help-49

[last slate]

anyone?

[fallow scarab]

Read

[midnight plank]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question!

[last slate]

must you reduce your matrix to REF in order to find the basis? Or can you just take the row vectors/column vectors as your basis?

[last slate]

i know i went to screenshot the theorem

[blissful pier]

as long as they are linearly independent

[last slate]

thats the thing like how do you know

and why do people do REF

[blissful pier]

A = nxn matrix and you can rref to identity (which I assume is what you mean) then A is invertible and columns of A are linearly independent, so columns of A form a basis of Rn

[last slate]

so your safest bet is to do REF?

[gaunt imp]

The rows/columns of a matrix form a basis iff they're linearly independent

Which happens iff the determinant is non-zero

[blissful pier]

-# (and spans Rn)

[gaunt imp]

(if I remember correctly, this had to do with the definition of the determinant being a multilinear function with det(I)=1)

[last slate]

you mean a subspace in R^n

[gaunt imp]

Well, the guy was talking about an nxn matrix in R^n

If the rows/columns are linearly independent, they will always span R^n

[last slate]

Btw im talking about any mxn matrices

not necessarily nxn

[gaunt imp]

Oh, that's different then

[last slate]

”mxn”

[gaunt imp]

Well, if you have an mxk matrix within R^n, first, the rows/columns of the matrix won't be vectors of R^n

(unless you add 0s)

[gaunt imp]

Alright, I see now

Note that the dimension of your whole space changes depending on whether you take the row space or the column space to fit the choice of m/n

[last slate]

yes

but now must you apply REF

to be safe

[gaunt imp]

And answering this directly, if your columns/rows are linearly independent, you can take them as they are

They will form a basis of the subspace regardless

When you need to apply REF is when they're not linearly independent

At least, if memory serves me right

[last slate]

Say i dont wanna spend time checking for linear dependency

can i just do REF for all cases

[gaunt imp]

Fairly sure it would work, yeah

[last slate]

i assume if there was dependency, ill have less rows/ columns after applying REF?

[gaunt imp]

Iirc, you'd get a row/column of zeroes

[last slate]

yeah, but whys that?

what does REF do that’s special that reduces redundancy ?

[gaunt imp]

Because REF is essentially applying gaussian elimination

That is, it's trying to generate zeroes

[last slate]

yeah but

whats the relationship with linear independence

[gaunt imp]

If a set of vectors say V_k is linearly dependent then there exists some combination of coefficients such that the sum of all a_k(V_k)s is zero

This is an if and only if

Through REF, what you're trying to do is find as many zeroes as possible in the last row, if the biggest number of zeroes is all, that's what you'll get

[midnight plank]

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[midnight plank]

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[twilit field]

$\mathcal{A}T=T^$. Let $v_1,v_2,\dots v_m$ be eigenvectors of $T^$ with $\lambda_1,\dots,\lambda_m$ being the corresponding eigenvalues.
\ Then $\mathcal{A} T(v_i)= \lambda_i v_i$.

[grand pond]

wai

[twilit field]

I'm unsure if it's lambda_i or 1/lambda_i

[prime hornet]

I'm not quite sure where you're going with this

the eigenvalues of T* may have nothing to do with the eigenvalues of A at all

[twilit field]

mhm

[prime hornet]

you'll have better luck examining A on its own, without thinking of eigenvectors of T*

[twilit field]

will do

tq

[prime hornet]

(one word hint if you need it: ||composition||)

[twilit field]

not sure how that helps

oh

[prime hornet]

think about ||composing A with itself. what is this map?||

[twilit field]

A^*

[prime hornet]

eh? how did you decide on that?

[twilit field]

oops

A^2

[prime hornet]

yes yes, but what is this map?

it's something special, not just A^2

[twilit field]

I'm quite unsure

[prime hornet]

let it act on an operator T

that is, find A^2(T)

[twilit field]

$AT^=T^$

[grand pond]

wai

[twilit field]

it is just T*

[prime hornet]

that's AT

what's A^2T?

[twilit field]

T

[prime hornet]

so what is A^2?

[twilit field]

identity map

[prime hornet]

yup c:

can you take it from here?

[twilit field]

all eigenvalues would be 1 , no( for A^2)

[prime hornet]

you're right, but we can use this to think about what the eigenvalues of A might be

[twilit field]

1 again

[prime hornet]

hmm, 1 certainly works, but it's not gonna be the only possibility

if lambda's an eigenvalue for A and T is nonzero, we're gonna have AT = lambda T, and hence (spoilers ahead! don't click on it if you want to think about it more) ||A^2T = lambda^2 T. with T nonzero, we get lambda^2 = 1, so lambda = +- 1.||

[twilit field]

Oops

Right

Got it

Thanks so. Much

[prime hornet]

ah, before you go

notice that I only gave a necessary condition on the eigenvalues of A

it's not necessarily the case that the eigenvalues are actually both values

A could have only one eigenvalue value, for example

but it does turn out that A has both; keywords: self-adjoint operator, skew-adjoint operator

[twilit field]

tq

.close

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[twilit field]

I think it is?

[solid iris]

can u think of a very short proof?

[twilit field]

just chose eps=delta

[lavish venture]

why even involve epsilons here

[twilit field]

🤦

[lavish venture]

do you know any theorems about continuous functions on compact sets?

[twilit field]

right

yeah, just realised

(0,1) isn't compact though

[0,1] sure

[lavish venture]

ok but if it’s uni cts on [0, 1] and (0, 1) is a subset then surely this is fine?

