I am taking phi from the vertical axis and making it rotate clockwise
#help-36
cedar obsidian
tired walrus
ok
your 30° does not look much like 30° visually speaking but OK
your phi is 3 right angles minus that 30°.
final saddleBOT
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restive sinew
f(x)=x^2 in the interval (0,1) it is uniform continous
final saddleBOT
f(x)=x^2 in the interval (0,1) it is uniform continous
whole halo
oh I thought you were going to use lipschitz
restive sinew
i meant after derivative it is bounded
whole halo
thats lipschitz
rocky tusk
how about
restive sinew
in the given interval
restive sinew
it is not UC in real
rocky tusk
can you read?
rocky tusk
i said [0, 1]
whole halo
[0, 1] and R are not the same thing
knief isnt saying R, knief is just saying [0, 1] instead of (0, 1)
rocky tusk
a continuous function on a compact set is uniformly continuous
whole halo
because of unboundeness
also you can still have an unbounded function be uniformly continuous, like f(x) = x
unless you mean like the derivative is unbounded
restive sinew
also you can still have an unbounded function be uniformly continuous, like f(x)...
please don't chagne the funciton
whole halo
well you see a common thing that people in conversations do is link one thing with another
restive sinew
the function is fixed only changing intervals
whole halo
if you cant keep both things organized in your head, we can do this slower
rocky tusk
why you guys are mixing two things?
what?
whole halo
f(x)=x^2 in the interval (0,1) it is uniform continous
now knief has just mentioned a way you can solve this problem
we know f(x) = x^2 is continuous
restive sinew
i am connecting it with derivative for fast solving some questions
restive sinew
this problem is a one liner
how?
whole halo
are you doing this at a high school or college level
restive sinew
where?
rocky tusk
it’s uni cts on [0, 1]
^
restive sinew
a continuous function on a compact set is uniformly continuous
ohh
whole halo
do better
rocky tusk
why would you need that
restive sinew
i am saying generally
whole halo
yea
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calm cloud
i dont really understand the b) question
final saddleBOT
i dont really understand the b) question
vestal crest
does the term "quartile" ring a bell in your head?
calm cloud
i am learning about it, Q1, Median(Q2), Q3 divide into 4 equal part, each 25%, but i dont know what the question wants me to do
vestal crest
have you learnt what they mean?
like, say, if I told you that my test scores are in the third quartile, what am I saying, exactly?
vestal crest
correct! and that's exactly what the question wants you to do!
okay, the other way around actually.
I gave you my test scores and told you that it's in the third quartile, and you inferred that I score higher than 75% of students. this is correct.
you are now given that 75% of students (aka, the third quartile) sleep for some number of hours, and you are to find that number of hours.
in other words, if I relate it to my question, it's like taking my position (of being better than 75% of students) and finding out how much I must have scored.
calm cloud
hmmmm
vestal crest
the process is very much like finding the median, but with a slight adjustment to the formula, so you can use the median formula as a starting point if you're feeling a little lost.
calm cloud
sooo i should find Q1 then im done right 🥹 still a little bit confused, what if they ask sleep at most
vestal crest
oh yeah, it would be Q1 here, since we're weeding out bottom 25%.
calm cloud
still wonder why Q1 but not Q3
vestal crest
sooo i should find Q1 then im done right 🥹 still a little bit confused, what if...
context matters.
for a metric to have 75% of people exceed it, that means that this metric is in the bottom 25%.
percentiles are counted from the bottom, so if this metric is topped by at most 75% of people... now that I think about it, this question makes not much sense, and I'll get back to why shortly.
vestal crest
still wonder why Q1 but not Q3
the reason is because you are told that 75% of students sleep at least this many hours. in other words, 75% of students sleep at least x hours or more.
since percentiles are counted from the bottom, if you take Q3, you'd be finding the amount of time where only 25% of students sleep at least this long, because Q3 is higher than 75% of the data.
vestal crest
context matters.
for a metric to have 75% of people exceed it, that means that t...
now, why does the "at most" question not make sense?
because let's say I ask for a test score that is exceeded by at most 75% of students.
well... a score that is at the first quartile is exceeded by exactly 75% of students. but "at most" means that if I picked a score that is exceeded by nobody, it still counts, because I only care that the score is not exceeded by more than 75% of students taking the test.
vestal crest
glad to have helped!
calm cloud
.close
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sharp wraith
hi i was wondering if there were faster ways of doing this which don't perhaps doesn't require manually calculating each coefficient, i noticed that the polynomial is an even function (since f(x) = f(-x)), which i think means there are no x terms of an odd degree, so i just assumed the polynomial is in the form ax^8 + bx^6 + cx^4 + dx^2 + e, and manually worked out each coefficient for each x degree, could have done it wrong tho
final saddleBOT
hi i was wondering if there were faster ways of doing this which don't perhaps d...
storm haven
Plug in x = 1
If x = 1, x raised to any positive integer power will just be 1
So what you get in the end is just the sum of the coefficients
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warm python
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wai
north juniper
i think it's 1 to 2-y
warm python
i think it's 1 to 2-y
why
north juniper
oh wait sry
warm python
,w $\int_{0}^{2-y} \frac{3}{4} (2-x-y)dx$
soft zealotBOT
warm python
cool
north juniper
soft zealotBOT
wai
warm python
I think it's this?
north juniper
i think you divide that one by $\int_{0}^{2-y} \frac{3}{4} (2-x-y)dx$
soft zealotBOT
성현
warm python
its the joint pdf by the mariginal that gives us the conditional pdf no
warm python
i think you divide that one by $\int_{0}^{2-y} \frac{3}{4} (2-x-y)dx$
hmm?
north juniper
oh divide by $\int_{0}^{1}\int_0^{2-y} \frac{2-x-y}{(y-2)^2} dxdy$ sry
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warm python
that's the conditional pdf though
north juniper
if 0<x<1 and 0<y<1 holds 0<x+y<2 does absolutely nothing
north juniper
that's the conditional pdf though
no that one is marginal
north juniper
you forgot Y<1
warm python
I still don't follow
north juniper
for getting marginal pdf, you need to integrate xy-plane not only x one
Y<1 implies 0<y<1 and 0<x<2-y , you need to integrate f in that region
north juniper
isn't P( X < 1 | Y < 1) = P(X<1, Y<1) / P(Y<1) ?
warm python
sure, but I'm finding it by first finding the mariginal
north juniper
joint density function itself consists of x,y why you only integrate in terms of x
warm python
and integrating the marginal
warm python
joint density function itself consists of x,y why you only integrate in terms of...
got it
thanks
.close
cedar obsidian
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cedar obsidian
In this derivation why k1k2/k1+k2 appears in the third step?
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old quarry
let length of focal chord PQ of the parabola y^2 = 12x be 15 units. if the distance of PQ from the origin is p then 10p^2 is equal to?
