its still wrong uh

i added these two

[urban jungle]

you should be inputting a and b

the legs of the triangle

not a and c

or just use a calculator, .5ab is simple enough

[solar goblet]

bad practice to rely on calculators

try calculating on paper instead

[urban jungle]

or use a calculator calculator, not an online formula calculator

[lament schooner]

yeah thats the mistake i found it

[lament schooner]

yeah true

😅

anyway

ty!

.close

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@silk ridge Has your question been resolved?

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@silk ridge Has your question been resolved?

[silk ridge]

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[weak cove]

Is the answer to this just [f^(-1(x))h(x)] since p(x) is irreducible in F[x] we know F[x]/p(x) is a field and f(x) nonzero will have a multiplicative inverse

[sick fulcrum]

whats theorem 5.10

[weak cove]

uh basically like TFAE if p(x) irreducible in F[x] then F[x]/p(x) is a field and an integral domain

[sick fulcrum]

if [f] isnt 0 it has an inverse in that quotient ring

[weak cove]

yeah since it's a field

[sick fulcrum]

ig

yeah

so take that inverse

and take h

set g = h * that inverse

[sick fulcrum]

so yeah

[weak cove]

[sick fulcrum]

yeye I just

braindumped

I was working it out as well

mb

[weak cove]

no worries lol

thanks for double checking

[sick fulcrum]

u are right tho

lol

also

how tf are you taking field theory

[weak cove]

I registered for it

and then went to the class

[sick fulcrum]

what year is field theory

4th year?

[weak cove]

I'm not sure what it usually is

I'm a freshman though

[magic pine]

they start them with rings and fields then move them to group theory ryb its wack

[weak cove]

I have group theory next quarter

:o

[sick fulcrum]

oh wow

[magic pine]

youre gonna learn the galois correspondence and then learn the axioms for a group

[weak cove]

galois theory would be the quarter after but the math dept is slow so they dont offer it

[sick fulcrum]

oh wow so they don't restrict the courses you can take

[weak cove]

so I'll just take grad algebra next year

[sick fulcrum]

at my uni they require you to have prereqs to register online

[weak cove]

the prereqs are like intro proofs and linear algebra i think

something like that

[sick fulcrum]

if you don't have prereqs you can still register but you gotta inform the prof so they don't automatically boot you from the course

[weak cove]

which i have

[sick fulcrum]

ig

you obviously have the mathematical maturity so

why not take it

[weak cove]

the advisors here disagree

[sick fulcrum]

idk man you seem to be handling it fine

[weak cove]

I wish u were in charge of the courses I could take haha

I asked to enroll in this last quarter, so I could take galois theory in the spring like I said abve

[sick fulcrum]

LOL

[weak cove]

but they told me no

[sick fulcrum]

no my university has pretty annoying rules about those too

luckily it's up to the prof

I've been lucky in getting waivers into courses I want to take so far

[weak cove]

I asked the prof first cus I had already dealt with advising being stupid so many times

[sick fulcrum]

but there's no escaping the freshmen courses

[weak cove]

and they forwarded my email straight to adviing

I got instrutor permission for both my courses last quarter cuase advising wouldnt let me take either of them

[sick fulcrum]

how it works here is one would first inform the prof of their missing prereqs and ask for a waiver

when the registrar's office sends them the list of students without necessary prereqs, the instructor will ignore your name when they start booting students

anyway

[weak cove]

if only

just finished my algebra HW now have to do my linear algebra

what a joy

ty again

[sick fulcrum]

the irony

[weak cove]

.close

[devout snow]

Channel closed

Closed by @weak cove

Use .reopen if this was a mistake.

[sick fulcrum]

"just finished my 4th year course homework time to do my 1st year course homework"

[weak cove]

well it's abstract linear algebra

XD

[sick fulcrum]

I've been finding multiple ways to cheese doing any actual linear algebra

in linear algebra

[weak cove]

too much algebra in my life

[sick fulcrum]

I would rather write a 3x longer proof that avoids linear systems

and matrices

than deal with any of those

https://math.stackexchange.com/questions/3906079/dim-w-dim-w-perp-n

i had something on a homework that would be solved by this

instead of dealing with matrices

I spent a large amount of time dealing with dual spaces

[weak cove]

omg

maybe you can explain dual spaces to me then

because we just got to those and

I have no clue whats going on

[sick fulcrum]

hmm

okay I can try but this is gonna be like

not the best explanation

because I have bad cointuition

so basically dual spaces are dual objects

uh

yeah this is a cursed understanding

Hom(V, F)

where F is the underlying field

[weak cove]

refraining from saying what is a dual object

what is Hom?