[subtle blaze]

Is this not even lipschitz continuous

[twilit field]

no idea what that is

[twilit field]

yeah, silly me

[subtle blaze]

It bounds the derivative

[twilit field]

( We haven't done derivatives yet, so checks out)

[solid iris]

i was kinda hoping wai could say this instead

[lavish venture]

🤔

[twilit field]

I think this has the exact same answer

xsin(1/x) is cont on[0,1] and thus uniform cont too

[lavish venture]

is it continuous on [0, 1]?

it’s not even defined at 0

[twilit field]

you can define it to be 0.

the limit is def 0

[lavish venture]

ok but i don’t see h defining it to be 0

[subtle blaze]

Just cos you can turn it into a continuous function doesn’t mean it is

Just cos I can turn ℚ into a complete field doesn’t mean ℚ is complete

[solid iris]

@lavish venture @subtle blaze try again

[lavish venture]

🤔

[subtle blaze]

Huh?

[lavish venture]

wut

mate it’s not even defined at 0

😭

i don’t see a piecewise definition here sir

[subtle blaze]

I’m so dumb

0 isn’t even in the domain bro

[solid iris]

it has continuous extension as wai was probably thinking but didnt say fully

[subtle blaze]

@lavish venture we’re both blind bro

wtf

[fallow scarab]

https://math.stackexchange.com/questions/1543771/prove-fx-x-sin1-x-is-uniformly-continuous-on-0-1#1543780

[lavish venture]

no but wai asked if we can use the same argument?

[lavish venture]

^

[fallow scarab]

Oh yes the same argument doesn't work

[subtle blaze]

Ah

[lavish venture]

that’s what all of this was about lmao

[subtle blaze]

I’m so completely out of it

Serves me right for trying to do ra

[fallow scarab]

Only assuming the function is defined differently at 0

[lavish venture]

@solid iris try again

[solid iris]

so same solution except dont be scared of continuous extensions!

[twilit field]

lemme prove that result

if a continuous function can be continuously extended to a closed bounded interval it's uniformly continuous.

[solid iris]

its not a very deep statement

[twilit field]

yea, just give me ~2 minutes, a bit stressed rn

[solid iris]

upgrading continuity to uniform continuity comes for free on a compact set

[fallow scarab]

But I bet the proof is still tedious

[solid iris]

the other thing is uniform continuity on a set implies the same on any subset

[lavish venture]

i mean this is trivial

it’s just built into the universal quantifier

[twilit field]

got it

tq

.close

[midnight plank]

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[solid iris]

np!

[midnight plank]

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[last slate]

I need help

[last slate]

part b

can anyone help

thanks

[lyric charm]

Nul(A) = Nul(B)

[last slate]

yeah idk how to get that

i dont even get the intuition its pissing me off i dont get what im doing with Ax=b

<@&286206848099549185>

[barren loom]

https://en.wikipedia.org/wiki/Kernel_(linear_algebra)#Illustration has a worked example

Here B has x=z, y=-z, so (1,-1,1)^t is in the null space as you can check by multiplying with A, col1-col2+col3=0

[midnight plank]

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[proud violet]

$T_2^n$ is composition n times right?

[grand pond]

Beth

[shut canyon]

I think it is iteration, composition then

f^n

f∘…∘f

[proud violet]

You can do the first part right?

[proud violet]

Yes got it thanks

[shut canyon]

upside down parabola

[proud violet]

Well,,, with a sharp point so kind of but not really

[shut canyon]

I'll try to draw it, I don't yet see what is going on

[proud violet]

When you apply the function twice it’s like making 2 tents across [0,1] and after that with another application 4 tents etc.

[shut canyon]

this is rx(1-x), r>4

I believe

so maybe similar?

[proud violet]

Ummm. It looks like a tent

This shape (of course with axes and that stuff)

[shut canyon]

Thank you for your help @proud violet I'm grateful.

T^n can be drawn? infinite drawing?

[proud violet]

Well, each T_2^n can be drawn as n is finite, but of course as n tends to infinity you can’t get an accuracy graph. I imagine it will look like a rectangle but still be a function

[shut canyon]

Look, I visualise it like so

[shut canyon]

do you know if phase portrait can show the periodic points? I don't yet know how to draw phase portrait

[proud violet]

I don’t know tbh

[shut canyon]

Use the graph of T^n to conclude T has exactly 2^n periodic points of period n.
I don't understand this just yet

[proud violet]

I’m sorry idk how to screenshot. But this is composing twice

[shut canyon]

how to conclude, with induction?

[proud violet]

Try first to get a general expression for T_2^n (you can show it works by induction)

[proud violet]

Yes

[proud violet]

But since it wants you to use the graph, you can also just use it

[shut canyon]

Oh, maybe not use the graph to conclude but reason algebraically also

[proud violet]

It’s a good exercise to do this but it’s also a pain lol

There is lots of symmetry in the graph as you would expect

[shut canyon]

group theory?

it's like iterated reflection , mirror^n

[proud violet]

Yeah a little bit

I’m thinking that it’s taking the line [0, 1] and pitching a tent and is doing the same across split intervals with repeated application

[shut canyon]

Nice analogy
⛺∘…∘⛺∘…∘⛺

⛺ : ⛺ + ⛺ ⟶ ⛺

[midnight plank]

@shut canyon Has your question been resolved?