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let length of focal chord PQ of the parabola y^2 = 12x be 15 units. if the dista...
old quarry
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native widget
So the problem is in applied optimization. Asked to relate an inscribed cylinder inside a sphere with radius 1 in order to maximize volume. I cant get a function for volume because I havent figured out how to relate the cylinder into the sphere properly
final saddleBOT
So the problem is in applied optimization. Asked to relate an inscribed cylinder...
soft zealotBOT
thorny canyon
your approach is to try using pythagorean theorem to relate radius of cylinder, height and radius of sphere if I'm reading right?
native widget
yeah, just trying to get r in terms of h or h in terms of r.
but it introduces even more variables
thorny canyon
yeah, just trying to get r in terms of h or h in terms of r.
you're pretty close actually
you can convert both of them in terms of radius of sphere times some variable function
parametrizing into a third variable essentially
remind you of smn?
native widget
does it use that r=rho-l where rho=1 and it turns into that quadratic and then i put something in for h?
thorny canyon
I'm afraid idk what rho refers to there
native widget
R=1 & h and r are both varibles
thorny canyon
ah alr
native widget
the radius of the sphere looks like the curly p
thorny canyon
does it use that r=rho-l where rho=1 and it turns into that quadratic and then i...
I was thinking of angles tbh, finding l seems to be a headache for this approach
native widget
well i thought that, and i assumed that 45 45 would be the max area for a triangle but then my volume was like 0.4 off from the max
thorny canyon
trigonometry
thorny canyon
take the marked angle as theta
represent r, h in terms of R and functions of theta
that'll reduce it to a differentiation problem
native widget
sintheta=h/2 & costheta=r ? then differentiate individually like d/dtheta sintheta is costheta=dtheta/2 and then d/dtheta costheta is -sintheta=dtheta?
im just so confused i dont know what to reduce with that
thorny canyon
sintheta=h/2 & costheta=r ? then differentiate individually like d/dtheta sinth...
yeah you got the thing
but don't diff individually
write the equation for volume, $V=\pi r^2h$
soft zealotBOT
donkey
thorny canyon
you'll get some $V=f(\theta)$
soft zealotBOT
donkey
thorny canyon
minimize f
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native widget
V=pi cos^2theta*sintheta, then dV/dtheta set to 0 get the max and a varible then go back to the orginal equation and get the other variable?
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shadow smelt
how to do part b
final saddleBOT
how to do part b
shadow smelt
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royal shale
how do we interpret anything from these bar graphs?
final saddleBOT
how do we interpret anything from these bar graphs?
echo plover
You have to calculate to know if the average and variance questions are true, no?
I mean you can reason it without calculation but u kinda have to get some number to know
cursive bough
generally, the less flat a histogram looks the higher the variance
to figure out which has a higher average, you can look at which histogram has taller bars toward the right
final saddleBOT
@royal shale Has your question been resolved?
royal shale
I mean you can reason it without calculation but u kinda have to get some number...
uhm, but the instructor at khan academy does not calculate anything but still manages to get an answer so i was wondering how
royal shale
generally, the less flat a histogram looks the higher the variance
i havent studied about variance yet
royal shale
to figure out which has a higher average, you can look at which histogram has ta...
so the answer should be option B right?
cursive bough
i havent studied about variance yet
variance is just how much the numbers vary
which one of those graphs looks more flat to you?
royal shale
variance is just how much the numbers vary
which one of those graphs looks more...
2nd one
marble one
cursive bough
like if I drew a flat line across each histogram, which one is closest to the line
royal shale
so it has less variance?
cursive bough
yes
cursive bough
yep, that is right
royal shale
and since the marble one has the taller bar towards the right
we can say option A is correct then?
cursive bough
exactly!
royal shale
what about option B though?
cursive bough
what about option B though?
let's pick a random movie from Marble. according to the graph, there are multiple movies that make between 3 and 4 million dollars
are there any DX movies that make more than that
royal shale
are there any DX movies that make more than that
yeah 2 DX comics
make more than 3 and 4 million dollars
cursive bough
make more than 3 and 4 million dollars
exactly, so what does that tell you about B
cursive bough
cool, happy to help!
royal shale
i love this server, people here are so helpful
royal shale
cool, happy to help!
also can you help me with this
i understand that option A is correct
but how do we figure out anything about option B?
cursive bough
if the data points are far apart, then they vary more
and if they're closer together, they vary less
royal shale
and if they're closer together, they vary less
thank you very much 💛
cursive bough
thank you very much 💛
no problem, but i think I may have been wrong about the histograms and varying.
after thinking about it, there is a higher variance if the values are more spread out. and since the second histogram isn't as spread out (a lot more is concentrated in one little area), it should actually have a lower variance
so it should be flipped from what I said earlier, sorry about that. ignore the line trick I did lol
royal shale
didnt you say the same thing earlier? 😭
oh ohh
i understand what youre saying
youre talking about the histogram
i see
royal shale
so it should be flipped from what I said earlier, sorry about that. ignore the l...
but the answer should be same, right?
since the variance of DX comics is still less by what you said
cursive bough
since the variance of DX comics is still less by what you said
the variance of Marble movies is less since theres a big spike there, and that means there are a lot of data points concentrated in one area
so they aren't as spread out, meaning they vary less
royal shale
the variance of Marble movies is less since theres a big spike there, and that m...
but you told me to notice the spread for variance not the spike
cursive bough
really? that's a bit shocking, sorry. I'm not sure how they'd expect you to know that by looking, variance is hard to tell with histograms 🤔
cursive bough
but you told me to notice the spread for variance not the spike
yeah but a spike means there are a lot of data points in the same range (like between 2 and 3 million). and since there are a lot that are together it means that if you put down the values as dots like in the other problem, they'd be close together
the issue with histograms is that we don't know the actual values, just whether theyre between 2 and 3 million or 3 and 4 million, etc
so determining variance is not easy to do by eye usually
royal shale
really? that's a bit shocking, sorry. I'm not sure how they'd expect you to know...
see, since the taller bar is shifted to the right in case of marble comics, the average is more for marble comics, option B is incorrect and for option C when we notice the spread, the spread for marble comics is much more than DX comics, thats all what i interpret from what you said
royal shale
so determining variance is not easy to do by eye usually
ohhh
warm charm
histograms a good candidate for pre civil rights literacy tests
worthy radish
can anyone tell me what this table does and is?
final saddleBOT
can anyone tell me what this table does and is?
worthy radish
for what is it used
tranquil pine
can anyone tell me what this table does and is?
You put in a value of x, you get an outuput y
Function's like a machine
Simply
worthy radish
is it in order to make a perfect line?
of parabola
cause he marked the points and dragged lines through them
tranquil pine
of parabola
No? Parabola is a curve, the equation is of second degree so it will give a parabolic curve
finite pasture
Its gonna be a smily face up shape as x^2 is positively and this is quadratic
Isnt that ur question
tranquil pine
i dont understand
What grade you are in?
finite pasture
i dont understand
Bro look now imagine this y side isnt visible
worthy radish
NANNAANANANANA\
tranquil pine
show me this
You give the function, in ur case its y, an input, and it gives the value "y" by applying a set if rules to it
finite pasture
You give the function, in ur case its y, an input, and it gives the value "y" by...
I think bro is here to troll
tranquil pine
I think bro is here to troll
😭how simply can i explain, i give up
finite pasture
Bro his account is newly made
tranquil pine
Bro his account is newly made
Oh you meant him,
finite pasture
Ye
finite pasture
Well idk
Js thoughts , many ppl trolled me b4 into explaining for them and they js came out trolling me and they aint here for this yk
tranquil pine
Js thoughts , many ppl trolled me b4 into explaining for them and they js came o...