[sick fulcrum]

we'll drop the category theory

the set of linear maps from V to F

[weak cove]

oh good

[magic pine]

austin uses L

[sick fulcrum]

sure

[weak cove]

ye

[sick fulcrum]

$\mc L(V, \mbb F)$

[sick fulcrum]

common notation, I'll use it

so

[weak cove]

.reopen

[devout snow]

✅

[sick fulcrum]

the dual space is "contravariantly isomorphic" to V

so

the intuition is

pick a basis of V

we'll assume it's finite for now to make life easy

say v1, thru vn

we can biject every vector in V with a linear transformation T: V \to F

elements of L(V, F) are called linear functionals btw

[weak cove]

yeah

that's what we call them too

[sick fulcrum]

so

[weak cove]

this is so refreshing from when I tried to watch a youtube video and all of the terminology was way different

[sick fulcrum]

I hate nonstandard terminology lol

we're first gonna make a basis of V^*

L(V, F)

[weak cove]

the dual basis?

[sick fulcrum]

the standard dual basis

intuitively what are these guys doing

I assume you know about the standard dual basis?

if not I can define it

(duality is a very confusing thing)

[weak cove]

I think I have it in my notes but maybe we could define it anyways because it hasn't really sank in yet

[sick fulcrum]

sure

[weak cove]

$$\phi_j(v_k)=\begin{cases} 1 \quad \text{k=j} \ 0 \quad \text{else} \end{cases}$$

[woven radish]

Austin

[sick fulcrum]

yes

thank you

so we're gonna see why this actually works

let $v = \sum_{i=1}^n a_i v_i$

[weak cove]

suppose that is equal to 0

[sick fulcrum]

and let $T: V \to \mbb F$ be any linear transformation whatsoever

[sick fulcrum]

Then we have that $T(v) = \sum_{i=1}^n a_i T(v_i)$ by linearity

[weak cove]

plugging in v_i gives just a_i

[weak cove]

so a_i = 0 for all i

[sick fulcrum]

yeah so the standard dual basis just picks out the scalars

[weak cove]

so the kernel is trivia

and their dimension is equal

[sick fulcrum]

so clearly we only need to understand how T acts on our basis

the intuition behind the standard dual basis is the fact that those phi_i's are just "indicators"

they're picking out the scalars

now

it's helpful actually seeing an explicit bijection

so for every $v \in V$, write $v = \sum_{i=1}^n a_i v_i$, and send it to the linear functional given by $\sum_{i=1}^n a_i \phi_i$

[sick fulcrum]

this is certainly injective, since \phi_i is a basis

and every vector can be written as a unique linear combination of its basis elements

[sick fulcrum]

to help with the intuition

the REAL confusing part is dealing with the double dual

Hom(Hom(V, F), F)

[weak cove]

is that the dual map

[sick fulcrum]

the dual of the dual

[weak cove]

Hm we haven't done that yet

we defined a dual map though

[sick fulcrum]

I want to introduce it just to make life less confusing ig

and so you can get a bit ahead

yk

if you're confused today

after a few days

it'll sink in

[weak cove]

sure

[sick fulcrum]

and make sense

[sick fulcrum]

so basically this thing here is isomorphic to V

it's actually an example in category theory

of naturality

but obviously no category theory here

[weak cove]

one day

[sick fulcrum]

now let's see why these guys are actually isomorphic

unironically i think the category theory intuitions will help

or shall I say "cointution"

[weak cove]

lol but I do not have any of them

they probably would

[sick fulcrum]

Let T be a linear transformation from L(V, F) to F

we would like to pick out a vector in V that T goes to

so we're gonna cheat a little bit

[weak cove]

doesn't T go to F

[sick fulcrum]

recall that for each $f \in \mc L(V, \mbb F)$, there is exactly one $v \in V$ such that $f$ is a linear combination of $\phi_i$'s with the same scalars as $v$ is a linear combination of the $v_i$'s

[sick fulcrum]

in V

[sick fulcrum]

so in other words we have

[weak cove]

T((L(V,F), F)

[sick fulcrum]

$T: \mc L(V, \mbb F) \to \mbb F$

[sick fulcrum]

so again we just need to study how $T$ acts on our basis

[sick fulcrum]

now $T(f) = T(\sum_{i=1}^n a_i \phi_i) = \sum_{i=1}^n a_i T(\phi_i)$

[sick fulcrum]

okay so it stands to reason that we can pick a similar basis for $\mc L(\mc L(V, \mbb F), \mbb F)$. Define

[

\varphi_i (\phi_j) = \begin{cases}
1 &\text{if i=j} \
0 &\text{otherwise}
\end{cases}

]

[woven radish]

Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[sick fulcrum]

ugh okay

ignore the compile error

[weak cove]

yes

np

[sick fulcrum]

but \varphi_i is representing v_i

[weak cove]

yes

[sick fulcrum]

