#help-13
cedar kilnBOT
marble ermine
m and n in N*
how do i show that if n and m are coprime, then Z/mZ x Z/nZ is cyclic
mighty drift
what does the CRT say?
marble ermine
it says that x = a mod m and x = b mod n has a unique solution mod mn if m and n are coprime
mighty drift
i.e. Z/mZ x Z/nZ ~ Z/(nmZ)
mighty drift
yes
cedar kilnBOT
cedar kilnBOT
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carmine otter
Hi
wraith daggerBOT
carmine otter
This is my working but its vastly different from the answer key
livid hound
you misinterpreted the notation
livid hound
sin^-1(x) isn't the same as (sin(x))^(-1)
tidal sierra
sin^-1(x) isn't the same as (sin(x))^(-1)
True
livid hound
$-1$ is an exception in $\sin^n(x)$ notation, when -1 is used, that indicates the function inverse.
wraith daggerBOT
ℝαμΩℕωⅤ
cedar kilnBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
@carmine otter Has your question been resolved?
wraith daggerBOT
carmine otter
Okay i got to here, how can i simplify the surds within the surd?
livid hound
firstly should've omit components like that,
and you haven't correctly applied the differentiation property for arcisin(x)
$\dv{x} \sin^{-1}(x) \redneq \frac{1}{\sqrt{1-x}} \ \
\dv{x} \sin^{-1}(x) = \frac{1}{\sqrt{1-x^{\blue{2}}}}$
wraith daggerBOT
ℝαμΩℕωⅤ
cedar kilnBOT
cedar kilnBOT
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cedar kilnBOT
polar prism
Hi guys, I need help thinking about ariving at the answers to each of these questions. I need some kind of method
I will be sending the 4 more others shortly
zealous fable
Which type of math is this?
polar prism
Which type of math is this?
Inductive reasoning
Usually we start by spotting patterns
Then reason towards the conclusion from there
For example
polar prism
FOr this one
The first block has two balls(solid) on the left and one square on the right
The second block has two balls(solid) on the left and two balls on the right
The third block has two squares on the left and one ball(non-solid) on the right
The third block has a square topped by two balls(solid) on the left and one non-solid ball ont he right
zealous fable
I think A is the answer for this
I think this type of problem doesn't have a method for it
polar prism
I think A is the answer for this
Why?
polar prism
I think this type of problem doesn't have a method for it
Please explain that
zealous fable
Why?
I think it have 3 patterns
1 is the number of vertices (the image to the left will have 2 less verticles)
2 is projection
3 is the pattern of diagonal line (which is \ | / | \)
zealous fable
What about this one?
I think it is more fun if you try to find it on your own, though.
polar prism
I think it is more fun if you try to find it on your own, though.
I did. But I got it wrong and I do not know why
zealous fable
I did. But I got it wrong and I do not know why
Why do you know it is wrong?
polar prism
Why do you know it is wrong?
The platform we're using marked it as incorrect.
This is the approach I used ⬇️
- There are black and white arrows
2.The arrows either point left or right
- The black arrows always occupy the first two spaces
- The white ones only occupy the last space
- Onto the first changing pattern:
(a) One arrow is removed on the second block
(b) Two arrows are removed in the third block
C: We expect three arrows to be removed in the fourth block
- Onto the second changing pattern:
(a) Row1 Block1: The leftmost arrow changes direction, the rightmost arrow is removed
(b) Row2 Block1: The middle arrow changes direction.
(c) Row3 Block1: The rightmost arrow changes direction
(a) Row1 Block2: The leftmost arrow changes direction, the rightmost arrow is retrieved
(b) Row2 Block2: The middle arrow is removed.
(c) Row3 Block2: The leftmost arrow is removed, the rightmost arrow changes direction
[Third block goes here]
(a) Row1 Block4: The leftmost arrow is removed, the middle arrow changes direction.
(b) Row2 Block4: The leftmost arrow is removed
(c) Row3 Block4: The middle and rightmost arrows are removed.
zealous fable
The platform we're using marked it as incorrect.
I am curious if my answer was correct 😂
polar prism
I am curious if my answer was correct 😂
The short answer (without looking at other patterns) is supposed to be A because we expect 3 arrows to be missing in that block
But the platform marked it as wrong
It's just so hard to figure out the other patterns 🥲
polar prism
Was it correct?
Yes 😄 💚
zealous fable
What about this one?
Wait, is this a new one?
polar prism
Wait, is this a new one?
no
polar prism
6. Onto the second changing pattern:
(a) Row1 Block1: The leftmost arrow changes...
This is my reasoning. Pretty hard to spot the pattern
zealous fable
This is my reasoning. Pretty hard to spot the pattern
"C: We expect three arrows to be removed in the fourth block". Isn't this enough to choose an answer 😂
zealous fable
I think I found the pattern for this
cedar kilnBOT
@polar prism Has your question been resolved?
polar prism
"C: We expect three arrows to be removed in the fourth block". Isn't this enough...
I don't know what the problem really is. But let's skip it for now shall we?
polar prism
I think I found the pattern for this
What is it?
zealous fable
What is it?
The purpose of this is to balance the scale
polar prism
The purpose of this is to balance the scale
Interesting 😄
zealous fable
Each shape (square, circle, hollow circle) has a value so you can use that to balance the scale
polar prism
Holy cow. You ||fucking|| genius
You know what leave that to me to figure it out
Can you please now help me with...
zealous fable
Can you please now help me with...
It is very easy to spot a value for each shape. You should try that first😂
polar prism
Ok.. just a sec..