Lmao
proper dagger
NANNAANANANANA\
<@&268886789983436800> lowkey trolling
barren hound
uhhh @worthy radish I think you need an actual teacher, or to pay closer attention to whatever video you were watching
we're almost certainly not going to be able to explain this to you in a language that you're comfortable in
final saddleBOT
@worthy radish Has your question been resolved?
worthy radish
It’s half past 3, im tired, im going insane now cause i have to do another 2 assignments which im very late with which ill do now, ill pass math for now. Now after doing all these equations I discovered there’s some table that is confusing for me as I’ve now wasted 15 mins trying to understand it.
clever gorge
Hey so for my math homework 1 question is basically that a population if rabbits triples every ten years, and this part specifically is asking for a calculator ready equation to find how long it would take for 5000 rabbits to emerge from a starting population of 500. Me and my friend disagree on the answer so can someone help me figure out if im right or not? (I alr found the constant of proportionality)
final saddleBOT
Hey so for my math homework 1 question is basically that a population if rabbits...
violet spoke
hi
mint orbit
thats a lot of random equations in your picture
clever gorge
only focus on the top until the boxed answer
violet spoke
ok
clever gorge
the rest is different stuff
sick berry
I'm stuck in this exercise, I need to solve this trigonometric function with the chain rule and I don't have any idea of how can I solve this one
mint orbit
I'm stuck in this exercise, I need to solve this trigonometric function with the...
sorry you have to open a new channel
violet spoke
I'm stuck in this exercise, I need to solve this trigonometric function with the...
go to #help-12 post it and ill help u
mint orbit
Hey so for my math homework 1 question is basically that a population if rabbits...
you can judge me but I think you'd want to use a discrete one, no?
use a difference equation, solve the relation, and then extend it to continuous
clever gorge
you can judge me but I think you'd want to use a discrete one, no?
nah it's continuous
mint orbit
then what is the implied underlying model for rabits
y' = ay?
and i guess, you're saying you solved for a
clever gorge
idk what you mean by that sorry. the information given is just a starting population of 500 rabits and that from any given population, the triple in 10 years
mint orbit
idk what you mean by that sorry. the information given is just a starting popula...
yea but from this information I'd assume it were a discrete equation, because we dont have some sentence like 'the population of rabbits grows proportional to the existing population'
anyways sure
clever gorge
im like 99% sure it's continuous but that isn't what me and my friend disagree on
mint orbit
okay one second
clever gorge
he got 10*log (base 3) 10
mint orbit
sure so something like
idk what we use for population
$P(t) = P_0 e^{\frac{\log 3 }{ 10 } t }$
soft zealotBOT
jan Niku
mint orbit
log here is natural
clever gorge
nah
clever gorge
but log is natural that's what i was saying yes to
i got (10 ln10)/ln3 but he got 10 log (base 3) 10
and usually when we disagree he's right
but i can't find anything wrong with my work
so i just need help figuring out if i got it wrong
mint orbit
it might help to use more symbols
i usually wouldnt say that
but if we call $P(t) = P_0 e^{\alpha t}$
soft zealotBOT
jan Niku
mint orbit
where P0 the initial population is known
then the time to triple is gamma, the solution to $3P_0 = P_0 e^{\gamma \alpha}$
soft zealotBOT
jan Niku
mint orbit
I get $\gamma = \frac{ \log 3}{ \alpha}$
soft zealotBOT
jan Niku
mint orbit
so the answer would be $\frac{ \log 3}{ \frac{ \log 3}{ 10 } }$
soft zealotBOT
jan Niku
clever gorge
mmmm idk about that
mint orbit
sorry, its not to triple
its to equal 5000
one sec
itd be
$\frac{ \log (5000) - \log (500) }{ \frac{\log 3}{10 } }$
soft zealotBOT
jan Niku
mint orbit
,w simplify (log5000 - log500)/( (log3)/10 )
clever gorge
yes
soft zealotBOT
mint orbit
so $\frac{ 10 \log 10 }{ \log 3}$
soft zealotBOT
jan Niku
clever gorge
yup that's what i got
mint orbit
clever gorge
okay ty
mint orbit
sorry took me a second to catch up
you know
this looks like change of base too
lmc
clever gorge
somehow my frind got $\frac{10 \log_3 10 }
did i do it
maybe
no
i failed to use the command
soft zealotBOT
jan Niku
mint orbit
clever gorge
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mint orbit
somehow my frind got $\frac{10 \log_3 10 }
you see why this is the same?
to be sure
but if you got it you got it
clever gorge
why what is the same
mint orbit
why the two answers are the same
mint orbit
**jan Niku**
this
mint orbit
**jan Niku**
and this
clever gorge
wait lemme do some maths
mint orbit
.reopen
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✅ Original question: #help-36 message
clever gorge
no but remember i got ln not log
mint orbit
to recall $\log _a b = \frac{ \log _c b }{ \log _c a }$
soft zealotBOT
jan Niku
mint orbit
change of base
mint orbit
**jan Niku**
I'm using log here to mean ln
clever gorge
ohhhh
mint orbit
so $\log _3 10 = \frac{ \ln 10 }{ \ln 3 }$ by change of base
soft zealotBOT
jan Niku
mint orbit
thats why theyre the same
clever gorge
which is considered more simplified?
mint orbit
i dont know you said calculator ready right?
clever gorge
yes
clever gorge
so ig it doesn;t really matter
mint orbit
if i was going to put this into a calculator id probably want ln
but yea probably doesnt matter
clever gorge
do your calculators accept logs of arbitrary base
we use desmos so yeah
clever gorge
alr
mint orbit
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warm python
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warm python
so I want to use this method
I have $P(Y≤y) = P(-\infty<e^{-x}<y)= P(0<-x<ln(y))=P(0<x<ln(1/y))$
right so far?
atomic moon
I might be asleep, but how are you going from X to x
soft zealotBOT
wai
warm python
I might be asleep, but how are you going from X to x
I'm just following an algorithm honestly
marsh temple
atomic moon
at the end its just X
warm python
I have $P(Y≤y) = P(-\infty<e^{-X}<y)= P(0<-X<ln(y))=P(0<X<ln(1/y))$
soft zealotBOT
wai
atomic moon
But then you use cdf or whatever and differentiate it
warm python
But then you use cdf or whatever and differentiate it
slight issue
atomic moon
Ah you don't have the cdf
Well you can now say that the last probability is a substraction of two probability no?
warm python
I can integrate the desnioty function from 0 to ln(1/y) and differentiate that
Okay, that does give me the answer
just maade a calc mistake
.close
final saddleBOT
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tall vector
Idk what to do next
tired walrus
!msgdel
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The original post of this help channel has been deleted, and it will abruptly close and possibly lock. (This is irreversible.) Please claim a new channel, and don't delete the first message of any future channel you claim.
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grand nimbus
Given matrix K that has a decomposition (?) $LDL^T$ such that L is a basic(?) lower triangular matrix and D is a diagonal matrix with positive elements. Prove that K is symmetric and positive definite.
final saddleBOT
Given matrix K that has a decomposition (?) $LDL^T$ such that L is a basic(?) lo...
soft zealotBOT
prograce
grand nimbus
Idk how to start, should i try to prove using definitions of PD and symmetric matrix?
desert mantle
yes
grand nimbus
Ok
final saddleBOT
@grand nimbus Has your question been resolved?
grand nimbus
How do I prove that zDz^t>0 for any vector z !=0 and diagonal matrix with positive elements
desert mantle
multiply out
soft zealotBOT
grand nimbus
Is there a shotcut or just multiply ..?
desert mantle
just multiply
grand nimbus
😭
sleek belfry
Hi, i have an exercise to prove binomial theorem using induction.