(this is the part that's hard to wrap your head around)

[weak cove]

disagree

the part that is hard to wrap my head around is what the point of this is

[sick fulcrum]

oh

the point is to confuse students

[weak cove]

lol

goal accomplished

[sick fulcrum]

genuinely I do not know what a double dual is used for

[weak cove]

no and I'm not saying like

[sick fulcrum]

but again my knowledge of linear algebra is limited

[weak cove]

like why do we care about it

or dual spaces

[sick fulcrum]

dual spaces are kind of important i think

[weak cove]

why

[magic pine]

they let you rigorously define higher dimensional tensors i suppose

[sick fulcrum]

because the idea is you can represent V with the space of linear maps from V to its underlying field

like in category theory if you can define the morphisms you can extract the objects

[sick fulcrum]

I see

[weak cove]

I have a maybe unrelated maybe related question

[magic pine]

also, i'd like to say that it is a disservice to give a good explanation of dual spaces without giving the naive intuition that V is F^n and V* is (F^n)^T

[weak cove]

yeah this is what Snow was saying earlier

[sick fulcrum]

you take your vector and you lie it down

ez

[weak cove]

🤷‍♂️ idk though

[magic pine]

it makes the double dual naturality much more obvious

at least imo, but i digress

[sick fulcrum]

it's natural in the literal categorical sense too

[magic pine]

well ofc

[weak cove]

Going to interrupt with a real problem now

if that's fine

[magic pine]

but the reason as to why becomes clearer from that intuition imo

ye my bad

[sick fulcrum]

what's U^0 lol

[weak cove]

the annihilator

[sick fulcrum]

U is a subspace?

ah yes this book

[weak cove]

It must be I think

axler

[sick fulcrum]

what's the prime here

[weak cove]

prime?

o

dual I think

[sick fulcrum]

right I see

[weak cove]

yeah the dual space

[magic pine]

define phi on half the domain
leave

https://tenor.com/view/mujikcboro-seriymujik-gif-24361533

[sick fulcrum]

draw a commutative diagram and then flip all the arrows

[weak cove]

😭

[sick fulcrum]

i wonder if the dual of the quotient map is actually an inclusion

or is it still a quotient map

[weak cove]

honestly what I see reading that question is just death

Idk how to even begin

[sick fulcrum]

try to mess around with V/U first

suppose phi is in (V/U)'

oh crap

we can't pick out a basis

because it could be inf dim...?

[weak cove]

i think it could be

[sick fulcrum]

right yes

ah okay

what's the codomain of pi'

V^* im guessing

yes

[weak cove]

yes

[sick fulcrum]

yeah it's just injecting Hom(V/U, F) into Hom(V, F)

sorry I use hom too much

I was doing category theory before this because I am 1.5 weeks behind

wheeee

[weak cove]

lol

L(V/U, F) -> L(V,F)

[magic pine]

austin have you tried this problem yourself

[sick fulcrum]

LMAO

holy shit no way someone actually solved this with homological algebra spam

[magic pine]

this is just showing that the Hom functor is left exact i think

makes sense lol

[sick fulcrum]

hom functor is very nice

ngl

[magic pine]

if you consult the def of pi' it should be almost straightforward

[weak cove]

yes

but

I can barely understand it

like the notation

so it is hardly much of an attmpt

[magic pine]

focus on the definitions

these are like very literate problems

[sick fulcrum]

i like to think about it by looking at the underlying set

[magic pine]

little in the way of intuition

[magic pine]

^, if you draw the arrows for the domains and codomains it should also make more sense

[sick fulcrum]

category theory helping me with freaking linear algebra

[sick fulcrum]

yeah look at this guy

also you know that if we have an f: V/U to F we have an \overline{f}: V to F, unique such that f = \pi \circ \overline{f}

[weak cove]

😭

[sick fulcrum]

i know this is a commutative diagram

but it just means

$f = \tilde f \circ \pi$

[sick fulcrum]

i messed up the composition order here

so what do we know now

this is true for every $f \in \mc L(V, \mbb F)$

[sick fulcrum]

look at the underlying sets

my professor who's teaching category theory now (topology last semester) always likes to expand out Hom sets

so you could write

{linear maps V/U, F} (relationship between these guys) {linear maps V to F}

and now stare at the kernel

[weak cove]

I know this

but Im sorry I really dont get anything else

[magic pine]

austin can you write pi'(phi) in terms of pi and phi

specifically pi and phi

not pi'

[weak cove]

Let me try

[magic pine]

if not then consult the definition

[weak cove]

pi'(phi)=phi composed with T

[magic pine]

what is T

[sick fulcrum]

gonna go to sleep best of luck

[weak cove]

goodnight

T is a linear map from V to W

[magic pine]

well it can't be any arbitrary map though

it's a specific map

[weak cove]

right

[magic pine]

like what you stated is the definition for T'(phi)

[weak cove]

pi'(phi)=phi composed with pi

[magic pine]

yes

so now consider what that means

let v in V
where does pi(v) take you

[weak cove]

I want to say the field but our phi is like weirder here

oh pi

wait

I thought you said phi

pi(v) takes us to V/U

right

[magic pine]

yes

and then phi

[weak cove]

to F?