[shut canyon]

let me try to write the proof

[shut canyon]

2-periodic means T₂(T₂(x)) = x and T₂(x) ≠ x.

x ≤ ½

→ T₂(x)=2·x

if 2x ≤ ½ → T₂(T₂(x))=4·x=x → x=0 ❌ (fixed)

• Case: x ≤ ½, so T₂(x) = 2·x
• Subcase: 2·x ≤ ½ → x ≤ ¼
• Then T₂(T₂(x)) = T₂(2·x) = 2·(2·x) = 4·x
• Solve 4·x = x → 4x - x = 0 → 3x = 0 → x = 0
  T₂(0) = 0, it’s a fixed point, not 2-periodic.

if 2x > ½ → T₂(T₂(x))=2-4·x=x → 3x=2 → x=⅔ ✅

x > ½

→ T₂(x)=2-2·x
→ T₂(T₂(x))=x → x=⅓ ✅

2-periodic points = ⅓, ⅔

So the 2-cycle is x = 1/3 ↔ 2/3 🏓⛺

[shut canyon]

.close

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[rough schooner]

why you get squrt(10)/2?

[midnight plank]

Please don't occupy multiple help channels.

[solar hearth]

@rough schooner were you given cot A

[solar hearth]

b=3a, so b^2=9a^2
a^2+b^2 = 10a^2=5^2=25

[grand flower]

@rough schooner

[midnight plank]

@rough schooner Has your question been resolved?

[vagrant shoal]

yo Can someone really cracked with maths and chill DM me please I need some help

[midnight plank]

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[twilit field]

show $\sqrt{x}$ is uniform continuous on $[0, \infty)$

[grand pond]

wai

[twilit field]

so if eps is given we have $\frac{x-y}{\sqrt{x}+\sqrt{y}}< \varepsilon$

[grand pond]

wai

[twilit field]

the issue is x-y doesn't have to be less tha this , does it

[quiet hinge]

I don't understand your concern

[twilit field]

my concern is , how does this allow me to chose delta

[runic hamlet]

well |x-y| < delta so the x-y is not the issue

the sqrtx+sqrty is an issue

[twilit field]

how do I fix that

[midnight plank]

@twilit field Has your question been resolved?

[twilit field]

tq

.close

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[smoky briar]

how do these type of functions usually work

[smoky briar]

the notation i mean

are those outputs?

like it outputs 0 if w*x + b <= 0?

[lyric charm]

thats a piecewise function

def f(x):
  if w*x + b <= 0:
    return 0
  else:
    return 1

[smoky briar]

oh wow that's interesting

o.w means otherwise i assume

[smoky briar]

didn't know this existed in math? basically a small program lol

.solved

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[last slate]

yo

[toxic prism]

yo

what question?

[last slate]

whats a good way of figuring out how many vectors C) is

[sudden yacht]

There are infinite vectors

[visual tiger]

talking about dimension?

Dimension is the "degree of freedom" of a subspace

in how many independent directions can you go

Since you need two variables, s and t, to describe W, that should set you on the right track

To prove it more rigorously, you can rewrite W as Span(some list of vectors)

all you have to do is to check that the list you wrote is LI (linearly independent)

[last slate]

can you check for that intuitively

[visual tiger]

well, for example, t acts on the 2nd coordinate while s doesn't

so t is "independent" from s

in most cases, you'll have to check manually for linear independence

as soon as you get three variables to describe a subspace, it isn't as easy as "one isn't a multiple of the other"

[last slate]

chat gpt split the coordinates into linear combination of 2 vectors is that a good way to do it?

[visual tiger]

that's what I suggested

W = Span(......)

[visual tiger]

but then there's still this step

[sudden yacht]

Also, it might help writing that "general form" in the following fashion:

s(1, 0, 1, 2) + t(4, 1, 0, -1)

@last slate

This helps both find the generators and the dimension

[last slate]

@visual tiger this is what you meant right

[visual tiger]

from this writing you can see W = Span((1,0,1,2), (4,1,0,-1))

[last slate]

but by just looking at it is there a way to tell in which dimention the set is

[sudden yacht]

"In which dimension" doesn't mean anything, fyi

Or, better, the answer would be R⁴

[bold knoll]

the dimension is the number of linearly independent vectors that span the set. You’ve found two linearly independent vectors that span the set, so the dimension is two

[sudden yacht]

#help-49 message

[last slate]

i mean is the whole idea how many variables there are?

can you not have 1 variable but have your set of vectors be linearly independent ?

[fallow scarab]

Stop using chatgpt and read a real book

[last slate]

i am reading a real book 💀

just because i asked it for something once or twice doesnt mean im using it to learn

[sudden yacht]

Sure, W = {(t, -2t, 3t, 0), for t real}

[last slate]

yeah so it’s wrong to just see how many variables you have right?

im trying to imagine it geometrically

[bold knoll]

What do you even mean by “variable” here?

[last slate]

input

like t

[sudden yacht]

You can't, because those vectors live in R⁴

[last slate]

not this particularly

[bold knoll]

No, it’s not sufficient to count the number of variables

[sudden yacht]

It's called parameter

[bold knoll]

You may miss linear dependence

of some of the vectors you find

you have to actually check that they are linearly independent

[last slate]

i mean in general, im trying to picture how the vectors would be graphed. It worked fine for me in part a

[sudden yacht]

I mean variable might be fine, bit honestly I never heard it in this context

[last slate]

I thought so too

[bold knoll]

You can’t picture 4 dimensional space, and you don’t need to. You just need to check the definition algebraically

[last slate]

the way im thinking of it is seeing how the vectors are drawn

[last slate]

yes but say its R^3

[bold knoll]

The human mind can only picture up to 3 spatial dimension, so you’re out of luck here

[sudden yacht]

Then you're changing the exercise

[bold knoll]

better to learn the general algebraic method

which doesn’t rely on visual intuition

[last slate]

im asking in general,

Is it better to imagine how the vectors are plotted then you would see in which span they live, or is it better to split it which I don’t fully understand

[last slate]

but I don’t fully understand how it relates to the original vector

[sudden yacht]

Huh? Which original vector?