Identifying shapes first:
- Solid Circles (Use symbol sc)
- Empty Circles (Use symbol ec)
- Solid Squares (Use symbol ss)
(a) If two solid circles can balance out one solid square, then: 2sc = ss
(b) If two solid circles can balance out two solid circles, then: 2sc = 2sc (which makes sense)
(c) If two solid squares can balance out one empty circle, then: 2ss = ec
(d) If two solid circles + one solid square can balance out one empty circle, then: 2sc + ss = ec
Based on these observations, we need to find an option that fits the pattern.
Lol I'm not sure how to allocate values to the shapes though 🥲
zealous fable
Lol I'm not sure how to allocate values to the shapes though 🥲
You are complicating it. Just use the solid circle to measure value of each shape
If a square can balance 2 solid circles then 1 square = 2 solid circles
polar prism
yes
uhh
wait a min
what about the empty cricle?
looks like it weighs more than just two squares
zealous fable
what about the empty cricle?
convert square to solid circle
cedar kilnBOT
@polar prism Has your question been resolved?
polar prism
convert square to solid circle
I think the answer is D
makes sense to me, based on the rules
zealous fable
I think the answer is D
Yeah, that correct
polar prism
This one
zealous fable
This one
I can't find a pattern to it. But I noticed that the position of telephones never changes so my guess is that F is the answer
Not sure, though
polar prism
Lemme check that
Yeah... this one's hard
Even if the position of the telephones did not change
I wouldn't understand why F is the answer
Oh wait
I see.. yea F makes sense
zealous fable
I see.. yea F makes sense
So was it correct, though 😂
polar prism
So was it correct, though 😂
Considering all the confusion going on, it would make so much sense that the position of the telephone does not change in none of the 9 quandrants
So I concur with F being the answer
zealous fable
So I concur with F being the answer
Do you have an answer to that question? A guess is just a guess
polar prism
Do you have an answer to that question? A guess is just a guess
Honestly I did not have an answer. I'm genuinely impressed you saw about the telephones
I was still trying to count the number of envelopes and voice notes
And it wasn't making sense
polar prism
Can you help with this one? @zealous fable
zealous fable
Can you help with this one? <@793428843871731732>
Yeah. Let's see that
polar prism
I had tried to use the analogy of the grey arrows as weak winds and the bold ones as strong winds going either to the west or east
Geography stuff 🙈
But it's a bit of a challenge to see which elements contribute to either outcome
zealous fable
Yeah, it is quite hard
polar prism
The arrows seem to indicate the direction of transformation or switching between the elements in the boxes. For each arrow, elements are either swapped horizontally or left unchanged depending on the direction. I think the answer could be F
Not sure though
zealous fable
Not sure though
I am wondering about its transformation rule if the rule is applied to the box directly above it. Then you would expect the output after the box gets transformed to be the box to the right of it.
But what about the last one
Like this:
zealous fable
So we get the box above based on the arrows bellow, and not the other way around...
Oh, that just means the box above is transformed with the arrow below to get the box on the right of it.
polar prism
Ahh, I see
Never thought about it that way though 😄
How did I miss that
Then F would probably not be the answer
zealous fable
Then F would probably not be the answer
We don't know the rule of this yet
zealous fable
!occupied
cedar kilnBOT
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
zealous fable
The arrows seem to indicate the direction of transformation or switching between...
Damn. The arrow doesn't make any sense
polar prism
Damn. The arrow doesn't make any sense
So you can see my predicament
zealous fable
So you can see my predicament
I don't think it is simply swap place because some shape pop up out of nowhere
polar prism
I've held you here for much longer than I should 🥲
It's probably time to let go
You've helped me more than anyone has, and I genuinely appreciate that 💚
Thank you so much for doing this. Means a lot to me
I will try to go figure it out overnight
zealous fable
Thank you so much for doing this. Means a lot to me
Ok. No problem
I think this inductive reasoning is for genius, though 😂
polar prism
I think this inductive reasoning is for genius, though 😂
Really? 😄
Our teacher told us that these are widely used in Psychometric tests for Jobs and such
That's interesting to know
zealous fable
Our teacher told us that these are widely used in Psychometric tests for Jobs an...
Well, then it is a very hard question
polar prism
Well, then it is a very hard question
😄 But not an impossible one
Alright, I'm closing the session. Thank you so much mate 🙏
Take care 💚
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
void sand
does this work? 
cedar kilnBOT
does this work? <:cat_happycry:636407813144576010>
void sand
the first ss is the problem, and the next two is my attempt at a proof
misty furnace
How is A bar defined?
void sand
How is A bar defined?
it is the closure of A
misty furnace
Union of ?
void sand
interior and boundary
misty furnace
interior and boundary
Boundary means all the limit points right?
void sand
that's not how Lee defined it 
misty furnace
interior and boundary
Unionn
void sand
Boundary means all the limit points right?
void sand
yes, it is
misty furnace
Not intersection
misty furnace
Lemme see
I think you have to start by taking a point form x
Not A and the boundary
void sand
I think you have to start by taking a point form x
X can be written as the union of Int A, Ext A, and \partial A
I covered all cases
void sand
I showed (I think) that if p is in Int A or \partial A, then p is a limit point
misty furnace
You only have to show the Ext A part
void sand
and if it's in Ext A then no
misty furnace
I showed (I think) that if p is in Int A or \partial A, then p is a limit point
Not required
void sand
well, I did it anyways 
misty furnace
Yeah
void sand
misty furnace
misty furnace
Yw once again
void sand
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cedar kilnBOT
void sand
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hot python
can someone explain step by step how to find the foci of an ellipse?
gaunt iron
I’ve got a problem my brain isn’t letting me solve this simple problem. If I have a 5/12 roof pitch, rise over run = 5/12, how much incline would I have per inch of run? And the goal is to find how much run I would have at 12 rise
hot python
.close
gaunt iron
I’ve got a problem my brain isn’t letting me solve this simple problem. If I hav...