[magic pine]

phi(pi(v)) -> F

yes

now let u in U

what does pi do

[weak cove]

nothing because pi takes from V

oh well

U is a subspace

[magic pine]

yes

[weak cove]

it maps to

0

well to 0 of the quotient space

?

[magic pine]

yes why

what is special about pi

[weak cove]

ph is the quotient map

pi*

it maps to the set of all translates of your vector

but if your vector is already in U

the set of all translates is just U

so we've done nothing

right

[magic pine]

yes that's a good geometric perspective

it reduces U to [0]

what now

[weak cove]

pi'(phi)=phi composed with pi

so pi'(phi) takes

x in V, maps it to V/U, and then

phi acts on it somehow

which sends it to F?

[magic pine]

yes

but we are interested in showing that pi'(phi) is in U^0

so we care about how pi'(phi) acts on the subspace U of V

[magic pine]

i made you say all this to make sure the order of maps and the domains/codomains were clear

fwiw i think this is a good picture of what is happening

[weak cove]

U^0 = { phi in V' : phi(u)=0 for all u in U}

[magic pine]

yes

so we want to show that if u in U

then pi'(phi)(u) = 0

[weak cove]

wait

pi'(phi) in U0 means
pi'(phi(u)) = 0 for all u in U

so that's why we want to show that

[magic pine]

hold on

it's pi'(phi)(u)

not pi'(phi(u))

[weak cove]

:o

pi'(phi) we want to show is in U0

so we have to show that

pi'(phi) is a linear functional that maps to 0 for all inputs from U

pi'(phi)(u)

I see

[magic pine]

inputs from U

[weak cove]

yes

meant to put that

[magic pine]

👍

[weak cove]

pi'(phi)(u) = phi(pi(u)) ?

[magic pine]

yes by def

[weak cove]

and

pi(u)=0

kind of

[magic pine]

pi(u) = [0] yes

[weak cove]

pi(u) is the zero in V/U

[magic pine]

because its coset is 0 + U

[weak cove]

so we're taking

phi(0)

and phi is linear so this has to go to 0?

[magic pine]

yes

[weak cove]

Okay

that makes a little more sense

am I writing these sets correctly?

[magic pine]

yeah looks good

[weak cove]

Okay

In the last one

where did we even use that

phi was in

(V/U)'

I feel like all we used was that phi was linear

not that it mapped specifically that set

[magic pine]

well pi : V -> V/U

pi' : L(V/U, F) -> L(V,F)

so pi' should take some element in L(V/U,F), which our phi was in

now the important part isn't that phi is in (V/U)', i guess

it's that U is a subset of ker(phi o pi)

[weak cove]

I guess I'm still a bit confused about the 0 part aswell because pi(u) sure it is 0 in V/U but like that's an entire set of elements right

it's the whole space U

[magic pine]

yes, but in V/U that's just a single element

[weak cove]

and we can just say

the whole space maps to 0 because phi is linear?

[magic pine]

wdym

U maps to 0 under phi o pi

but not V as a whole

[weak cove]

Like

the naive argument is

pi'(phi)(u)=phi(pi(u))=phi(0) which since phi is linear = 0

but the 0's are like

not actually 0

so I'm getting al ittle confused

[magic pine]

well but they are

they are additive identities in their respective vector spaces

[weak cove]

so writing this is just A okay

I feel like I am missing

something

[magic pine]

i would write it as pi'(phi)(u) = phi(pi(u)) = phi([0]) = 0_F

but this is really it

[0] is the additive identity in V/U

0_F is the additive identity in F

this is all valid

[weak cove]

yeah

[magic pine]

also, this generalizes a little bit
if phi is in L(V/U, W), then pi'(phi)(u) = 0_W for any u in U, and it is the exact same argument

the takeaway is that if phi is injective, U contains the entire kernel of (phi o pi)

once you get to group theory you'll see why this is more relevant

it touches on the firs isomorphism theorem

[weak cove]

alrighty

Are these sets fine?