<@&268886789983436800>

[last slate]

like generally

the splitting part

can we do b

[sudden yacht]

Sure

t(5, -3, 1, 1)

So you have only 1 generator, meaning the dimension is 1

[last slate]

But was t always meant to be a scalar?

or is it just the input like in (x,y)

thats what’s confusing me

is t the t axis or what

[sudden yacht]

Yeah of course. (It's also written in the definition of W)

[sudden yacht]

No no

It's just a parameter

[last slate]

What about (x,y,z)

Thats the definition for R^3

is it the same

or no

[sudden yacht]

The same compared to what??

[last slate]

like when they say ℝ ^3 = { (x,y,z) : x,y,z ∈ ℝ }

see how similar it looks

x direction right, y direction left, and z up

you know

[sudden yacht]

Well, yeah, you can view it as 3 parameters giving you dimension 3

[last slate]

But then with this problem

youre going t some direction?

[sudden yacht]

Exactly

[last slate]

but x,y,z arent scalars right

[sudden yacht]

Here #help-49 message you mean?

[last slate]

yes

they are just coordinates

right

isnt t also a coordinate like i dont get that

[sudden yacht]

Of course they are, you've (correctly) written that x,y,z ∈ ℝ}

[sudden yacht]

I don't know how you can visualise it as a coordinate

[last slate]

can i restate my question?

say you have (x,y) you think of this as “go x a certain direction, then go y a certain direction “

similarly here i see the parameters as just going a certain direction

so I don’t understand the splitting part and treating t like its scaling some vector (5,-3,1,1)

get what i mean @sudden yacht

[sudden yacht]

Yes and no

[sudden yacht]

It's just to "extract" the said direction

Which is exactly the generator vector

[last slate]

like how did you suddenly go from:

“Go 5t some direcrion, go -3t some direction, go t some direction, then t some other direction” to “thats just (5,-3,1,1) scaled by t

[sudden yacht]

Huh? I don't get your doubt

The direction is only one in this exercise!

And it's the vector (5, -3, 1, 1)

[last slate]

i think im over complicating it. For part C though, how do you know how to rewrite it

[sudden yacht]

Wdym how I know? 🤔 I just factor out the parameters, by using vector space properties

$$(s+4t,t,s,2s-t) = (s,0,s,2s) + (4t,t,0,-t) = s(1,0,1,2) + t(4, 1, 0, -1)$$

[grand pond]

Alberto Z.

[lyric charm]

!noai

[midnight plank]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[bold knoll]

The fact this is so common that it needs its own dedicated shortcut to avoid having to write it every single time is very funny

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

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[small warren]

Can I please get help with this question? I'm stuck at being unable to factor x^2+8x-593=16, I'm not sure how i got to 21 so i wasn't sure if i was right so i drew a diagram and multiplied and this is what i got.

[small warren]

G is just a variable that i devised.

I think i'm supposed to solve this with the quadratic equation and take the positive value?

[fresh leaf]

625 is the area of the painting right?

[small warren]

Yes but that includes the frame

I believe the question is trying to find what value of x when multiplied by itself results in the length of the bare canvas.

[fresh leaf]

i hate worded questions ;d gimme a bit idm helping, but an official helper can take over not sure of the rules here

[small warren]

Ty same but i just watched a khan academy video that motivated me XD now i really enjoy questions like these

i hope i don't get spoiled the answer

Maybe with my own drawn diagram I can subtract 4 from each side to find just the x area? i don't know

The reason I added them both is because that is equivalent to the total area of the frame (625 cm^2)

i was thinking of then subtracting the area of the frame or something like that.

[fresh leaf]

so one side of the painting is (4+x+4)

[small warren]

Ignore everything other than the diagram and the stuff below it and on the top right, those are what i got, but the other stuff is doodles

[small warren]

You're right!

I'm going to try that, 1 second

[fresh leaf]

so we have that one side is (x+8) that means the area will be (x+8)^2

[small warren]

ooo

and square rooting both sides yields $(x+8)=+/-25$

[grand pond]

🩷Aurora💜

[fresh leaf]

im trying to find where you made the mistake

[small warren]

And we only care about the positive because it's a real life canvas

[small warren]

it's when i only added 4

instead of 4+4 to x

[small warren]

So the solution is -8 + 25

x=17?

[fresh leaf]

yes

[small warren]

Thanks!!! ❤️

can i get help with this too

apparently this is wrong

this is my work

[hallow mantle]

Did u complete the square

[small warren]

i'm so dumb i think it should be x-8 right?

[small warren]

Yea i added half of b and squared it and added it to both sides, i think

[hallow mantle]

Wait whats the question

Ill just do it

[small warren]

this

if i weren't lazy and wrote out my answer then i would've wrote x-8

because 64 squared is something totally different XD

i got 95%

[hallow mantle]

Yep its 8

[small warren]

because of that silly mistake xd

[small warren]

Awesome

[hallow mantle]

😭🙏

[small warren]

is that good

[hallow mantle]

Its great

[small warren]

Yayyy

[small warren]

Cool!!

[hallow mantle]

What year are u in?