<@&286206848099549185>
cedar kilnBOT
<@&286206848099549185>
olive thorn
cedar kilnBOT
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.
Show your work, and if possible, explain where you are stuck.
gaunt iron
I got it, it’s 28.8
cedar kilnBOT
@gaunt iron Has your question been resolved?
glacial cypress
I am asked to find Fourier series for the given function. I am confused, because when I graph them on desmos it only matches for l = pi. Is it supposed to match only for 1 specific interval or what. I am confused??
cedar kilnBOT
I am asked to find Fourier series for the given function. I am confused, because...
@glacial cypress Has your question been resolved?
random shale
Well for one you will always need to truncate a Fourier series when plotting them digitally so you expect errors.
You expect bigger errors at the points of discontuity (see Gibbs phenomenon)
However in this case you shouldn't expect it to be too inaccurate, perhaps try plotting more terms, if that doesn't work it's likely you have made a mistake in your calculation
Also considering you have identified where the error has come from, you can probably guess where the error in calculation was
glacial cypress
I have tested the coefficients with online calculators and they are correct. Maybe there is some logic that I do not understand and It's probably something to do with the interval or the period
w.e. I just don't geit
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cedar kilnBOT
cedar kilnBOT
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wise aurora
Hi
wise aurora
Add the following vectors using components.
a. 9 m/s [N30°E] and 2 m/s [N 60°E].
Am i right?
Someone help pls
<@&286206848099549185>
never mind .close
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
cedar kilnBOT
@eager hemlock Has your question been resolved?
cedar kilnBOT
@eager hemlock Has your question been resolved?
cedar kilnBOT
@eager hemlock Has your question been resolved?
hot python
Nabil is writing a science fiction novel that takes place in another
galaxy. In her galaxy, all the planets travel in an elliptical orbit
around their star. The majority of the plot of Nabil’s story takes
place on the planet Tanus which orbits a star named Ini. The
length of Tanus’ major axis is 150 million miles and the length of
its minor axis is 75 million miles. The star Ini is located at a
focus of the elliptical orbit.
Now, write an equation that models the movement of Tanus around its star.
hot python
i'm not sure if im doing this right but heres what i have so far
so
a=75 because major axis is 150
b= 37.5 because the minor axis is 75
so 75^2= 5625
and 37.5^2= 1406.25
rounds to 1406?
which would give me
x^2/ 5625 + y^2/ 1406.25 =1
is this correct?
cedar kilnBOT
@hot python Has your question been resolved?
hot python
<@&286206848099549185>
hot python
<@&286206848099549185>
strong stratus
are h and k zero?
hot python
oh no i forgot
hot python
are h and k zero?
wait how do i find those
strong stratus
I am not entirely sure if this is correct, but h and k are used in the offset of circle formula
basically h, and k are used to determine the circle's position on a graph
if the elipse is centered on the position 0,0 then h and k are zero
cedar kilnBOT
@hot python Has your question been resolved?
sturdy ferry
How’s it not c
cedar kilnBOT
How’s it not c
sonic mesa
humble karma
You need to take the 1/3 out first.
The issue is the original expression you have is not in the form $$\frac{1}{3}(x^2 - k^2),$$ but rather $$\frac{1}{3}x^2 - k^2$$
wraith daggerBOT
Azyrashacorki
humble karma
You must force the factorization
So that it can be written as $$\frac{1}{3}(x^2-k^2)$$ and $k$ won't be $2$.
wraith daggerBOT
Azyrashacorki
sturdy ferry
So -1/3k^2
humble karma
Yeah well if you compare the terms, you'll have $\frac{1}{3}x^2 - 2 = \frac{1}{3}x^2 - \frac{1}{3}k^2$.
So you have to solve for k in that
wraith daggerBOT
Azyrashacorki
humble karma
Yheah
sturdy ferry
OHHHHHHHHHH
humble karma
It's like factorization, but you're forcing it.
Say you have $(a+b)$, then I could write that as $\frac{1}{2}(2a+2b)$.
What I showed you is a more systematic way of dealing with it, but if you think of it as factoring it's more natural.
wraith daggerBOT
Azyrashacorki
humble karma
Yeah
sturdy ferry
Got it
stiff crane
For making a table of values for this equation: y= 9-x^2 is this correct ?
X Y
-2/ 5
-1/ 8
0/ 0
1/ 8
2/ 5
upper laurel
whats 9 - 0^2?
stiff crane
whats 9 - 0^2?
9
upper laurel
change the (0, 0) in your table
upper laurel
other than that, the table is correct
stiff crane
Cool thx
upper laurel
np
hot python
is this correct?
cedar kilnBOT
is this correct?
hot python
Nabil is writing a science fiction novel that takes place in another
galaxy. In her galaxy, all the planets travel in an elliptical orbit
around their star. The majority of the plot of Nabil’s story takes
place on the planet Tanus which orbits a star named Ini. The
length of Tanus’ major axis is 150 million miles and the length of
its minor axis is 75 million miles. The star Ini is located at a
focus of the elliptical orbit.
i used this equation which im pretty sure is correct, to make the ellipse for this problem-
. Create a graph of Tanus’ movement around Ini using your equation from question 1. Label
your graph with the position of the star and the planet.
using the equation x^2/ 5625 + y^2/ 1406.25 =1
h and k are not specified i dont think
or maybe im wrong
cedar kilnBOT
@hot python Has your question been resolved?