[magic pine]

that last one is confusing me

[weak cove]

same

[magic pine]

V/U = {v + U | v in V}
where v + U = { v + u | u in U}

alternatively, V/U = {[v] | v ~ w iff v - w is in U}

[weak cove]

I'm trying to make the sets because I have to show V is isomorphic to U X (V/U)

but I don't know how to express U X (V/U)

[magic pine]

what exactly confuses you abt it

[weak cove]

I guess the step after the green

I'm trying to express it in terms of vectors from V and U

alone

but I can't get to what the purple set should actually be

[magic pine]

you should see V/U as V with every vector in U sent to 0

the exact definitions i wrote above and you have as well

[weak cove]

V/U = { v +U| v in V}

[magic pine]

yes
now if v is in U, then v + U = 0 + U

so any vector in U is in the same coset/equivalence class as 0

im trying to see what the best way to define this isomorphism is, though i think showing existence is the best way funnily enough

[weak cove]

is it like going to be taking a vector in V and splitting it into a component from U and a component not in U?

[magic pine]

sort of yeah

[weak cove]

blunt

but here's wht I found on SE

[magic pine]

oh we have finite dimension

[magic pine]

i forgot

[weak cove]

lol ignore that

[magic pine]

i think what they said still works

they edited it after that comment

[weak cove]

but they did not edit the part about U intersect U' ={0}

what is AC?

[magic pine]

the old answer just had the wrong definition for U'

the new answer is correct

[magic pine]

axiom of choice

but that's for the case of infinite dimension

[weak cove]

Okay I'll just steal their isomorphism then

amazing

[magic pine]

yeah just show it is a linear bijection and you're done

[weak cove]

that leaves me with just the counterexmaple one

[magic pine]

(and show it doesn't depend on a representative)

[weak cove]

a representative?

[magic pine]

so phi(u + u') = (u, u' + U) = (u, \tilde{u} + U) for any other representative \tilde{u}

cause remember you can have multiple representatives for an equivalence class

if that isnt clear lmk

[weak cove]

it isn't sorry

[magic pine]

let's say we take a homomorphism from Z -> Z_2

or the other way

Z_2 -> Z

[weak cove]

also is this supposed to be a comma

[magic pine]

no

they're saying phi(v) = phi(u + u') = ...

since v = u + u'

[magic pine]

define it by phi(n) = 2n

this is "not well defined" in the sense that
phi(0) should be the same as, say, phi(2)

since 0 and 2 are the same mod 2

so which representative you use for the class makes the map change

[weak cove]

okay

that makes sense

[magic pine]

there is a nice counterexample here i believe

to the dim thing

[weak cove]

is it like

the zero map

[magic pine]

oh that works

really well

nice

you can crank out the details but that should be good enough for any map coming from a nontrivial vector space

[weak cove]

lmao okay now help me explain why because like

range T = 0

dim 0

[magic pine]

well just take 0 : V -> V

[weak cove]

seems good

[magic pine]

where V is not 0-dimensional

so ye like you said

dim range T = 0

now you want the annihilator of range T, so the set of linear maps T : V -> F that take 0 to 0

[weak cove]

oh

all of them

lmao

[magic pine]

yup

which has dimension > 0

[weak cove]

i think my sets were wrong thn earlier

but thats fine

should I know what the dimension is

I feel like I might've learned that before

the dimension of the set of linear maps from W->F

[magic pine]

the set of linear maps is a vector space like any other

so you should think of it as F^n

[magic pine]

recall that W* is isomorphic to W

so the dimensions should match

[weak cove]

ah right

I’m not sure I’ve actually shown that last implication

Trying to prove injective Ty

[magic pine]

u' + U = v' + U implies u' - v' is in U

[weak cove]

I know that

[magic pine]

but recall that we constructed our vectors in the form u + u' for a reason

u is in U

what is special about u'

[weak cove]

It isn’t

So they have to sum 0

So they’re equal

[magic pine]

it isn't (or is 0)

[weak cove]

Yeah

Now is finite dimension going to allow me to be done here

Injective => invertible

Or is it not really finite

[magic pine]

do you have equal dimension

[magic pine]

i think it is from the question

[weak cove]

Doesn’t matter

[magic pine]

please elaborate

[weak cove]

Nvm

It does matter

But they’re isomorphic so surely they have the same dim

😂

[magic pine]

based

[weak cove]

How to show surjective 😭

[magic pine]

take (w, v + U) in the codomain

that v in the v + U can be written by our basis as (u + u')

so that v + U = u + u' + U = u' + U

[weak cove]

So

Whatever v = u + u’?