[small warren]

9th grade :<

im an adult though

i kinda dropped out

[hallow mantle]

I reccomend using a site called maths genie it has great exam questions

[small warren]

but im back in school

[hallow mantle]

Rip

[small warren]

ye

[small warren]

Oooo so i can practice?

[hallow mantle]

Its got basic stuff like this + super hard question

[small warren]

Oooooo

[hallow mantle]

Yea they even have video tutorials

[small warren]

Niceee

Thanks, maybe i'll remember it sometime

[hallow mantle]

Anyways cya

[small warren]

Cya thank u both for the help!!! ❤️

i wouldn't consider the earlier thing cheating because i just made a simple error in not adding another 4 to the quadratic equation

i was considering submitting the answer and if it were wrong and i wanted to know why i could try and examine

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[valid sinew]

https://arxiv.org/pdf/2105.13098
Hi! I'm trying to undestand this algorithm

[valid sinew]

On the 6th page, it explains the blocked part of the algorithm

[formal orchid]

Bro is studying 50 years into the future 🙂‍↕️

[formal orchid]

Let me take a look.

[valid sinew]

So I successfully wrote a non-blocked version of this algorithm which correctly tridiagonalizes the matrix:

However, right now I'm trying to write the blocked version, but I don't quite understand the mathematical notation used in the paper ...

So how are we supposed to combine multiple ak and ek vectors together and then multiply them? It's very confusing

Note: The same algorithm is also described here (perhaps simpler since it's the original): https://arxiv.org/pdf/1102.3440

So here for-example, I'm storing four rows ak and ek in separate matrices to perform operation on four rows/columns at once , what do I next? I'm also confused because in the non-blocked version we were doing the outer product of ak and ek vectors

[midnight plank]

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@valid sinew Has your question been resolved?

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@valid sinew Has your question been resolved?

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[final wharf]

What is derivative of complex number?

[astral canyon]

What's the derivative of a constant?

[orchid fossil]

0

[astral canyon]

Thank you mr/ms unrelated individual

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[severe grail]

Hi, how is 7pi/6 not a solution?

[toxic prism]

pretty sure that's 1/sqrt(3)

[severe grail]

ohh im being dumb

alr thanks

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[inland patio]

Consider a sequence of independent random variables with values in $[-\infty,\infty]$ and define $$Z:=\limsup_{n\to\infty}\frac1{n}(X_1+X_2+\cdots+X_n).$$Then this random variable is $\sigma(X_n:n\geq1)$-measurable. The claim is that also $$Z=\limsup_{n\to\infty}\frac1{n}(X_k+X_{k+1}+\cdots+X_n)$$for any $k\in\mathbb{N}$. Why is this true? First I would have said $$\frac{S_n^{(k)}}{n} = \frac{S_n - S_{k-1}}{n} = \frac{S_n}{n} - \frac{S_{k-1}}{n},$$ and $\frac{S_{k-1}}{n} \to 0$ as $n \to \infty$, so the limsups are equal. But I'm not sure about $\frac{S_{k-1}}{n} \to 0$.

[inland patio]

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[sinful torrent]

Ok I feel stupid here

[sinful torrent]

IK the log is inverse to the normal equation

But I am unsure about how this relates to the gradiant

[toxic prism]

that's right

how does the transformation look like on a graph tho

[sinful torrent]

Ik what they are for a normal curve

+- b to move up and down

(x+-z) to move left or right

[toxic prism]

right

but functions and their inverses have a very special relation

[sinful torrent]

Erm y = x is the diagonal the reflect

[toxic prism]

exactly

you can use that property to relate the gradients

[sinful torrent]

I would suppose that they would have the same gradiant at the same x value?

[toxic prism]

not exactly

slopes at coresponding points are reciprocals

what I mean is, suppose f(x) has some point (x,y) on it with slope m

inverse f will have (y,x) on it with slope 1/m

[sinful torrent]

So slope of the inverse will be 1/m

[toxic prism]

on the corresponding point

[sinful torrent]

So if the rate of change at x is 2, the inverse will have 1/2 (at that specific point

[toxic prism]

no, its not fixed based on x

[sinful torrent]

Its based off m, at x, no?

[toxic prism]

no

suppose f(x) has a point (x,y) on it, whats the reflection of that pt about the line y=x?

[sinful torrent]

The inverse of that

So -(x,y)

[toxic prism]

no

[sinful torrent]

1/xy?

[toxic prism]

that's not a point

think in coordinates

[sinful torrent]

Well if its reflected on y = x, it would be shifted to the corresponsing coordinates on the other side of the divide

[toxic prism]

yes

and what would the coordinates be?

assume the blue one is (x,y), what's the coords on the black?

[sinful torrent]

Oh

Ok

So it switches?

[toxic prism]

yup

[sinful torrent]

If the orgigonal is at 8,12, the inverse would be at 12,8

[toxic prism]

for reflection abt y=x, yes

those kinda points are what I'm referring to as corresponding

so slope of f at x, will be equal to 1/(slope of f inverse at y)

[sinful torrent]

Ah, i see

[toxic prism]

you can prove this by taking $f[f^{-1}(x)]=x$ and differentiating using chain rule

[grand pond]

donkey

[sinful torrent]

So the tranformation is the switching of the x and y coordinates, and this allows us to find the gradient with 1/m

[toxic prism]

right

[sinful torrent]

and how can this transformation be used to find the

gradient of y = loge x at its x-intercept?

Well you derive it, no?

Itll be 1/x

What transformation maps y = lnx to y = ln(−x), and how can this transformation also be interpreted as a dilation?