hot python
<@&286206848099549185>
cedar kilnBOT
@hot python Has your question been resolved?
distant kite
Next year is the finale term of my university life, and I need to submit a math thesis. I studied analysis and number theory thing. I have no ideas what to write. Can anyone suggest some ideas or direction of writing a thesis?
crimson delta
get an advisor
distant kite
how
crimson delta
talk to professors at your university
mighty drift
Next year is the finale term of my university life, and I need to submit a math ...
might wanna ask in #advanced-lounge too
cedar kilnBOT
@distant kite Has your question been resolved?
crimson sedge
cedar kilnBOT
crimson sedge
I got stuck
worldly cove
you mean 16m^2 + 16mn + 4n^2
crimson sedge
Yeah
Ignore the n at the first
What property or law do can I apply on the -3(m-n)(m+2n)
worldly cove
just use distributive property: (x + y)a = xa + ay
in this case you can set a = m + 2n
and then repeat it
twilit escarp
Distribute the -3 in the (m-n)
crimson sedge
Like this?>
twilit escarp
Then use double distributivity
crimson sedge
is this the result
worldly cove
Like this?>
(-3m + 3n)(m + 2n) = -3m(m + 2n) + 3n(m + 2n)
twilit escarp
Oh waot
worldly cove
(-3m + 3n)(m + 2n) = -3m(m + 2n) + 3n(m + 2n)
then you can do -3m(m + 2n) + 3n(m + 2n) = -3m^2 - 6mn + 3nm + 6n^2
crimson sedge
This is what I got
The answer is supposed to be: 13m2+13mn+10n2
is 3nm same as 3mn?
Or what do I do there
worldly cove
-6mn + 3nm is clearly -3mn, and 16mn - 3mn = 13mn
yes, 3nm = 3mn
5 * 3 is 3 * 5
4 * 2 is 2 * 4
crimson sedge
-6mn + 3nm is clearly -3mn, and 16mn - 3mn = 13mn
So, should I just add the 3nm to the 10mn?
worldly cove
yes
cedar kilnBOT
@crimson sedge Has your question been resolved?
wanton isle
A side and a diagonal of a parallelogram are 12 inches and 19 inches respectively. The angle between the diagonals, opposite the given side, is 124°. Find the length of the other diagonal.
wanton isle
i tried solving this and I'm getting different values against the correct answer.
The answer is 7.48 in.
livid hound
can you show your work
cedar kilnBOT
@wanton isle Has your question been resolved?
inner locust
cedar kilnBOT
inner glacier
What have you tried till now?
cedar kilnBOT
@inner locust Has your question been resolved?
sand cradle
Sry if this is rather physic-y but why do we have U_(Capacitor) = |U_0| after turning the switch?
grand shale
Sry if this is rather physic-y but why do we have U_(Capacitor) = |U_0| after tu...
could you tell what U means?
is that energy or voltage
sand cradle
could you tell what U means?
Voltage
We charge this capacitor C
Then we turn the switch
And then the stuff that is written on the bottom of the picture hold
First one is clear
But why the second one?
grand shale
its an LCR circuit with a DC source right, so can you write down the differential equation ?
sand cradle
its an LCR circuit with a DC source right, so can you write down the differentia...
No that comes later, first we want the starting conditions
To be able to solve that diffeq later
grand shale
r so the Vs are just voltmeters right?
sand cradle
Yeah
vagrant elbow
Sry if this is rather physic-y but why do we have U_(Capacitor) = |U_0| after tu...
It's just an initial condition
I mean it is convention to start with all charge in the capacitor before the oscillations begin
grand shale
yeah so it should be given as an inital condition
sand cradle
I mean it is convention to start with all charge in the capacitor before the osc...
Wait so at the very beginnung, U_C = 0, right?
Before charging
grand shale
or else you are to assume that U_c =0
sand cradle
Then we charge and it slowly reaches |U_0|?
And then we use the switch, and that is at t = 0
vagrant elbow
Wait so at the very beginnung, U_C = 0, right?
Depends on what you mean by very beginning
grand shale
you had LC oscillations going on before t=0, is that relevant?
sand cradle
yeah then you have to assume charge at C is 0
So in this picture?
grand shale
unless something else is specified
sand cradle
Where the switch is left
grand shale
yep, if you can send the full q it might help
sand cradle
yep, if you can send the full q it might help
There is no q, this is just a presentation I'm doing on electromagnetic oscillations and I'll have to do this entire experiment myself
Ok, after we turn the switch
Will U_L also be |U_0|?
Or only U_C
grand shale
Ohhhh, so you had RLC circuit till t-> infinity, then you had LC oscillations
solve the RLC circuit first
sand cradle
What
vagrant elbow
I'm so confused
So you start with RLC and after all the energy is over you switch over to LC?
sand cradle
So you start with RLC and after all the energy is over you switch over to LC?
I start with this
After having charged the capacitor C up, we go over to this
grand shale
so first you had a battery supplying voltage to the capacitor and inductor, till infinite time ( implying fully charged).
so basically figure out the voltages and charges by solving the differential equation using voltage of capictor at t=0 is 0 and voltage at inductor at t=0 is 0
vagrant elbow
Ah okay
sand cradle
Now there will be electromagnetic oscillations
grand shale
then you turn the switch and you have LC oscillations
sand cradle
so first you had a battery supplying voltage to the capacitor and inductor, till...
In the end, I get these
But only with the use of the starting conditions
THese are the starting conditions
sand cradle
so first you had a battery supplying voltage to the capacitor and inductor, till...