[magic pine]

yes

also i shouldnt be reusing the u

there

so now you have (w, u' + U)

whose preimage is obvious

if you look at your defn for the isomorphism

[weak cove]

Literally love tou

You

I definitely would not have finished this without you 😭

I’m washed gotta study over the weekend

[magic pine]

L

[weak cove]

Bro this class it went from literally baby cakes

To now I don’t know anything

In like 2 days

[magic pine]

dual n quotient spaces do be like that

also ive never done quotient spaces in linalg before but they are really similar to quotient spaces in group theory and topology
so it is very good that you are seeing them now and building experience w them

when you get to the group theory ones it'll be a breeze sorta

[weak cove]

i hope so

[magic pine]

also before we go, i wanted to say the important thing about the other thm we proved with the pi'(phi)

it tells you that any linear map L : V -> W can be turned into an isomorphism
phi : V/U -> L(V)
where U = ker L, and phi is defined by pi'(phi)(v) = phi([v]) = L(v)

we proved that this lets us get injectivity, and surjectivity comes from restricting to the image of L

this is similar to the first isomorphism theorem in algebra like i said, which is super important

[weak cove]

yeah ryb sent a picture of this earlier

but like for our problem

I don't think I did the thing about

[weak cove]

this

I didn't really get at what step that came in

[magic pine]

you kind of do in the surjective part

actually in the injectivity part

[weak cove]

so it's fine then to just prove injective, surjective, linear map?

[magic pine]

ye should be

the representative part is actually important when going the other way, so quotient space -> some other space

my b

[weak cove]

no problem

adequate?

[magic pine]

yes, but start the counterexample by T : W -> W, T(w) = 0, where dim W > 0

or if you want it simpler just say
T : R -> R, T(x) = 0

[weak cove]

👍

okay tysm

gotta go

.close

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[boreal vine]

Please help me understand how the 3 part piecewise function was derived from the interval table?

[restive river]

here?

[boreal vine]

Yes

[restive river]

wanna know it?

[devout snow]

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[subtle wyvern]

is this unsolvable

[restive river]

ez

[solar goblet]

try writing it out algebraically

let a be the first number, b the second

[restive river]

algebra = this is possible

[subtle wyvern]

x^2 + (x-3)^2 = 117, x^2 + (x+3)^2 = 117

in both equations either one of thee answers are negatibve

[restive river]

a square and b square = 117

now just do algebra

[subtle wyvern]

oh nvm im retarted

[solar goblet]

its a trinomial

there are 2 roots

[subtle wyvern]

I alr solved it

[restive river]

subsitute and expand the squares

[subtle wyvern]

but I didnt think properly

[solar goblet]

good job

[restive river]

then solve quadriatric expression

[subtle wyvern]

because I solved for x

the other number is x+3 or x-3

depending on the equation

mymistake

[restive river]

solve for a and b

bro help me

[subtle wyvern]

what

[restive river]

#help-26

[subtle wyvern]

+close

.close

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[restive river]

hello

[ivory isle]

thanks for stealing the channel as im typing

[restive river]

I am new on this app I dont know how to exactly use it

[restive river]

sorry I don't get you sir

[ivory isle]

what is your question

[restive river]

tbh I do have questions but I dont know how to send

[restive river]

I am very noob on this app

[ivory isle]

just upload a picture or write it as text

[restive river]

ok wait

[restive river]

question number 17

[winter patrol]

what have you tried

[atomic glade]

Number of black cards remaining/ Total number of cards remaining

[devout snow]

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[devout snow]

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[smoky girder]

Expand and simplify (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca).
Hence, show that if a + b + c = 0, then a^3 + b^3 + c^3 = 3abc.

[smoky girder]

Im not sure how to do the second one

[sturdy yew]

did u do the first part

[smoky girder]

ye i did it with a calculato

[sturdy yew]

what did u get

[smoky girder]

a^3 + b^3 + c^3 = 3abc.

[sturdy yew]

u mean -3abc?

[smoky girder]

oh yea

its expression so just
a^3 + b^3 + c^3 - 3abc.

[sturdy yew]

so now u have a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca)

what happens to the eqn when a+b+c = 0

[smoky girder]

ohhhh

u would get a^3 + b^3 + c^3 - 3abc = 0

[sturdy yew]

yep

[smoky girder]

and then u just move the 3abc?

wow

tyvm

[sturdy yew]

also note that the same thing happens when a=b=c

i wud leave it to u to prove that if u r interested

[smoky girder]

ok ty

really appreciate it

.close

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[boreal vine]

Please help me understand how the 3 part piecewise function was derived from the interval table?

[dim moat]

I would do it in another way:
Recall that:

$|x| = \begin{cases}
x & x\geq 0\
-x & x \leq 0\
\end{cases}
$

[woven radish]

cristorenzo99

[devout snow]

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[deep vortex]

Let $f: \mathbb C^{5} \to \mathbb C^{2 \times 3}$. Can this be injective, surjective or bijective?

