well studying the solutions of a_1v_1+...+a_nv_n-b_1w_1-....-b_mw_m=0 also involves row reduction
#help-0
mortal trellis
hmm that feels like it should have a decently nice solution
toxic ferry
I've reached the conclusion that:
(a1 - b1)u1 + ... + (ak - bk)uk = w(a1 + ... + ak)
And... I'm not entirely sure what to make of it
mortal trellis
well the left is an element of U
so the element on the right also has to be
but w is not in U so the only possible option is that the RHS is 0
toxic ferry
RHS?
mortal trellis
right-hand side
toxic ferry
Ah, gotcha'.
but that would mean that (an - bn) = 0 no?
So... 0 = 0.
I'm still not sure what exists at the intersection if I'm honest ^^;
mortal trellis
hmm so we need a_i=b_i and sum(a_i)=0
toxic ferry
Oh sorry - I've mentioned earlier that dim(W) = k
I don't know that just yet. and I'm not entirely sure how to explain it either.
mortal trellis
I don't think we actually need it atm
toxic ferry
Maybe, haven't thought this through entirely
mortal trellis
from sum(a_i)=0 we can find the vectors that are in the intersection
toxic ferry
Oh sorry - I've mentioned earlier that dim(W) = `k`
I don't know that just yet. ...
I went about proving it, as expected it is k
mortal trellis
spanned by the vectors {u1-u2, u1-u3, ..., u1-uk} I think
toxic ferry
spanned by the vectors {u1-u2, u1-u3, ..., u1-uk} I think
u1? Are you sure?
Wait, if there's a way to prove that w belongs to U + W, that does it.
toxic ferry
Oh, it didn't occur to me haha
mortal trellis
w = u_1 - (u_1-w)
mortal trellis
so yeah the intersection is k-1 dimensional and spanned by {u1-u2, ..., u1-uk}. clearly all of those are in U and u1-ui = u1-w-(ui-w) so they are also in W
and intersection can't be k dimensional because U+W is at least k+1 dimensional
toxic ferry
so yeah the intersection is k-1 dimensional and spanned by {u1-u2, ..., u1-uk}. ...
You basically took both bases and subtracted one from the other?
Could you explain the process again?
mortal trellis
I need that a1+...+ak=0
that is satisfied by the vectors which are 1 at a position and -1 at a different position
here I chose (1, -1, 0,...,0), (1, 0, -1, 0,...,0) etc
which then turn into u1-u2, u1-u3 etc
toxic ferry
Sheesh.. would've never thought of that honestly
mortal trellis
next time you will
toxic ferry
next time you will
Could you examine a solution I came up with and see if it makes sense really quick? It's pretty short
I'm trying to figure out if my intuition is.. there.
mortal trellis
sure
toxic ferry
Basically:
U + W = {(a1 + b1)u1 + ... + (ak + bk)uk - w(b1 + ... + bk) = 0 | an, bn belong to R}
So:
(a1 + b1)u1 + ... + (ak + bk)uk = w(b1 + ... + bk)
Wouldn't that necessarily prove that w belongs to U + W?
mortal trellis
no
U + W = {(a1 + b1)u1 + ... + (ak + bk)uk - w(b1 + ... + bk) | an, bn belong to R}
without the =0
toxic ferry
Oh, right.
mortal trellis
and I don't know how you want to get that w is in U+W from that
toxic ferry
Wait.. so uh
(a1 + b1)u1 + ... + (ak + bk)uk - w(b1 + ... + bk) = w
mortal trellis
(a1 + b1)u1 + ... + (ak + bk)uk = w(b1 + ... + bk)as mentioned earlier can only be satisfied when b1+...+bk=0 as otherwise w would have to be in U
toxic ferry
(a1 + b1)u1 + ... + (ak + bk)uk = - w(b1 + ... + bk + 1)... Yeah idk where I'm going with this
You're right, shucks
mortal trellis
what you want to do is choose a1=b1=1 and ai=bi=0 for i >= 2
mortal trellis
w = u_1 - (u_1-w)
which is just this
toxic ferry
Wait what?
mortal trellis
sorry a1=1, b1=-1
mortal trellis
`U + W = {(a1 + b1)u1 + ... + (ak + bk)uk - w(b1 + ... + bk) | an, bn belong to ...