My logic:
1) for n=1 the binomial theorem is true(selfexplanatory)
2) we assume n=n holds and prove n=n+1 using n=n (see picture), i guess we can do
(a+b)^(n+1) = (a+b)^n * (a+b)^1, but not sure how to continue
floral nova
There really isn't any magic other than multiplying it all out first
final saddleBOT
@sleek belfry Has your question been resolved?
main mirage
See above message. In particular, it is important to note the Pascal's identity:
nCr + nC(r + 1) = (n + 1)C(r + 1)
If you want, you can kinda simplify things by using summation notation.
main mirage
If you want, you can kinda simplify things by using summation notation.
I should have mentioned that using summation notation is a little more tedious than actually expanding it out (it's doable but setting it up has been a little weird).
main mirage
we would then have
Wait
You're trying to prove the binomial formula for (a + b)^(n + 1). That means it should be (a + b)^n ***** (a + b).
Also, you are missing your binomial coefficients.
sleek belfry
oh true
sleek belfry
You're trying to prove the binomial formula for (a + b)^(n + 1). That means it s...
thats what i have, no? except the missing coefficients
lets say we want to multiply first factors of (n+1)^n and (n+1)^1 with each other, we would then have
(n choose 0) * (n choose 0) doesnt simplify into anything, its just (n choose 0)^2
main mirage
we would then have
um I see no coefficients here
The coefficients are very important in the binomial proof
sleek belfry
ok lemme fix
main mirage
Ngl I think you should actually write down the expansion rather than do the summation strategy
It is significantly easier, significantly easier to conceptualise
main mirage
You just need it for the actual binomial expansion :)
sleek belfry
ok so something like this...
i guess we can factor out the combination number but not sure how we would get the pascal law from that
final saddleBOT
@sleek belfry Has your question been resolved?
viral kite
using (n+1)C(r+1) = nCr + nC(r+1) would be good
final saddleBOT
@sleek belfry Has your question been resolved?
final saddleBOT
final saddleBOT
Available help channel!
solar crest
can someone help
final saddleBOT
can someone help
solar crest
i dont udnerstand this
i thought the matrix f would be 3 x 2?
the answer seems to be 2 x 3 instead...
vital crag
the answer seems to be 2 x 3 instead...
correct
vital crag
solar crest
i thought you only have 2 columns because there's 2 variables, x and y
vital crag
how matrix-vector multplication is defined
if you haven't seen a video yet, here's a good one https://www.youtube.com/watch?v=Awcj447pYuk
solar crest
okay makes sense
ty
is my kernel incorrect
if i rref the matrix i get y as the free variable instead of z
vital crag
show the rref matrix
supple trout
I have a problem. So, I was messing around with my drone delivery idea, and I ran into this problem. I want to spend as little on energy as possible, but it’s not that simple.
I realized there are three things I can mess with:
How fast I fly the drone.
How high I fly it.
What size battery I use.
Here’s the thing: flying faster drains more power. Flying higher drains more power too. And somehow, the faster I go and the higher I fly, the more the battery struggles. Also, smaller batteries need more recharging, which is annoying and eats into my time.
Of course, I’ve got limits:
Can’t fly slower than 10 m/s or faster than 30 m/s.
Can’t fly lower than 50 meters or higher than 150 meters, safety first.
Battery size has to be reasonable, somewhere between tiny and huge.
And I realized there’s this weird combo: if I go too fast and too high, the drone starts acting sketchy.
So basically, I’m trying to figure out the best combination of speed, height, and battery so each delivery uses the least energy, without the drone dying or me having to constantly swap batteries.
What should I do?
final saddleBOT
I have a problem. So, I was messing around with my drone delivery idea, and I ra...
late rose
I have a problem. So, I was messing around with my drone delivery idea, and I ra...
what's stopping you from running the drone at 50m with 10m/s speed?
that would be "least energy", would it not?
frail moat
can't say without specifics, this is abit like rocket science
supple trout
what's stopping you from running the drone at 50m with 10m/s speed?
Running it at 50 m and 10 m/s sounds nice, but that’s not exactly safe. At 50 m, you’re flying close to building tops in some areas, and air turbulence around them can mess with stability. Plus, at that speed, the drone’s motors are pulling way more power, so the battery drains faster, and heat builds up real quick. You could try it, but you’d probably overheat or run out of juice halfway through.
late rose
Running it at 50 m and 10 m/s sounds nice, but that’s not exactly safe. At 50 m,...
so you're saying that the drone, at minimum speed, already takes up too much energy?
supple trout
so you're saying that the drone, at minimum speed, already takes up too much ene...
Yeah, pretty much. Even when it’s moving slow, the drone still has to fight gravity just to stay in the air. Hovering burns a lot of power by itself, it’s not like a car idling. The motors are constantly working to keep it stable, and if there’s any wind or if it’s carrying a package, that baseline energy use shoots up fast.
Btw for a username suggestion, I would say "foxy" is a pretty decent name. It's simple & matches your profile.
late rose
Yeah, pretty much. Even when it’s moving slow, the drone still has to fight grav...
Can I know why the rationale of the minimum speed being 10m/s?
late rose
Btw for a username suggestion, I would say "foxy" is a pretty decent name. It's ...
hmmm okay
violet spoke
look at this mfs pfp
supple trout
Can I know why the rationale of the minimum speed being 10m/s?
It's the lowest speed the drone stays stable & efficient in forward flight.
supple trout
look at this mfs pfp
Mine?
violet spoke
do NOT ❌ ❌ let him give you username suggestions
supple trout
do NOT ❌ ❌ let him give you username suggestions
Why?
late rose
do NOT ❌ ❌ let him give you username suggestions
please, if you have nothing meaningful to add, don't say anything
violet spoke
I'm sorry
vital crag
look at this mfs pfp
!topic
final saddleBOT
look at this mfs pfp
Please read the channel description before posting, and stay on topic.
supple trout
please, if you have nothing meaningful to add, don't say anything
Btw, should I rephrase my problem in a more mathematical way if that could help?
late rose
Btw, should I rephrase my problem in a more mathematical way if that could help?
It's fine, i can understand it perfectly well
supple trout
It's fine, i can understand it perfectly well
Very well, then. Shall we continue?
late rose
It's the lowest speed the drone stays stable & efficient in forward flight.
Alright so right now you have the fact that the drone flying with the theoretically lowest power consumption already takes too much energy. I would assume that this alone, unless rectified with a more efficient motor etc, would make it unviable for usage.
supple trout
Alright so right now you have the fact that the drone flying with the theoretica...
Yeah, that’s exactly the issue. Even at its most efficient speed, the drone still eats up a lot of power. Without better motors, lighter materials, or some way to cut energy use, it’s hard to make this work for real deliveries, the battery just won’t last long enough to be practical.
late rose
Nevertheless, some data would be useful. In particular, I believe you should see how some factors affect the drone's flight time, which I believe is the primary issue here. In particular, gathering data on
- Effect of drone speed on flight time
- Effect of drone height on flight time
- Effect of package mass on flight time
- Effect of battery mass on flight time
would be useful
frail moat
Nevertheless, some data would be useful. In particular, I believe you should see...
aerodynamics would be factor but we moving into more physics
late rose
aerodynamics would be factor but we moving into more physics
Yes that too.