[restive river]

what do you think

[deep vortex]

So for f: V -> W we have that dim V = dim Kern(f) + dim Image(f)

Also, f is injective iff Kern(f) = {0_V}

[restive river]

yes

[deep vortex]

So we have that dim V = 5

5 = dim Kern(f) + dim Image(f) and so 5 - dim Kern(f) = dim Image(f)

[deep vortex]

So dim Image(f) <= 5, right

Never 6

Which is dim C^(2 x 3)

So this is not surjective

And so not bijective

Now what about injective

[restive river]

indeed

now think about the definition of injectivity

[deep vortex]

Also, f is injective iff Kern(f) = {0_V}

I guess we have to use this

[deep vortex]

But how do we use that?

[deep vortex]

Are you saying dim Bild(f) = 0 if dim Kern(f) = 0

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[ripe grove]

I need to prove that angle n is 90 degrees

[tawdry wave]

Angle N being 90° is a given

Because LN is perpendicular to NP

[ripe grove]

Thanks, i somehow missed that

[tawdry wave]

It happens, no worries

[ripe grove]

.cloee

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[pulsar ermine]

I kind of understand why the Cauchy criterion is written like that in set logic, but I was wondering whether if it’s always the case that we can exchange for all as intersection, there exists as union?

[devout snow]

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@pulsar ermine Has your question been resolved?

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[burnt token]

i have CDF, i found its PDF. I now need to find PDF of Y=ln(x). Do I need to start with CDF and go to PDF or can I just start at PDF and do transform there?

[tall knoll]

depends on which method you were taught

(in actuality the two "methods" are the same, just one uses the result of the other)

[burnt token]

so i can just transform PDF f(x) into f(y)?

usually with transforms you are supposed to start with F(X) so that you can F(X) = P(X <= x) = P(Y <= x) etc

but i am missing on how to fro directly from f(x)->f(y)

[tall knoll]

if you apply this method on some general function g(X) you can figure out a way to go directly between pdfs

[devout snow]

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[burnt token]

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[lapis pelican]

can someone help me with this question pls

[devout snow]

Please don't occupy multiple help channels.

[lapis pelican]

!status

[devout snow]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[lapis pelican]

2

[final storm]

!show

[devout snow]

Show your work, and if possible, explain where you are stuck.

@lapis pelican Has your question been resolved?

[lapis pelican]

<@&286206848099549185>

[rustic jetty]

You know Ava's speed, and how long it took her to get from point G to point F. Can you find the distance between the points?

[lapis pelican]

16 miles

@rustic jetty

<@&286206848099549185>

[rustic jetty]

Bro calm down

[lapis pelican]

t-16?

[lapis pelican]

i need to finish off this question today

oh

how

[rancid flare]

I thought it had been solved?

[lapis pelican]

when?

t+16

[rancid flare]

In another help room?

[lapis pelican]

no i guess it's been solved half way thru

what do i do next?

[rustic jetty]

call ava's speed some $S_a$, and ben's speed some $S_b$

[woven radish]

Dork9399

[rustic jetty]

and same for their time, $T_a$, and $T_b$

[woven radish]

Dork9399

[lapis pelican]

that's what i did^^^

[rustic jetty]

mb im just thinking out loud

So in terms of $T_a$, what is the distance EG?

[woven radish]

Dork9399

[rustic jetty]

Assume that $T_a$ is the time taken to travel distance EG, and $T_b$ is time taken to travel GF

[woven radish]

Dork9399

[lapis pelican]

^^^

[rustic jetty]

You know that EG is T_a/60

mb

and what is EG in terms of S_b?

[lapis pelican]

^^^

as long as T_a is measured in hours

would their time be equal?

<@&286206848099549185>

[devout snow]

@lapis pelican Has your question been resolved?

[lapis pelican]

i guess you need to use the same units

[devout snow]

@lapis pelican Has your question been resolved?

[devout snow]

@lapis pelican Has your question been resolved?

[devout snow]

@lapis pelican Has your question been resolved?

[devout snow]

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[modern badge]

is there a some kind of "x cannot be 1" rule in these type of questions or what?

[modern badge]

because i get two values for x

[topaz axle]

yeah they clearly both work

[faint hearth]

You need to see what happens if you stop the fraction at a certain point

For example, 4-3=1
4-(3/(4-3))=1

You can keep continuing this

And you will see it always equals 1

[modern badge]

let me try it

[faint hearth]

So the correct value here should be 1

More formally, you are finding $\lim_{n \rightarrow \infty} (a_{n})$ where $a_{n} = 4-\frac{3}{4-a_{n-1}}$ with $a_{1}=1$ and $ n \geq 1$

[woven radish]

smidgin

smidgin

[modern badge]

great explanation thank you

.close

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[tender tulip]

Last 2 digit of
$\lfloor \frac{2^0}{3} \rfloor + \lfloor \frac{2^1}{3} \rfloor + …. + \lfloor \frac{2^{1000}}{3} \rfloor$

[woven radish]

r

[feral agate]

!status

[devout snow]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[tender tulip]

1

Modulo 3 or 100

[feral agate]

Why would you want to use 100?