(here)
toxic ferry
I'll have to think this through for a little, bit of a hard concept to grasp I guess
Tysm for the help though!
(I think I get what you did btw, don't think your efforts were for naught ^^;)
You just checked for which values of an and bn the equation would make sense for w does not belong to U and LHS = w.
That's the best way I can sum this up I think
mortal trellis
more LHS is in U compared to RHS is in span(w), so the vector x=LHS=RHS has to be in the intersection of U and span(w) which is only {0} as w is not in U
that type of argument comes up a few times
arguing that the LHS is in some span and the RHS is in some other span, so the vector has to be in the intersection
and often the intersection is {0} so you get some nice conditions
inner verge
help
lone heartBOT
mortal trellis
you're welcome 🙂
lone heartBOT
Available help channel!
lament pilot
lament pilot
<@&286206848099549185>
lone heartBOT
@lament pilot Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
topaz palm
lone heartBOT
topaz palm
someone explain the steps
elder vine
.rotate
topaz palm
quick
elder vine
Add ten to both sides then divide both sides by 3
elder vine
Don't rotate
topaz palm
oh ok
elder vine
Just add 10 to both sides then divide by 2
topaz palm
oh ok
elder vine
.rotate
elder vine
Is it back to normal now
topaz palm
it’s upside down
elder vine
rotate,
topaz palm
,rotate
ocean sealBOT
topaz palm
elder vine
I see
topaz palm
ik u do
elder vine
It literally tells you what to do
topaz palm
ik it does
elder vine
Then why do you need help
topaz palm
cause idk what to do
elder vine
What year are you in
topaz palm
i’m homeschooled
topaz palm
idk how that helps with my hw tho
elder vine
Isn't all work hw?
topaz palm
+10 then /3
hm
topaz palm
Isn't all work hw?
bro i need maths help not philosophical help lol
elder vine
You said you were homeschool
elder vine
So all work is homework
elder vine
What you want to do is get all the numbers on one side and all the x on the other
topaz palm
why did the guy/girl put brackets
topaz palm
What you want to do is get all the numbers on one side and all the x on the othe...
hm
but he added 10
to both sides
elder vine
Yes, to cancel out the minus ten
elder vine
Then you are left with 3x, so to find x just divide both sides by 3
topaz palm
why tho
elder vine
Because if three times a number equals 21, then one times a number is 21/3
elder vine
And you need one times a number
wanton nova
You want to find x, right? Not 3x
topaz palm
why do we want x alone
it didn’t tel me@what i need to find
it just gave me an equations
wanton nova
I tell you
elder vine
Well what else are you going to find
wanton nova
You need x
topaz palm
why do i have to do what u tell me
elder vine
Use your initiative
topaz palm
Well what else are you going to find
idk
wanton nova
Bro
topaz palm
that’s why i don’t understand the question
elder vine
Find out the unknown variable
topaz palm
oh ok
elder vine
In this case x
elder vine
No
elder vine
If you saw 3z+5=4 solve to find z
elder vine
Just find out the letter
wanton nova
Cryptical
topaz palm
MasterOogway
wanton nova
Are you trolling?
elder vine
Mate it's not that difficult
topaz palm
3-3
elder vine
Once you get the hang of it, it'll be fine
topaz palm
idk
elder vine
$5x^2+lnx+5sin3x+4=0$
ocean sealBOT
TactfulMirror15
elder vine
What's x?
topaz palm
WOW
wanton nova
🤣🤣🤣
elder vine
Don't worry mate
topaz palm
why is there words now
elder vine
sin is trigonometric function on ln is natural logarithm
I have to go but good luck with maths
If you need any more help, don't hesitate to ask me
topaz palm
trigo who
topaz palm
natural when
wanton nova
How old are you?
topaz palm
I have to go but good luck with maths
thanks
elder vine
You won't need to worry about sin and ln until much later on
topaz palm
thanks
wanton nova
You won't need to worry about sin and ln until much later on
5th grade
elder vine
now find it
He's cooked you there pal
topaz palm
i guess u need a separate help channel too huh
elder vine
He sounds like he is the same age as limit as x approaches 0 of sinx/x
wanton nova
Fr
elder vine
See ya later bois
topaz palm
what’s that
elder vine
find it
topaz palm
😂🔥😂😂
wanton nova
I think
tacit arch
why is this channel so long and no one's helping
topaz palm
EXACTLY
alpine sable
Because it’s what you are trying to find
tacit arch
can you post the instructions to the problem?
topaz palm
can you post the instructions to the problem?