My question is really just how we get to the starting conditions after C is fully charged
It makes sense why Q = C * |U|
It makes sense that I = 0A (we stopped the battery)
But why U_C = |U_0|
Is also U_L = |U_0|?
vagrant elbow
But why U_C = |U_0|
It's arbitrary
vagrant elbow
Is also U_L = |U_0|?
The amplitude yeah
sand cradle
It's arbitrary
I could've picked U_C = 0 aswell?
vagrant elbow
It would be that
vagrant elbow
I could've picked U_C = 0 aswell?
Sure
vagrant elbow
But you'd work that in with a phase difference
sand cradle
I mean is it not U_C = |U_0| because teh capacitor is charged
vagrant elbow
Not the value of the amplitude
vagrant elbow
I mean is it not U_C = |U_0| because teh capacitor is charged
Yes
vagrant elbow
In the end, I get these
Notice how it's a cosine here you'd use sine instead
sand cradle
Because we first charge it, only then use the switch
vagrant elbow
So we do need |U_0|
Always
vagrant elbow
The amount of charge in your capacitor at an instant solely depends on your phase
When t = 0, the phase is 0, at t = 0 we have maximum charge in the capacitor because that's how we let the system progress
And it is most convenient for an LC circuit
sand cradle
When t = 0, the phase is 0, at t = 0 we have maximum charge in the capacitor *be...
Why are we talking about charge here, that'd be Q, I'm fine with that being C * |U_0|
I'm asking about the U_C
That's the voltage of the capacitor
vagrant elbow
If you start t = 0 after a quarter cycle, then the initial phase is pi/2, you can introduce a phase shift and have the capacitor have no charge at t = 0
vagrant elbow
Why are we talking about charge here, that'd be Q, I'm fine with that being C * ...
Ah shit sorry I'm used to having U represent energy
Well anyways q = CV
It's proportional
sand cradle
Ah shit sorry I'm used to having U represent energy
Why is the voltage in the capacitor at t = 0 equal to |U_0|?
I'm fine with the charge in it being C * |U_0|
vagrant elbow
q = CU?
sand cradle
Yes
vagrant elbow
U = q/C
vagrant elbow
I'm fine with the charge in it being C * |U_0|
Follow from this
Charge is C|U_0| then potential difference is U_0
sand cradle
Charge is C|U_0| then potential difference is U_0
Uh
cedar kilnBOT
cedar kilnBOT
Available help channel!
lone lily
cedar kilnBOT
crystal raptor
You can check your own answer by plugging your x and y back into the equations
cedar kilnBOT
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eternal wyvern
hii, i'm doing some review before my final exam for linear algebra and i'm kinda confused about this one in particular, why is this not a subset? doesn't it include every rational number which also includes 0?
eternal wyvern
if this does fail what axiom does it not pass?
uhhh
you should ask in a different channel
but you need the y to be seperate in it's own side
bro
this is my channel with my question, you can move to a different one
pale jacinth
chill out
eternal wyvern
hii, i'm doing some review before my final exam for linear algebra and i'm kinda...
unless you can help with this you shouldn't be chatting in here
pale jacinth
read the rules...
sorry, I misread the name
eternal wyvern
okay anyway back to topic
eternal wyvern
sorry, I misread the name
ahh
pale jacinth
but yeah, this is your channel rn
austere hull
okay anyway back to topic
explain how you know
eternal wyvern
explain how you know
this is the correct answer based off the willey and we have 3 attempts so i prolly just guessed until i got it correctly but:
since the a0 a1 a2 and a3 are rational numbers, then you can add them up in any combination and its still going to be rational, you can multiply them in any way and they're still rational, and you have a zero vector by using all zeroes
so it is a subset?
the willey is inaccurate
or does it think 0 is not a rational number 😭
had to google that LOL
cedar kilnBOT
@eternal wyvern Has your question been resolved?
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@eternal wyvern Has your question been resolved?
eternal wyvern
.close
loud ermine
cedar kilnBOT
loud ermine
Where did that e^-λ go?
i know it is like that
but how did they make the e dissapear?
cedar kilnBOT
@loud ermine Has your question been resolved?
steel heart
the base of log is assumed e ig
cedar kilnBOT
@loud ermine Has your question been resolved?
marsh lake
cedar kilnBOT
dusk finch
Let's call the matrix "A" for a moment.
Your equation is:
Ax = 3x
we can write 3x as (3I)x, now its a matrix multiplication
so Ax = (3I)x
Ax - (3I)x = 0
(A - 3I)x = 0
where 0 is the 0 vector
dusk finch
(A - 3I)x = 0
now it's a linear system
marsh lake
do i let x = (a,b,c)
dusk finch
you can compute the matrix of coefficients A - 3I easily
dusk finch
do i let x = (a,b,c)
that's unnecessarily complicated
dusk finch
(A - 3I)x = 0
TLDR: subtract 3 from each entry on diagonal, and then do gauss method
here it's explained in video
dusk finch
so from Ax=lambdax, i should start by factorizing and rewriting it?
yes
generally, you will be solving this system:
(A - lambda I)x = 0
dusk finch
now you can just do gaussian reduction btw
and solve the system
and since the determinant is 0, it will have infinitely many sols
marsh lake
now you can just do gaussian reduction btw
if you dont mind can you demonstrate it? since how i did it was really just let x = (a,b,c) and solving for each row, thanks
marsh lake
and since the determinant is 0, it will have infinitely many sols
i got a=b=c=1
wraith daggerBOT
MæthIsAlwaysRight
dusk finch
it's essentially solving this system
wraith daggerBOT
MæthIsAlwaysRight
\[
\left[\begin{array}{ccc|c}
5 & 9 & -14 & 0 \\
1 & 1 & -2 & 0 \\
1 & 3 & 1 & 0 \\
\end{array}\right]
\]
MæthIsAlwaysRight
dusk finch
wdym?