But I think a drone itself would already be aerodynamic by design.
supple trout
Nevertheless, some data would be useful. In particular, I believe you should see...
Oh okay.
I have another problem tho. It's a different one:
I’ve got this weird problem at my school. So, I’m volunteering at this charity, and they’ve got this huge storage room full of boxes with coins. Some boxes are near the front, some are in the back, and there are shelves stacked high.
Here’s the thing: the amount of money in each box isn’t the same. The front ones usually have like $100-ish, the back ones have a bit more because people dump bigger donations there, and the higher shelves get even more coins for some reason.
And I was thinking how much money is actually in this room? Like, if I were to somehow add up every single box and all the coins inside, what would the total be? I don’t want to open every box, obviously, but I need some way to figure out the total without just guessing.
Basically, I need to somehow account for all the boxes, in all the corners and up on the shelves, and come up with the full total.
late rose
I have another problem tho. It's a different one:
I’ve got this weird problem ...
Assuming all the boxes and shelves are of the same type, and you were sufficiently strong, you could weigh the mass of each box and shelf in the room, and subtract that with the mass of an empty box or shelf in order to arrive at the mass of coins in the room.
Then, divide that mass by the mass of a single coin. This should give you an estimate --- not precisely exact --- of the total number of coins in the room.
supple trout
Assuming all the boxes and shelves are of the same type, and you were sufficient...
Ye, but this is time consuming & labor intensive. My team would probably not agree to do this since the shelves are stacked high, & the storage room is huge.
late rose
Ye, but this is time consuming & labor intensive. My team would probably not agr...
Are the boxes arranged neatly in rows and such?
supple trout
Are the boxes arranged neatly in rows and such?
No, boxes are stacked at different heights, some on the floor, some on shelves, and even some in corners that are hard to reach.
late rose
No, boxes are stacked at different heights, some on the floor, some on shelves, ...
Is it possible for you to estimate, perhaps off memory, how many boxes there are?
supple trout
Is it possible for you to estimate, perhaps off memory, how many boxes there are...
No, I really can’t estimate, because the boxes are all over the place, stacked differently, in corners, on different-sized shelves, it’s too irregular to just guess.
late rose
No, I really can’t estimate, because the boxes are all over the place, stacked d...
alright so maybe the solution I might recommend here is to identify different 'groups' or 'clusters' of boxes, take three from the group and get the average of the coins inside these (using the weighing tactic earlier) and assume that every box in that cluster has the same mass as that average. Repeat this for every cluster you identify. Shelves and corners would count as their own clusters, for example.
supple trout
alright so maybe the solution I might recommend here is to identify different 'g...
Yeah, I see what you mean, that could work for a rough estimate if the clusters are well-defined. The thing is, though, the boxes aren’t really uniform even within a cluster. Some corners are tight, some shelves have gaps, and the height and placement vary a lot.
late rose
Yeah, I see what you mean, that could work for a rough estimate if the clusters ...
I don't think there is any other non-labour-intensive way.
supple trout
I don't think there is any other non-labour-intensive way.
Yeah, that’s what I was thinking too. The thing is, since the boxes are all over the place, different heights, shelves, corners, gaps, any method that actually accounts for all of that without physically checking each box would have to treat the room as a sort of continuous space and sum everything up somehow. Or maybe somethin else Idk.
late rose
Yeah, that’s what I was thinking too. The thing is, since the boxes are all over...
Yes, and any method which does that would lose enormous amounts of precision. The method I suggested, would work even if boxes were placed at different heights along a shelf or if there were stacks of boxes at corners etc. If you want more accuracy, you'll need to measure more boxes.
supple trout
Yes, and any method which does that would lose enormous amounts of precision. Th...
I see, what I’d do then is I’d try to make it realistic, like front boxes have less money, back boxes have more, higher shelves have more, and corners are stacked differently. I’d split the room into zones, floors, shelves, corners, and treat each one separately. It still won’t be perfect, but at least it gives a solid estimate without having to weigh every single box
late rose
I see, what I’d do then is I’d try to make it realistic, like front boxes have l...
Yes that is literally what I said just now. Take the average for three boxes hailing from a single cluster and assume every box belonging to that cluster has that same average.
supple trout
Yes that is literally what I said just now. Take the average for three boxes hai...
ok. Imma have to go eat rn.
Please don't close it I'm not done.
final saddleBOT
@supple trout Has your question been resolved?
late rose
ok. Imma have to go eat rn.
The channel will close by itself given sufficient time.
final saddleBOT
supple trout
.reopen
final saddleBOT
✅ Original question: #help-36 message
supple trout
The channel will close by itself given sufficient time.
I reopened it.
So, I saw something called "Volume integrals" & they seem that they help solve problems like mine. So I would like to know if it could actually help?
late rose
i dont really see how
supple trout
Ok, I see.
I have a few other problems.
Every morning I have to deliver bread from my bakery to three different cafes. Each cafe orders a different amount, one wants 50 loaves, another 30, and the last 40. My van can only carry 100 loaves at a time, so I can’t just load everything in one trip.
The roads aren’t simple either. Some are faster but really narrow, others take longer but are easier to drive on. Fuel costs money, and I want to spend as little as possible. On top of that, all the cafes need their bread before 9 AM, or they’ll start complaining.
So every morning, I have to figure out how many loaves to load, which cafes to visit first, and which roads to take so that:
- Every cafe gets all their bread.
- I don’t overload the van.
- I spend the least fuel and time possible.
- I finish all deliveries before 9 AM.
late rose
Every morning I have to deliver bread from my bakery to three different cafes. E...
If you want to spend the least fuel and time possible, you're going to have to be a good driver and willing to drive on narrower roads.
That said, I would suggest routing every possible road between the supply source and cafes, along with each roads' time to travel.
Without that information, I can't really help you plan an optimal route.
It also seems like fuel grows proportionally with time, so minimising time should really be the concern.
bold turtle
You can grow fuel with time? /s
But yeah, all of what the fox hath spoken
This otherwise feels like a graph theory networking problem, which I'm not really sure anyone can help further on unless you somehow have a graph
supple trout
Without that information, I can't really help you plan an optimal route.
Ok. For the first cafe, there’s a main road that takes about 15 min and a smaller side street that’s faster at 10 min but narrower. The second cafe can be reached via a riverside road taking 20 min or a busy market lane that’s 12 min but slow because of traffic. The third cafe has a hilltop route of 18 min or an older street that takes 25 min but is easier to drive.
supple trout
This otherwise feels like a graph theory networking problem, which I'm not reall...
Well, how do I do get this graph O wise doodleman?
bold turtle
...I guess the addendum to that would first be to learn what graph theory is
[which for the record, does not refer to "graphs" as in the graph of a function, but to the discrete maths concept of a stucture made with vertices and edges]
supple trout
...I guess the addendum to that would first be to learn what _graph theory_ is
Where do I learn graph theory?
late rose
Well, how do I do get this graph O wise doodleman?