Ah coz of the

Last 2 digits

You can calculate that value exactly actually, so you don’t need to use mod 100

[green kelp]

geometric progression

[feral agate]

Use $x=\lfloor x\rfloor + {x}$

[woven radish]

kheerii

[feral agate]

Yes and that

[tender tulip]

the decimal part is 1/3?

[patent thistle]

2^k/3 either leaves 1/3 or 2/3 decimal part and you can count how many instances of each show up among 2^0/3 through 2^1000/3

[feral agate]

Close

It will alternate between 1/3 and 2/3

[feral agate]

Yeah smth like this

[tender tulip]

For every even k, the decimal part is 1/3
For every odd k, the decimal part is 2/3

Can someone explain? I still dont understand

[patent thistle]

i think what was being suggested was computing 2^0/3 + ... + 2^1000/3 and then subtracting the decimal parts to get the sum in the question

then you can try to work it mod 100

[tender tulip]

Ok thanks

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[solar heron]

hey there! i'm trying out some of the IJMO questions to prepare for an upcoming exam, how would i begin thinking about this?

[solar heron]

in my head, i'm imagining something like this, and i've noticed that if you split the triangle down the middle, you get a triangle with angles of 90, 60, & 30

but that's all i've got so far

[sturdy yew]

join the 2 points B and C to the centre

[solar heron]

and where would i go from there?

[sturdy yew]

drop a perpendicular to the line BC from the centre of the circle

[solar heron]

i'll have a little stare at this for now, thank you!

should i close this channel or keep it open while i stare, just in case i need to come back?

[sturdy yew]

keep it open ig

[solar heron]

i don't think i'm quite getting anywhere, what would be next?

[devout snow]

@solar heron Has your question been resolved?

[sturdy yew]

a perpendicular dropped from the centre of a circle to any chord, bisects the chord

use this and trigonometric ratios to find 1/2BC

also u can find angle BOC(O is the centre of the circle) using angle sum property of a quadrilateral on the KITE ACBO

[solar heron]

ah, i got it

thank you!

how do you start thinking about getting to that point though, would it just be faffing around, seeing what points you can connect and what you get from those?

[devout snow]

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[devout snow]

@fluid axle Has your question been resolved?

[devout snow]

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[muted barn]

Whats the cube root, and fourth root and so on as exponents

[muted barn]

?

[restive river]

,, \3[m]{x^n} = x^{\ff nm}

[twilit comet]

cbrt x = the number which must be cubed to get x

fourth root x = number which must be raised to an exponent of 4 to get x

so on

[restive river]

so [
\s[3]{x} = x^{\ff13}
]
and you can apply the same logic to the rest

[twilit comet]

let's say that $t = \sqrt[4]{x}$... then $t^4 = x$

[muted barn]

Oh. And fifth root is x^0,2

[twilit comet]

yes

[muted barn]

Aight thank you ren

.close

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[woven radish]

ren

[twilit comet]

np

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[high zephyr]

Is this possible

?

[solar goblet]

i cant make out the faint writing

[high zephyr]

Is it possible to write a natural number at each vertex of the cube so that all eight numbers are different, but the product of the numbers at the vertices of each face is the same? If it is, give an example and if not, explain why.

[solar goblet]

i meant the numbers

[high zephyr]

I need to find the numbers

[manic plank]

center: 3
down left front face: 4
top left front face: 2
down right front face: 1?
top right front face: 24
bottom side face: 16
top side face: 5
last one: 8

That's what I can make out

[devout snow]

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• Show your work, and if possible, explain where you are stuck.
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[shy fossil]

Hello everyone, i need help with the Compound interest calculation. I havent figured it out how to calulate the beginning captial of 12500€ for the duration in 4 years, the interest rate is 2.2%, but i have to covert it to the interest factor. Idk what the interest factor on that is. It would be nice if i could get some help on this.

[shy fossil]

i got 13530€ on that one, but idk if thats correct

i used the formula
Kn: K0 •qn

K4=12500ۥ 2,02%=13.530,402 yeah

the sloution is different tho idk if I calculated it correctly

[sage burrow]

12500 x 2.02 cant be 13530... your writing differs from your calculation.

you are calculatiing 12500 x (1.0202)^4 = 13530.402

and yes, thats right (in my opinion).

[devout snow]

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Remember:
• Ask your math question in a clear, concise manner.
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• After 15 minutes, feel free to ping <@&286206848099549185>.
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• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[merry sundial]

I tried solving the diff equation

[merry sundial]

It looks like I made a mistake in that step

[stuck field]

well your rhs looks kinda weird, both yours and the one in the image above

It's (x^2-5)^2/|x^2-5|

[merry sundial]

Ou

?