dusk finch
yeah
marsh lake
would u do r2-5r1
dusk finch
where exactly?
dusk finch
oh i know what's wrong
marsh lake
yeah?
dusk finch
i forgot to turn your 5 to -5
wraith daggerBOT
MæthIsAlwaysRight
marsh lake
sorry i copied it wrong 😭 its 2am rn
dusk finch
[
\left[\begin{array}{ccc|c}
2 & 9 & -14 & 0 \
1 & 1 & -2 & 0 \
1 & 3 & -4 & 0 \
\end{array}\right]
]
dusk finch
Yes, now it works out
wraith daggerBOT
MæthIsAlwaysRight
dusk finch
[
\left[\begin{array}{ccc|c}
1 & 0 & -1 & 0 \
0 & 1 & -1 & 0 \
0 & 0 & 0 & 0 \
\end{array}\right]
]
wraith daggerBOT
MæthIsAlwaysRight
dusk finch
so b - c = 0
and a - c = 0
or b = c and a = c
so a = b = c
so eigenvector can be [1, 1, 1] or [2, 2, 2] or anything in this form
marsh lake
so eigenvector can be [1, 1, 1] or [2, 2, 2] or anything in this form
ohh i see
can i ask one more question? about matrix
thanks for that btw
dusk finch
can i ask one more question? about matrix
sure
marsh lake
dusk finch
part d I suppose?
marsh lake
yep
dusk finch
well so do you already have the characteristic equation?
marsh lake
i dont know what that is sorry, im prelearning all of this
dusk finch
well then you should probably learn about that
marsh lake
well so do you already have the characteristic equation?
is it det(A-lambdaI)=0
dusk finch
just the left side of it
anyway, let's just assume the polynomial would be say
$\lambda^3 + c_1 \lambda^2 + c_2 \lambda + c_3 I$
wraith daggerBOT
MæthIsAlwaysRight
dusk finch
then $A^3 + c_1 A^2 + c_2 A + c_3 I = 0$
marsh lake
why is it like that
wraith daggerBOT
MæthIsAlwaysRight
dusk finch
why is it like that
it will be a third degree polynomial, i dont know which one yet, so ill show you the general case
marsh lake
it will be a third degree polynomial, i dont know which one yet, so ill show you...
because we have 3 eigenvalues?
dusk finch
because we have 3 eigenvalues?
mainly because it's 3 x 3 matrix
dusk finch
**MæthIsAlwaysRight**
Now we can multiply that by A^-1
dusk finch
assuming c3 is non-zero
marsh lake
Yeah that makes sense
marsh lake
which is det(A-lambdaI)
what lambda would i use
dusk finch
what lambda would i use
variable lambda
the letter
you mainly need to find the coefficients of the polynomial
what eigenvalues did you find btw?
marsh lake
what eigenvalues did you find btw?
-1 3 5
dusk finch
,w expand (lambda+1)(lambda-3)(lambda-5)
wraith daggerBOT
dusk finch
okay, here is your characteristic polynomial
roots of characteristic polynomial are the eigenvalues
so i used that to get it
marsh lake
ohhh
dusk finch
this would also mean that A^3 - 7A^2 + 7A + 15 = 0
now multiply by A^-1 and solve for it
dusk finch
yeah, its called cayley hamilton theorem iirc
or sth like that
it states that every matrix satisfies it's characterstic equation
dusk finch
np
cedar kilnBOT
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urban timber
Can someone tell me if I’m even right or just completely off the mark. Sorry for the incoherent work. I’m trying to simplify the expression leaving them in algebraic form without the trig functions at the end
cedar kilnBOT
Can someone tell me if I’m even right or just completely off the mark. Sorry for...
@urban timber Has your question been resolved?
urban timber
<@&286206848099549185>
cedar kilnBOT
@urban timber Has your question been resolved?
urban timber
<@&286206848099549185>
wanton sail
btw it's not too hard to check your answer to questions like this by using a graphing calculator like Desmos
If I graph the thing you're trying to simplify and your final answer, I can see the two graphs differ, so something went wrong
Lemme look through your work
you're missing parentheses
the first two terms should be in parentheses (.. - ..) · sqrt(1-x^2)
Once you fix that and simplify, your answer is right, and you can double check it by graphing
@urban timber
cedar kilnBOT
civic sand
we know that the rectangle as paralel sides right?
that means that the angles must be the same
so the triangles (big triangle - rectangle) must be similar to eachother and similar to the big triangle
we could try starting from that
dense island
1/2(base)(height)
Let me think, ty btw
Surely it's not y=x(b + h)
But that seems right? @civic sand
dense island
Yeah
civic sand
ok
the triangles ABH and BIC are similar to the Triangle AGC
I think ratios can be a clue
$b / x = h / h_y$
civic sand
@dense island How does the question need to be answered?
civic sand
**Max\-Cat**
something like this?
dense island
Yeah, I didn't intend to do it fully
I had no clue how to translate that into a function
dense island
**Max\-Cat**
How did you get this?
dense island
I had no clue how to translate that into a function
Which is the important part for me
civic sand
the angle BAH is equal to the angle CBI since the side of the rectangle are paralel, the same can be said for the angles BCI and ABH
The triangles are similar which means that their sizes and lenghts can be expressed as ratios
dense island
BCI?
dense island
Yeah
civic sand
the angles are identical thus the sides must be ratios
dense island
That's not where I'm having difficulties, I wonder where the algebraic motivation came from per se
How you got to the exact equation
civic sand
so the side BH is to CG, the same that BI is to AG and so on, that's where the ratio came from
dense island
the angles are identical thus the sides must be ratios
This is clever and I realized for myself, I had no clue how to make it a function
dense island
so the side BH is to CG, the same that BI is to AG and so on, that's where the r...