A graph in the mathematical sense is a set of nodes (also called vertices), and connections between nodes (called edges).
In your case, the nodes are the cafes and the supply source. The edges are the roads between nodes.
Each edge can have weight (also called a cost) to them. This is just a value associated with each edge. In this case, the weight would be the travel time.
By representing a problem this way, we can apply graph algorithms to problem like this.
bold turtle
The Internet is an amazing invention sometimes
Conceptually, you can think of graphs like maps
supple trout
_The Internet is an amazing invention sometimes_
Indeed it is, What a wonderfully weird world the internet is.
supple trout
Conceptually, you can think of graphs like _maps_
Well, How do I solve my problem?
supple trout
A graph in the mathematical sense is a set of nodes (also called vertices), and ...
Well, How do I solve my problem?
bold turtle
By drawing a map
supple trout
_By drawing a map_
Well, How so, my good sir?
bold turtle
-# Are you kidding me.
This is something we can't do for you
We don't have the prerequisite information beyond what you've provided, and fundamentally it's not really much of anything to provide a map, let alone a graph, sufficient enough for the problem at hand
supple trout
-# Are you kidding me.
Wait, How'd you write this?
bold turtle
No.
late rose
Well, How do I solve my problem?
It's a bit complicated... this is a form of the Travelling Salesman Problem which probably involves stuff that I don't think I could explain well, and I don't think you could understand very well.
The first step, which I can tell though, is to map every road between the cafes and supply source, along with information about those roads travel time and (i forgot to mention this earlier) whether or not they are bidirectional/what direction they travel in.
supple trout
We don't have the prerequisite information beyond what you've provided, and fund...
What are the information you need? everything else, what are you saying?
supple trout
It's a bit complicated... this is a form of the Travelling Salesman Problem whic...
What about lagrange multipliers?
late rose
Ok. For the first cafe, there’s a main road that takes about 15 min and a smalle...
Like what you said here wasn't bad, but you need to be more comprehensive (are you sure those are all roads between cafes and the supply source?) and specific. You could help by representing your map in an adjacency list.
supple trout
Like what you said here wasn't bad, but you need to be more comprehensive (are y...
What???
late rose
What about lagrange multipliers?
not really useful here
supple trout
not really useful here
oh okay. What about optimization?
bold turtle
Not even practical here, since there're no functions here yet to even consider optimising
This might be an optimisation question, but since this is firstly a question about graphs, well, you need a graph
supple trout
This might _be_ an optimisation question, but since this is firstly a question a...
I don't even know how to make a graph.
late rose
_The Internet is an amazing invention sometimes_
^
bold turtle
I also have the nagging suspicion the level of maths involved is (academically speaking) far higher than your current understanding
late rose
It's a bit complicated... this is a form of the Travelling Salesman Problem whic...
that is basically what i said here
supple trout
that is basically what i said here
Where do I map all of those?
late rose
Where do I map all of those?
you'll need to get out your stopwatch and local map.
supple trout
you'll need to get out your stopwatch and local map.
Ok, Ok.
For how much time have we been talking together?
late rose
For how much time have we been talking together?
You can check that yourself
supple trout
You can check that yourself
Ok your the message was sent at 8:22 p.m. while the first message was sent 5:54 p.m. We've been talking for 2 hours straight.
Our conversation really was intriguing I guess.
late rose
Ok, Ok.
Once you've gathered the data for the travel times between nodes, you'll want to represent that data in an adjacency list.
late rose
Our conversation really was intriguing I guess.
It was.
supple trout
It was.
Well, quite happy to have had this conversation with you, Foxy.
I hope we'll meet again.
.close
exotic sandal
final saddleBOT
exotic sandal
can someone help explain this to me? im not quite sure what its asking for in terms of partial fractions
drowsy epoch
do you know pfd at all
wraith glen
partial fraction differentiation
drowsy epoch
decomposition
wraith glen
oh shit
drowsy epoch
differentiation 
exotic sandal
i feel like i have learned it, but im blanking on it rn
green pebble
maybe look over it again then try the integral
green pebble
show your working
tulip coyote
(taking the partial fractions that you found as correct, which they may or may not be, note the mention of the absolute value too)
tulip coyote
Likely -4 - 3?
exotic sandal
what -3
bold turtle
show your working
[If it helps, we have a bot command to show a factoid for this]
!show
final saddleBOT
Show your work, and if possible, explain where you are stuck.
bold turtle
(I'd presume so too, but since we've neither the answer's working out nor OP's, I can't be sure)
tulip coyote
(taking the partial fractions that you found as correct, which they may or may n...
I agree with what "A" and "B" would be, implied from the values, so I strongly think it may be because of the lack of absolute values(?)
tulip coyote
i dont see how i worked this out
And I'd suggest trying again, maybe if you'd like, you can walk through it here and we'll guide you if needed?
exotic sandal
step one: look at question:
step two: ?
bold turtle
I agree with what "A" and "B" would be, implied from the values, so I strongly t...
The hint strongly suggests this
bold turtle
step one: look at question:
step two: ?
Step three: show your working out
bold turtle
So...
tulip coyote
After all, you somehow got 7/4 and -3/4, didn't you - how?
bold turtle
!status
final saddleBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
You were status 1, then?
(And simultaneously you had an answer?)
At least one of these assumptions I'm making is wrong
At best, you're being dishonest with yourself somewhere
exotic sandal
i am going back to stuff i have already done before, i didnt get the answer written there this week
bold turtle
Ah, then this needed to be stated from the get-go
exotic sandal
if your help is gona boil down to "well you have done it already so clearly you already know" just tell me so i can just go look for help elsewhere
bold turtle
No, if you'd have let me finish...
...I was about to explain that you're more likely not sure how to actually find partial fractions
tulip coyote
if your help is gona boil down to "well you have done it already so clearly you ...
It isn't, the point is we want to know where you are at, and what exactly we should be guiding you with, so that we don't either go over stuff you already know, nor do we go too far ahead and you're sat there like 
bold turtle
-# Jeez, you have cat emojis for every possible scenario I swear 
Regarding the hint it gives you here, do you know how to get A and B here?
(Or alternatively why we're even doing decomposition in the first place)
exotic sandal
.close
blazing nymph
could I please have some help trying to decipher this
final saddleBOT
could I please have some help trying to decipher this
blazing nymph
So we let there be 2 left closets then just define a map from one to the other
And if there exists a bijection then they are the same size
But I don’t see the bijection
OHH
WAIT
Wait I get it
😭
Sorry for taking a channel
.close
final saddleBOT
final saddleBOT
Available help channel!
tranquil pine
what are you actually doing when you integrate a vector valued function (visually)
final saddleBOT
what are you actually doing when you integrate a vector valued function (visuall...