Erm Well considering ln-x is just the negative of lnx

Its a simple flip

I dont see how that is a dilation

[toxic prism]

you can find the differential of e^x at point (0,1) [because x intercept of lnx is (1,0)] and then take reciprocal

[sinful torrent]

.

[toxic prism]

yeah that's how we usually do it

[toxic prism]

this one's js because they want you to use the transformation to find it

[toxic prism]

scale factor -1

[sinful torrent]

Ah I see

[toxic prism]

I think we call those directed dilations

[sinful torrent]

Last bits just the sketch

I should be fine on that

Tysm man

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[shut canyon]

What is the fixed point of cos(tan(x))=x?

[midnight plank]

Please don't occupy multiple help channels.

[shut canyon]

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[ocean chasm]

well do u want the exact ans?

[ocean chasm]

or the step to find the ans?

[runic hamlet]

for most equations like this you will never be able to give an "exact" answer

[ocean chasm]

well i meant by the numerical approximate

[runic hamlet]

i.e. a "formula" maybe involving some roots and known constants

[ocean chasm]

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[lyric charm]

"approximate" and "exact" cannot coexist

[ocean chasm]

ok

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[vernal field]

how to find eqn of common transverse tangent between 2 circles which have a pt of contact

[gleaming latch]

welp find the point of intersection first

[vernal field]

no there is some formula and it says S1-S2..
its in a video and its in ihindi

[gleaming latch]

!original

[midnight plank]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[vernal field]

can u understand hindi

[gleaming latch]

read bot

[vernal field]

@gleaming latch https://youtu.be/KEsOtGdiU88?si=Ob13LpYCBO7aIG8R 10:52.

[gleaming latch]

whats th emath problem

[vernal field]

no math problem.

conceptual problem.

[modern sapphire]

the situation is like this

[vernal field]

yes

[modern sapphire]

suppose that the blue equation is S_1 = 0

and the red is s_2 = 0

[vernal field]

what is S1 and S2

[modern sapphire]

those are the circles

[vernal field]

do we sub the common intersection points in eqn of circle 1 and 2

burh

ok then i got it

another doubt

[modern sapphire]

this expression is called s1

coz that is equal to 0 in the equation of the circle

[vernal field]

@10:57 he gives another eqn.. what is that for

[vernal field]

ohhh we find out common pt

[modern sapphire]

what? no, you dont find anything

s1 is directly the equation of the circle taken as is

[zealous schooner]

(with 1 as the coefficients for x^2 and y^2)

[modern sapphire]

your general form of equation is x^2 + y^2 - 2fx - 2gy + c = 0

then x^2 + y^2 - 2fx - 2gy + c this part is called S

[modern sapphire]

I think he is telling the condition for the circles to intersect. C1 and C2 seem to be the centers of the two circles, so C1C2 is the segment that connects them, and its used there as the length of that segment aka the distance between centers

Also, that seems to be a crashcourse

if you have no clue whats going on in there, thats a sign that you should take a full length course to understand the stuff

[midnight plank]

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[vernal field]

idk hindi ✅
but i do understand now

[dapper scarab]

would you mind .close

the channel opened because you responded lol

[rain wasp]

do you have any further questions?

[pearl hull]

.solved 👍

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[pearl hull]

Have a good one 🧡

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[twilit field]

kind of confused here

[twilit field]

nvm, my bad

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[edgy crater]

have i gone wrong somewhere, especially the last case where n = 2 (mod 3), since wouldn't i have to show that 4 divides n-1, which i dont think works? like if n was 11, n-1 is 10 which isnt divisible by 4

[edgy crater]

question is show 12 | n^4 - n^2 for all positive integers n

[fallow scarab]

Factor n^4-n^2

[edgy crater]

n^2(n+1)(n-1)?

[naive bridge]

why is n odd in the last case?

or does the problem arise when u consider the case that n is odd

[edgy crater]

i thought if something = 2 (mod 3) then its odd

[naive bridge]

unfortunately no

remember its 2, 5, 8, ...

[edgy crater]

ohh

😭

[naive bridge]

however you still have to deal with this case

would you like a hint or you wanna try a little more

[edgy crater]

il try thanks

yh think i got it😭

appreciate ur help

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[trim sun]

What am i supposed to do if the second graph doesn’t have exact points

[near cliff]

quotient rule?

[graceful drum]

take them as 0.5s

[trim sun]

I did that for h(3) but then how do i find the equation of the line to do the prime

Or slope or whatever

[graceful drum]

do you know the formula for slope of a line?

[trim sun]

Y=mx+b?

[graceful drum]

well i meant the formula from a slope using two points on the line

[trim sun]

Y-y1=m(x-x1)?

[graceful drum]

well thats the equation for a line given the slope and a point

[trim sun]

😭

[graceful drum]

i mean $m = \frac{y_2-y_1}{x_2-x_1}$

[grand pond]

CherryMan

[trim sun]

Ohhhh yeah

[graceful drum]

so we can calculate this from the graph right

[trim sun]

I just choose any 2 points right

[graceful drum]

yes

two convenient points

[trim sun]

Ok

[graceful drum]

(btw the slope is only constant for a line)

[trim sun]

Am i cooking

Got it 😆

[graceful drum]

nice

[trim sun]

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[lyric scroll]

does anyone know what u type in ur calculator

[lyric scroll]

the ( 5 0 41)

part

wasnt there a really easy way to do it with a calculator

[umbral timber]

that translates to : \ \
$\frac{50!}{(41!)\cdot((50-41))!}$

[lyric scroll]

like all u need is the 3 variables

[grand pond]

T&C

[lyric scroll]

yeah

also can u help with another thing

or should i create a new channel

[umbral timber]

i was told you should always create a new channel for a different problem

[lyric scroll]

okay bro thanks

,close

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[jaunty oriole]

Hello , i want to start learning math olympiad problems solving and i have only med school experience , from where i can start and with what .
I am in 8th grade
Thanks.