OH
civic sand
after that I used a rule of 3 simple to solve for H_y and just putted that in the equation
so
dense island
I think I understand, thank you
civic sand
no need with anything else?
cedar kilnBOT
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fierce laurel
cedar kilnBOT
fierce laurel
Confused by bounds here
Why are the z bounds not just where the cylinder and sphere intersect?
Here is my thought ^ z-planes of intersection.
Because you are just going to take that integral over the proper radius.
(but my thoughts are wrong apparently)
so why integrate with bounds from entire top half of sphere and bottom half?
i do not understand the logic. you are just finding the area essentially with the cylinder at a height where they intersect.
muted bear
because youre missing the little curved bits at each end
fierce laurel
because youre missing the little curved bits at each end
what little curved bits?
are we not just essentially finding the volume of the cylinder from the height of intersection? maybe i do not understand what it is asking
muted bear
take the cylinder off your graph really quick and ypull notice a bit sticking out of the planes
those are still inside the cyliner and the sphere
muted bear
that height varies based on the radius hence the z bounds being based on r
fierce laurel
i see
thank you
that was the issue, i did not understand!
much appreciated
.close
cedar kilnBOT
muted bear
youre welcome
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Available help channel!
plucky mirage
how would I fully factor 2(x^4−81)?
cedar kilnBOT
how would I fully factor 2(x^4−81)?
plucky mirage
the original question was 2x^4 - 162
dull oxide
how would I fully factor 2(x^4−81)?
Difference of squares
plucky mirage
I got to that but idk how to keep going
plucky mirage
Difference of squares
how tho?
dull oxide
$a^2-b^2=(a+b)(a-b)$
wraith daggerBOT
SWR
dull oxide
I tried to google it and I got this(x^2)^2-9^2
That's a start. Yeah
dire steppe
Confused by bounds here
As denoted from the equation from the sphere r^2 + z^2 = 64 so z = +- sqrt(64 - r^2) if you wanted to solve it analytically
dire steppe
how would I fully factor 2(x^4−81)?
I would substitue u for x^2
plucky mirage
That's a start. Yeah
thanks
cedar kilnBOT
@plucky mirage Has your question been resolved?
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marsh lake
cedar kilnBOT
marsh lake
first thing i have to do is find A^2 right?
humble karma
I think so, and then you'd find eigenvalues/eigenvectors for the diagonalization.
marsh lake
I think so, and then you'd find eigenvalues/eigenvectors for the diagonalization...
I found A^2
then i have to do det(A^2-λI) = 0 right?
for eigenvalues
crimson delta
look at how A^2 looks. do you actually need the char poly to tell the eigenvalues?
marsh lake
marsh lake
look at how A^2 looks. do you actually need the char poly to tell the eigenvalue...
idk sorry 😭
marsh lake
i got P = zero matrix which obviously doesnt make sense
humble karma
idk sorry 😭
Well as you can see from the computation you did, the eigenvalues are the entries of the diagonal.
This is true for diagonal matrices in general.
As for the eigenvectors, can you show your work for finding them?
humble karma
Well eigenvectors won't be (0,0,0) ever, so no
marsh lake
Well eigenvectors won't be (0,0,0) ever, so no
What did i do wrong
noble granite
why do you have 2D eigenvectors?
humble karma
What did i do wrong
You should REF the matrix you get
marsh lake
why do you have 2D eigenvectors?
wdym?
marsh lake
You should REF the matrix you get
which
wraith daggerBOT
nameless individual
humble karma
why do you say $x = \begin{bmatrix} 0 \\ 8 \end{bmatrix}$
It's (0,0,0), not (0,8)
noble granite
oh nevermind then
humble karma
which
When you subtract the identity matrix times the eigenvalues
You get a matrix
You have to REF this matrix
noble granite
are you sure that's all the solutions for $x$?
wraith daggerBOT
nameless individual
humble karma
To get the eigenspace
noble granite
for instance, when $\lambda=25$
wraith daggerBOT
nameless individual
noble granite
you have $\begin{bmatrix} 0 & \frac{22}3 & 48 \ 0 & 11 & 0 \ 0 & 0 & -24 \end{bmatrix}x=\mathbf0$
wraith daggerBOT
nameless individual
noble granite
but for instance, $\begin{bmatrix} 5\0\0 \end{bmatrix}$ is also a solution for $x$ that you have not considered.
marsh lake
Ye
wraith daggerBOT
nameless individual
marsh lake
how do i do it? 😭
marsh lake
**nameless individual**
how do i find this as a solution
noble granite
let $x=\begin{bmatrix}a\b\c\end{bmatrix}$
wraith daggerBOT
nameless individual
marsh lake
yep
noble granite
and you will get a system of equations
You'll get $b=c=0$ but nothing for $a$, for instance, when $\lambda=25$
wraith daggerBOT
nameless individual
marsh lake
yes
noble granite
and thats because $\begin{bmatrix}a\0\0\end{bmatrix}$ are all eigenvectors with eigenvalue $25$, no matter what $a$ is
wraith daggerBOT
nameless individual
noble granite
By definition of eigenvector and linear transformation
$\begin{bmatrix}0\0\0\end{bmatrix}$ is always an ``eigenvector", so we usually don't even include it
it's not useful, and eigenvectors are meant to be useful lmao
marsh lake
Ohhhh
noble granite
(not to torture you)
wraith daggerBOT
nameless individual
marsh lake
You'll get $b=c=0$ but nothing for $a$, for instance, when $\lambda=25$
what about when λ = 36?