@tranquil pine Has your question been resolved?
hybrid heath
If you integrate a velocity vector function, you get a position vector function
If you path integrate a force field, you get work done on that path
If you volume integrate an electric field, you get enclosed charge
What you get depends on what you are starting with
hybrid heath
It's not simple like area under a curve anymore
tranquil pine
if i sound clueless its cause i am
hybrid heath
It has become more abstract
tranquil pine
It's not simple like area under a curve anymore
so its just an antiderivative
and its not worth thinking of it like integration
?
formal trail
3blue1brown had a nice visualization in his new laplace transform video where it was vectorially adding up the average value of the function over a bunch of small intervals
scaled by the length of the interval ofc
final saddleBOT
formal trail
it was a complex-valued function really but it's the same as 2D vector integration really
and 3D is the same except being in 3D
final saddleBOT
Available help channel!
wicked horizon
Can someone tell me if this is correct
final saddleBOT
Can someone tell me if this is correct
wicked horizon
primal sparrow
You may want to check C and D again 🙂
wicked horizon
ok
hybrid heath
,rccw
soft zealotBOT
primal sparrow
I would presume it is 5
wicked horizon
I would presume it is 5
ok
primal sparrow
Ah yeah lol
mint jetty
yes
wicked horizon
Would it be y=5, x=9?
primal sparrow
Yep
final saddleBOT
@wicked horizon Has your question been resolved?
sonic crescent
how do i solve this problem?
final saddleBOT
how do i solve this problem?
pliant shore
how do i solve this problem?
just look at the meandiff column
but be careful, some of the rows have 'reject = False'
(as in, reject the null hypothesis)
dapper cloak
please make sure this is right please i go on 2 kneees
final saddleBOT
please make sure this is right please i go on 2 kneees
dapper cloak
please make sure this is right please i go on 2 kneees
yah @frail moat check my hw it's right correct?
frail moat
All
tranquil pine
hi
tranquil pine
i really need some help understanding something right now
regarding linear algebra
ill post the question but i really need someone to help me understand
ill wait for someone to come so i can explain my concerns exactly
cause it needs to be a back n forth dialogue
<@&286206848099549185>
final saddleBOT
Please do not ping individual helpers unprompted.
frail moat
Lmao imean wait a bit
storm haven
ill wait for someone to come so i can explain my concerns exactly
Just explain your first concern
tranquil pine
Just explain your first concern
okay so you know how the null space is the space orthogonal to the row space?
yea?
tranquil pine
Yes
okay so know how in order to get there
we take the row space vectors
Basically they act as normal vectors to the solution vector
which creates a plane
planes in this case which are orthogonal to each row space
right?
Now those planes
Their intersection is the orthogonal null space to the row space
what im trying to understand is the row operations you do on the plane equations, and how they give you the solution which is the intersection, like the geographical aspect of that
hollow niche
Ig if u want a basis for the solution space then you'd start with reaching rref
And then solve for the pivot variables in terms of the free variables. Then write the solutions in terms of some vectors and then those vectors would be your basis for the null space
tranquil pine
And then solve for the pivot variables in terms of the free variables. Then writ...
i know bro but im trying to understand how that works geometrically
tranquil pine
like how does applying RREF give me the solution , which is also a hyper plane in this case , which is also a space spanned by 2 independent hyper vectors, which is also a linear combination of those 2 vectors
like
tranquil pine
Yeah idk, I wasnt taught it with geometry in mind
ur good
ive been stuck on this for a long time i just want someone to explain it to me
storm haven
what im trying to understand is the row operations you do on the plane equations...
Could you rephrase this all
I don't get what you are saying
tranquil pine
Could you rephrase this all
ngl this feels better to say on call
if anyone is willing to join up a call im down
theres a vc
in here
<@&286206848099549185> forget the vc can anyone help pls?
uncut geyser
<@&286206848099549185> forget the vc can anyone help pls?
do you already know how to row reduce a matrix?
tranquil pine
do you already know how to row reduce a matrix?
yes
algebriaclly
i just don’t understand the geometric intuition
specifically here
in this question
uncut geyser
i just don’t understand the geometric intuition
okay so a homogeneous system like this (Ax⃗ = 0⃗) asks "which vectors x⃗ get mapped to the 0⃗ vector by the matrix A?"
these vectors form the null space of the matrix A
geometrically, the null space is a subspace of ℝⁿ (here, ℝ⁴)
tranquil pine
geometrically, the null space is a subspace of ℝⁿ (here, ℝ⁴)
yes but
this is so frustrating I wish i can explain in a vc
uncut geyser
yes but
dw just answer my questions it will make sense
can you picture what a subspace of ℝ⁴ might look like?
for example, what would a 1d or 2d subspace mean in that context?
tranquil pine
can you picture what a subspace of ℝ⁴ might look like?
yes
a cube or under
can you vc? If not its fine
uncut geyser
can you vc? If not its fine
kinda late for me sorry
tranquil pine
its ok
uncut geyser
but hopefully my explanation will make sense
tranquil pine
im confused on the part where you solve for the null space via the generated planes
like beginning at when you start doing elimination, to the result you get
uncut geyser
yes
good so in ℝ⁴, the full space is 4d (like a hypercube), but the solution space to a homogeneous system is a flat subspace inside that 4d space
tranquil pine
good so in ℝ⁴, the full space is 4d (like a hypercube), but the solution space t...
its a plane
uncut geyser
it could be:
- a line (1d)
- a plane (2d)
- a 3d hyperplane
- or just the zero vector (0d), if the only solution is trivial
tranquil pine
it could be:
\- a line (1d)
\- a plane (2d)
\- a 3d hyperplane
\- or just the ze...
here a plane right
uncut geyser
its a plane
yes but let's work from the bottom up for your geometric intuition
tranquil pine
ok
uncut geyser
the key idea is that the number of free variables = dimension of the solution space
when you solve the system algebraically, you'll reduce the matrix and find how many variables are free (a.k.a. not leading)... that number tells you the dimension of the solution space
so now let's predict, the system has 4 variables and 3 equations, "what's the maximum possible dimension the solution space could have?"
a plane, which is a 2d subspace, so there we are now
now let's row reduce the matrix
from the coefficient matrix, you gotta row reduce this to row echelon form, i would do the following:
- use row 2 as the pivot row (it's already simplified lol)
- eliminate row 1 and row 3 using row 2
tranquil pine
from the coefficient matrix, you gotta row reduce this to row echelon form, i wo...
yeah i get the algebra and the general rule for it but just not how it works geometrically
tranquil pine
so now let's predict, the system has 4 variables and 3 equations, "what's the ma...
For example youre saying 4 variables 3 equations but in my mind idk why this is the case
whats the correlation n stuff
uncut geyser
yeah i get the algebra and the general rule for it but just not how it works geo...
alrighty wait we'll get there but let's make sure you got the proper matrix
soft zealotBOT
∫_M dω = ∫_∂M ω
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uncut geyser
should be something like this
tranquil pine
okay let me explain how i see this tho
so you agree that this comes from the system
The system is planes
Each plane is orthogonal to its normal vector, its normal vector is a row vector
uncut geyser
okay let me explain how i see this tho
wait bro
uncut geyser
**∫\_M dω = ∫\_∂M ω**
Compile Error! Click the <:errors:504266556054962196> reac...
this tells us x_1 and x_3 are leading variables and that x_2 and x_4 are free variables
since there are 2 free variables, the solution space is 2-dimensional, a plane in ℝ⁴
that means every solution to this system lies on a flat 2d surface inside 4d space
you can think of it as a sheet floating through the 4d world, defined by two independent directions ("basis vectors")
you can find the basis by expressing the general solution in terms of the free variables, so you'll end up with two vectors, those are the directions that span the plane
tranquil pine
since there are 2 free variables, the solution space is 2-dimensional, a plane i...