[umbral timber]

The book Mathematical Circles is a great start id say but these questions are better for #discussion #competition-math

[jaunty oriole]

Ok , thx

[jaunty oriole]

But i see that this book is a little bit hard to understand
Someone says igcse book , is it good.

[floral apex]

@jaunty oriole always good to get exposure to problems

if you're just trying to get into competition math I'd suggest AOPS

https://artofproblemsolving.com/wiki/index.php/Category:Algebra_Problems

they have courses and books, but their free resources are pretty helpful too if you're just getting started

[jaunty oriole]

Ok , i will , thanks 🙏

Is igcse book good

Please

Please

[midnight plank]

@jaunty oriole Has your question been resolved?

[jaunty oriole]

no

<@&286206848099549185>
pls

[fallen sparrow]

Id say yes

But Igcse will only get you si far

Aops i think is a step further

Depending on how basic ur math is, i'd suggest starting with the igcse book

If you feel thats easy

You can move on with aops but start the questions order wise

Since the difficulty is aimed likewise

[midnight plank]

@jaunty oriole Has your question been resolved?

[undone yacht]

Yes

Its a good way to start also try looking for edexcel igcse , i recommend it more

Cambridge mostly starting to depend on non calculators exam

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[onyx tide]

Isn’t this perfect? 🤩

[onyx tide]

y = -x^2 + 4

[junior wedge]

Yikes scribbles

what did your problem say?

[onyx tide]

Just calculate everything

[junior wedge]

also would be more perfect if there were dots at (-2, 0) and (2, 0)

[onyx tide]

Yeah I’d drag the line on them

It’s kinda curved

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[midnight plank]

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[onyx tide]

Can i find x1 and x2 from f(x) = a(x-0p)^2?

[subtle blaze]

Do you have more context

[onyx tide]

So I need to draw parabola from y = -2x^2 + 3x + 2

[subtle blaze]

And what have you tried

[onyx tide]

@subtle blaze i need zero of a functions

So I know how to place

The x

Like

[subtle blaze]

Quadratic formula?

[onyx tide]

(x,x)

[onyx tide]

Yea but can i use something different

[subtle blaze]

Complete the square

Which is really just quadratic formula with the steps

[onyx tide]

Can I use canonical form of quadratic formula?

[subtle blaze]

What’s that

[onyx tide]

To skip solving d?

It’s like

a(x-p)2+q

[subtle blaze]

Yeah

Wait what

Wdym skipping d

[onyx tide]

I want to solve it

Without doing d

Is it possible

Like i need to find x1 and x2

So i know where the lines will cross on x-axis

[subtle blaze]

Why not doing d

What’s wrong with using quadratic formula

[midnight plank]

@onyx tide Has your question been resolved?

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[tawdry slate]

Help pls

I don't know how you get the height of the triangle

or how to do the question in general

The answer

[midnight plank]

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[tawdry slate]

.reopen

[midnight plank]

✅ Original question: #help-49 message

[tawdry slate]

sry mate

What were you going to say?

anyway I'll do a youtube in the mean time, if anyone can help me here I'll leave the channel open

[strong leaf]

Yo

How do u get the height

I get the rest of it

[tawdry slate]

I've got the same issue mate

I've got no clue on how you get the height

[strong leaf]

What class is this

[tawdry slate]

This is NZ level 3 statistics

[strong leaf]

Interesting

[tawdry slate]

https://www.youtube.com/watch?v=SUk3HZh3sDk

I've found this, the question is at 16:34

[strong leaf]

Probably something related to being a cdf function

Oh idk bro I did not go this deep it into stats

[tawdry slate]

h = 2/b-a

is the start of it

[strong leaf]

Why 2 tho

[tawdry slate]

I dunno, its the done thing

the formula for working out the height

[strong leaf]

Shouldn’t the height just be .7

[quaint creek]

hi

what is this

[tawdry slate]

I've worked it out, the answer is 1/3

[quaint creek]

my initial instinct is that the area of the triangle should be 1 because that covers all of the probability

[tawdry slate]

this is the code to crack the question

[quaint creek]

I've never learned this stuff before sorry

[tawdry slate]

that's correct

the whole thing is 1

[quaint creek]

ah oaky

so the base multiplied by the height

divided by 2

will be 1

and we know that it occurs most often at 130

so we figure out the height and draw it at 130

[strong leaf]

Yea that makes sense but the height weird

[quaint creek]

and then you calculate

wtv is to the left of 130

so its 10 x 0.6666 / 2

so like 33.3?

1/3 chance

[strong leaf]

Oh I get it

Area = 1/2 bh

H = 2* area/b

area/b = 1/30

@tawdry slate

Area is assumed to be 1

And the Base of the triangle is 150-120 =30

Plug it in and it gets height

[tawdry slate]

That's the answer

Thanks gents

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[quaint creek]

hey

[narrow gull]

hey wassup

[quaint creek]

so

that image is so bad

but anyways

f(g(3))

[narrow gull]

I can zoom it in if that helps?

[quaint creek]

so we want to find the value of g(3) first dont we?

[quaint creek]

no, its fine

[narrow gull]

you sure?