(0,k,0)?
wraith daggerBOT
nameless individual
noble granite
is it easy to intuitively understand that eigenvectors comes in lines?
marsh lake
is it easy to intuitively understand that eigenvectors comes in lines?
uhh not really sorry
noble granite
i.e. if $x$ is an eigenvector, then $kx$ is also an eigenvector for constant scalar $k$
wraith daggerBOT
nameless individual
noble granite
exactly
marsh lake
**nameless individual**
Thank u so much
noble granite
its nice helping people out
noble granite
Can you prove that the eigenvectors of $A^2$ and $A$ are the same, and the corresponding eigenvalue for $A^2$ is the square of that of $A$?
(and that $A^2$ doesn't have any more eigenvectors)
noble granite
(haha im not sure either i havent really learned any linear algebra lma)
lemme get gpt
wraith daggerBOT
nameless individual
noble granite
gpt gave some crap answers
wraith daggerBOT
nameless individual
noble granite
@marsh lake
noble granite
**nameless individual**
i meant, what do you have by definition?
marsh lake
A^2x=λ^2x?
marsh lake
yes! *you forgot an I*
λ^2Ix?
wraith daggerBOT
nameless individual
noble granite
yah, sorry, made the same mistake
marsh lake
ok yeah makes sense, since were working with matrices
noble granite
It would be nice if we could apply the difference of squares... but there's an I.
But we have $I=I^2$, by definition.
wraith daggerBOT
nameless individual
marsh lake
Yep
noble granite
so $(A^2-\lambda^2I^2)x=0$
marsh lake
(A+λI)(A-λI)x=0
wraith daggerBOT
nameless individual
marsh lake
oh so we can drop the A-λI
wraith daggerBOT
nameless individual
marsh lake
and therefore (A+λI)x=0
noble granite
You can't really drop the $A-\lambda I$
wraith daggerBOT
nameless individual
noble granite
I'm like decently sure that if $ABx=0$ then $Ax=0$ or $Bx=0$ but I have no good argument
wraith daggerBOT
nameless individual
noble granite
I thought it was "trivial" lma
marsh lake
i have no knowledge on linear algebra 😭
noble granite
<@&286206848099549185>
noble granite
I'm like decently sure that if $ABx=0$ then $Ax=0$ or $Bx=0$ but I have no good ...
nah that doesn't hold, sadly
flint plinth
is this still the question under consideration?
marsh lake
**nameless individual**
no, we are discussing this
noble granite
https://discord.com/channels/268882317391429632/373217754666500097 Well that went as well
flint plinth
ah gonna copy it here just for reference to reduce scrolling
the last part (and A^2 doesn't have any more eigenvectors) isn't necessarily true
the rest is
noble granite
the last part (and A^2 doesn't have any more eigenvectors) isn't necessarily tru...
(see the help thread
I think this is out of my league here lol, ima stay outta this
marsh lake
oh ok, thanks for helping me anyways lol
noble granite
twas nice
flint plinth
if $Av = \lambda v$ then $A^2 v = A(Av) = A(\lambda v) = \lambda (Av) = \lambda(\lambda v) = \lambda^2 v$
wraith daggerBOT
Bungo
flint plinth
conclusion, if (lambda, v) is an eigenvalue/eigenvector pair for A, then (lambda^2, v) is an eigenvalue/eigenvector pair for A^2
noble granite
it applies to higher integral powers
And in this case we know that's all the eigenvectors for A since there can only be three lines of eigenvectors for A
(unless its just a whole plane of eigenvectors, and that's another story)
marsh lake
what does that mean sorry
flint plinth
not to higher powers
you can just repeat what i did for powers greater than 2
A^3 v = A^2(Av) = A^2(lambda v) = lambda(A^2 v) etc
noble granite
what is this? STEP?
marsh lake
what is this? STEP?
marking scheme
noble granite
marking scheme
I never really could understand how you read those. What's A, B, and M?
marsh lake
I never really could understand how you read those. What's **A**, **B**, and **M...
I have no idea
those are how they give the marks i think
noble granite
lma nobody knows how they read those
marsh lake
lma nobody knows how they read those
this is a levels btw
noble granite
I've seen it on STEP as well
noble granite
it doesnt matter
peak bough
What's the problem here?
cedar kilnBOT
What's the problem here?
noble granite
probably some formatting issue: the left exponent is visibly lower than the right one
cedar kilnBOT
noble granite
haha lol 69420
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Available help channel!
carmine anchor
i have a Bernoulli equation i am solving it is dy/dx+(7/x)y=xy^2 I came to an answer of y = 5/x^2+5Cx^7 and have verified the solution i just wanna make sure im actually correct (unable to get image of my work to upload since im on a farm in Lithuania rn) thank you
noble granite
do you mean $y=\frac5{x^2}+x^7C$?
wraith daggerBOT
nameless individual
carmine anchor
do you mean $y=\frac5{x^2}+x^7C$?
i got 5 over x^2+5Cx^7 but i am not totally confident in my work
wraith daggerBOT
nameless individual
carmine anchor
do you mean $y=\frac5{x^2}+x^7C$?
cedar kilnBOT
@carmine anchor Has your question been resolved?
errant locust
Guys I need a computer
cedar kilnBOT
Guys I need a computer
crimson delta
buy one