well the thing is i know the solution space is 2D but not from this, matter of fact i dont even know why this is the case
tranquil pine
so you agree that this comes from the system
The system is planes
Each plan...
thats why i sent this
uncut geyser
well the thing is i know the solution space is 2D but not from this, matter of f...
ah ic lemme give a visual metaphor
tranquil pine
bc i know its 2D but from a different perspective
and basically im trying to see if my visualization is on the right track
uncut geyser
the solution space is 2d because there are 2 directions you can freely move in while still satisfying all the equations
imagine you're standing in a 4d room (pretend you can visualize it)
the equations are like walls, each one restricts your movement
but they don’t block you completely... instead, they slice the space, narrowing down where you can go
tranquil pine
unfortunately i can’t understand what ur saying i wish i can get on vc so bad
because in my head there’s something specific
and i want you to understand what it is
so then maybe we can connect your analogy with it and then i would understand it
uncut geyser
but they don’t block you completely... instead, they slice the space, narrowing ...
each equation reduces the number of directions you can move in, so:
- start with 4 dimensions (from x_1, x_2, x_3, x_4)
- each independent equation removes one degree of freedom
- if 2 equations are independent (and the third is a combo of them), you lose 2 dimensions
- that leaves you with 2 dimensions of freedom, a plane
uncut geyser
so then maybe we can connect your analogy with it and then i would understand it
i'll keep trying to explain i think you'll get it
but does this idea of "each equation slicing away a direction" help you sorta see why the solution space ends up being 2d?
or do i still need to elucidate something?
then i can show you how to find the basis vectors
tranquil pine
bro the thing is i know why the subspace is a plane and i can imagine it 100% but not from what you’re saying exactly
uncut geyser
bro the thing is i know why the subspace is a plane and i can imagine it 100% bu...
ah ic, you gotta think in terms of degrees of freedom tho
you start with 4 variables: x_1, x_2, x_3, x_4
tranquil pine
okay
uncut geyser
that's like having 4 knobs you can turn to make any vector in ℝ⁴
now, the system gives you 3 equations
but those equations are not independent, one is a linear combo of the others
tranquil pine
wait so I know like if the planes are all independent youre gonna have 1 exact solution right
uncut geyser
so really, only 2 of them are doing work
uncut geyser
so really, only 2 of them are doing work
each independent equation ties some of the knobs together... it says "if you turn this one, you have to turn that one too"
so instead of 4 free knobs, you're left with 2 knobs you can still turn freely
that's your 2d solution space, every solution is a combo of turning those 2 knobs
uncut geyser
wait so I know like if the planes are all independent youre gonna have 1 exact s...
yup
tranquil pine
yup
But like
when we talk about linear independence
know how this applies to vectors
and I noticed it applies to lines
and planes
Now is linear independence inherent to one object of those
or is it just a characteristic that we can apply on things if they are compatible
uncut geyser
Now is linear independence inherent to one object of those
it's not a property of one object (like a single vector, line, or plane), but rather a relationship among multiple objects
a single vector can't be called "independent" or "dependent", it just is
but a set of vectors can be called linearly independent if none of them can be written as a combination of the others
same goes for lines and planes
linear independence is a characteristic we apply to sets of objects when they're compatible, meaning they live in the same space and can be compared
tranquil pine
linear independence is a characteristic we apply to sets of objects when they're...
I see
so 2 planes independent = line solution
3 planes independent = point
okay i get this
but like
im talkin about when you start eleminating
via RREF
which is the same as elemination method
literally
so
During that process
what happens that gives you that intersecting space
whats the magic behind the elemination part
uncut geyser
whats the magic behind the elemination part
when you do row operations, you're rebuilding the system to expose its structure
each row in your matrix is a vector in ℝ⁴, so when you eliminate variables, you're transforming these vectors to make their relationships visible
if one row becomes all zeros, it means that equation was dependent, it didn't add new restrictions
if a row has a pivot (leading 1), it's an active constraint, it's slicing away a dimension
bound variables are tied down by pivots, so they're determined by the system
free variables are untouched, they can vary freely, and they span the solution space
essentially, the two things you're doing are:
- aligning the equations to show dependencies
- revealing which variables are bound and which are free
elimination transforms the system into a form where the geometry is obvious
you can see how many dimensions are left, and which directions the solution space stretches in
@tranquil pine does that answer your question?
tranquil pine
<@456226577798135808> does that answer your question?
ill read it
might take a minute to digest
final saddleBOT
@tranquil pine Has your question been resolved?
tranquil pine
each row in your matrix is a vector in ℝ⁴, so when you eliminate variables, you'...
each row is a vector or each row is a plane??
wait wait
are we taking the normal vectors to the pane equations and visualizing them?
i still don’t understand the 0ing part too
final saddleBOT
tranquil pine
ugh
tranquil pine
i'm lost in calculations
Consider the function f defined on [0,1] by (by the way (x,y) \in \mathbb{R}^2 and 0<x<y)
$t \rightarrow f(t) = \ln (tx +(1-t)y) - t ln(x) - (1-t)ln(y)$
By Studying f deduce that for all t $\in ]0,1[$ :
$t \ln(x) +(1-t)ln(y) \le \ln (tx + (1-t)y)$
Interpret the result geometrically
soft zealotBOT
Drk
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final saddleBOT
@tranquil pine Has your question been resolved?
granite tapir
This seems to be AM-GM in disguise, so if you use a,b,½ and solve for f(½) maybe that's AM-GM and proof of the exercise?
√ab ≤ (a+b)/2
tranquil pine
This seems to be AM-GM in disguise, so if you use `a,b,½` and solve for `f(½)` ...
but i need to proveit for every t
in 0,1
not just 1/2?
also i'm not really sure how i would the the am-gm inequality?
granite tapir
also i'm not really sure how i would the the am-gm inequality?
f(t) = ln(t·x + (1-t)·y) − t·ln(x) − (1-t)·ln(y), t ∈ [0,1]
t = ½: f(½) = ln(½·x + ½·y) − ½·ln(x) − ½·ln(y).
½·ln(x) + ½·ln(y) = ln(x^½·y^½) = ln(√(xy))
f(½) = ln((x+y)/2) − ln(√(xy)) = ln((x+y)/2 ÷ √(xy))
f(½) ≥ 0 ⟹ (x+y)/2 ≥ √(xy)
a=x, b=y
(a+b)/2 ≥ √ab
tranquil pine
f(t) = ln(t·x + (1-t)·y) − t·ln(x) − (1-t)·ln(y), t ∈ \[0,1]
t = ½: f(½) = ln(½·...
ohhh
that's neat
but that one value of t
granite tapir
whole halo
unfortunately no, this is not what they are going for
for one you can choose any other value and then the geometric root isnt very rooty anymore
granite tapir
Oh okay, sorry then, it doesn't proof it?
whole halo
its not the intuition they asked
I remember you dont seem to draw a line to indicate a difference between what something wants to say and what you can get out of it
whatever intuition they are asking for, it must work for any value for t
the arithmetic average can still be seen as a weighted sum but the geometric average loses meaning if you try out the same intuition as in t = 1/2
there is something simpler involved here
first
where have you generally seen t A + (1 - t) B from?
tranquil pine
where have you generally seen t A + (1 - t) B from?
linear combinations of something
whole halo
@tranquil pine hello?