[quaint creek]

ye I can see it

[narrow gull]

yeah

[quaint creek]

dw

[quaint creek]

do you know how to find that

[narrow gull]

I have an idea

3 is the x val right?

[quaint creek]

so

in this case, y = f(x)

since its a graph

so by plugging in x into the function f(x), you get a y value

so when we plug in the x-value 3 into the function g(x), what value of y do we obtain?

[narrow gull]

-1/2

[quaint creek]

when we zoom in

you can see that when the x-value is 3

or the third dashed line

the y value is the point on the third dashed line

so what's the y value, or the vertical value of that point

[narrow gull]

oh i though it was that because of the neg 2

[quaint creek]

ye

but its clear it goes down in intervals of -0.5

so that point is gonna have a y value, or g(3) = -1.5

so now, we solved f(g(3)) into f(-1.5), since g(3) = -1.5

so now we solve f(-1.5)

[narrow gull]

it doesn't show it

well I mean it's not clear

[quaint creek]

yeh thats why the question is kinda ass

but -1.5 is in the middle of the two dashed lines, so

actually wait

notice that the blue line goes from -5 to 10

(1, -5) to (-2, 10)

[narrow gull]

oh so I would use the midpoint formula?

[quaint creek]

well yeah I guess

if you learned it

so we get 2.5 for the y value

so the first question f(g(3)) = 2.5

try the second one yourself lmk if any help needed

[narrow gull]

okay thanks!

[quaint creek]

should be alot more straightforward than this one

[narrow gull]

I got -1/2 but it is saying it is not right

[quaint creek]

uh

what did you get for f(2)?

@narrow gull

[narrow gull]

I got 3

[quaint creek]

yes

[quaint creek]

so since f(2) is 3, what is g(3)

(we already did this in the previous question)

[narrow gull]

-1/2

[quaint creek]

.

we did g(3) in the previous question

so basically what is the y-value of the dot

at an x-value of 3, so the third dashed line

[narrow gull]

-1/2

no?

[quaint creek]

so the bottom is -2

and the top is 0

[narrow gull]

0hh mbmb

[quaint creek]

yeh

[narrow gull]

-1 1/2

[quaint creek]

yeah, so simplified to -3/2

[narrow gull]

oh okay

THanks man!

i'm good now

have a good night!

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[quaint creek]

you too

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[lone dock]

my answer was (6 - root 3)/ 40

im not sure if it was correct tho...

[subtle zinc]

eh

[lone dock]

i assumed a and all of those are squares

cant find the solution anywhere so super confused TwT

should i send what approach i used?

[graceful drum]

sure

[astral canyon]

Let the smallest square be of side length 1, then the side length of the medium square will be sqrt2 You now have a triangle with two known lengths (1 and sqrt2), and one known contained angle (135). That's how I would start.

[lone dock]

aight gimme a sec

i got the sides

then

[modern sapphire]

I am getting a pretty nice answer ||1/10||

[lone dock]

WHAAT

how

[lone dock]

can u please tell me ur approach?

[modern sapphire]

I made this figure from the two squares. Let the blue line to be 1

so small square has side sqrt2, middle one has side as 2

red side, which is side of big square is thus sqrt10

now area of the highlighted part of the figure would be 0.5 * sqrt2 * 2 * sin(135) = 1

so ratio 1/(sqrt10)^2 = 1/10

[lone dock]

wow

thank you so muchh

ill try and close this, thanks

[lone dock]

i didnt get this, how that sqrt 10?

[modern sapphire]

pythag on the triangle with blue and red sides

[lone dock]

GOTITTT

GOTIT

thankuu

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[edgy crater]

bit stuck on part (iii), i tried using the recurrance relation at the start, asserting that x_n+2 = kx_n+1 (1-x_n+1), then subbing in x_n+1 in terms of x_n into the relation, then setting both x_n+2 and x_n as equal to a, but when i expand it out etc it does't match the "show that part"

[midnight plank]

@edgy crater Has your question been resolved?

[dense heart]

hey guys

I think I have solved the collatz conjecture

[frail carbon]

show work

[latent wadi]

You can go to #math-discussion, this is a help channel

[edgy crater]

.close

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[small warren]

Can I get help with this transformation question? these are the options, I chose option c because i know that when the value of the output is multiplied with a negative number that the graph is reflected over the x-axis, and that because it's multiplied by 2 that it's going to become closer to the y axis because each value will get bigger, I eliminated options B and A because their asymptotes don't match the expected inversion of the graph that should be at y=2.

[thorn sorrel]

the -2 just means that all the y values of the function will be muktipled by -2

what is one known y value on your original function

[small warren]

0, -1

[rain wasp]

what c does is just moving everything rightwards

[thorn sorrel]

so what will the transformed coordinates be

[small warren]

0, 2?

[small warren]

i dont see that

[thorn sorrel]

yup

which option is that

[small warren]

Option b?

[thorn sorrel]

yup

[small warren]

i was under the impression that it wouldn't change the asymptote that much but just its values

the original asymptote approaches the value of -2

thanks a lot!!!!

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[twilit field]

any element of order 9 in Z_9 must be co-prime to 9

so there are 6 such elements in Z_9

the element of Z_3 must have order 1 or 3

so 2

so 12 such elements exist

[runic hamlet]

which element of Z_3 does not work?

[runic hamlet]

why do you only care about those elements?

[twilit field]

huh, all do work

right

[twilit field]

I need the order ot be 9 or 3

[runic hamlet]

then why did you exclude 3?

[twilit field]

I didn't realise then