?
alpine sable
Because that’s the question
topaz palm
why
alpine sable
It’s like asking ‘why do I have to do homework’
topaz palm
u are just following orders
wanton nova
😑
tacit arch
u are just following orders
yes indeed. are you unable to?
topaz palm
why can’t we live freely
tacit arch
okay i see why this channel went on so long now
alpine sable
why can’t we live freely
You are welcome to close the channel and move on
tacit arch
can you not troll please
wanton nova
Thats ehat i told him
tacit arch
Thats ehat i told him
yea i understand now
topaz palm
that’s why i’m trying to learn
wanton nova
He sounds like he is the same age as limit as x approaches 0 of sinx/x
Do you agree with this?
topaz palm
why are you following the math logic
it’s not an order
topaz palm
Do you agree with this?
yh
wanton nova
Do you agree with this?
Riemann
topaz palm
yh
wanton nova
Thats what you did in that exercise
wanton nova
3x and x3 is the same
topaz palm
what does it mean in english
alpine sable
or x3
Well they are the same thing, but it’s just a convention to write the number first
topaz palm
Well they are the same thing, but it’s just a convention to write the number fir...
where are u coming from
alpine sable
where are u coming from
Rude…
wanton nova
3x is just 3×x
alpine sable
Yes
lone heartBOT
tacit arch
lol 3 lots of x
lone heartBOT
Available help channel!
topaz palm
3x and x3 is the same
according to oogway
lone heartBOT
according to oogway
lone heartBOT
abstract fractal
Don't do anything special with the M
Just distribute it
Like you would with normal numbers
lone heartBOT
@acoustic plover Has your question been resolved?
abstract fractal
After distribution and some other steps, yes
But not immediately after distribution
First, tell me what you get directly after distribution
How do you get that?
That's vague
It's correct, but i want you to understand it
Can you see that M(w + 2u) = Mw + M(2u) = Mw + 2Mu
tacit arch
Use definition of eigenvector
Also it's better to say I don't know rather than guessing
What is an eigenvector
alpine sable
Basically, an eigenvector is a vector which is only scaled by some constant by a transformation
tacit arch
Do you have a book that explains eigenvector
alpine sable
I have to recommend 3Blue1Brown
This playlist gives great understanding about Linear Algebra
Skip the first few videos because you probably know them, they just explain what a vector is and what linear transformations are
lone heartBOT
@acoustic plover Has your question been resolved?
sterile stone
lone heartBOT
sterile stone
so im learning polar coordinates
and this dude is showing how to graph but didn texplain how S is put there on the graph
so Im just asking why does S go to there on the graph
.close
lone heartBOT
worn fox
lone heartBOT
Available help channel!
abstract salmon
How would I go about proving that BC1 and DA1 are equal length? AC is the diameter. I know that CDE and ABE are similar. BCE and and ADE are similar.
lone heartBOT
How would I go about proving that BC1 and DA1 are equal length? AC is the diamet...
@abstract salmon Has your question been resolved?
abstract salmon
<@&286206848099549185>
waxen flame
Don't know the specific answer but I would start with the Intersecting Chords Theorem. A consequence of that theorem is that triangle AED is similar to triangle CBE.
abstract salmon
I already know that the triangles are similar. I don't know what to do after that.
waxen flame
Use the fact that there is a constant, c, factor difference between similar sides.
abstract salmon
No, still stuck
I was trying to get at the problem using the radius. If I can prove that FGE is an isoceles triangle the problem is done.
waxen flame
One moment, I'm making a graph.
So we know the two lengths are congruent regardless of the position of B and D.
abstract salmon
Okay. But how do we know?
waxen flame
I'm working on an addendum to that graph with the two triangles.
Might take a minute, I'm trying to wrap my head around the different orientation of the two triangles.
waxen flame
A few more minutes, almost got this worked out.
Refresh that page to see what I'm doing.
abstract salmon
This?
Still don't know how to prove this though. I can see that they are equal. But the question is to prove it using math, not geogebra.
waxen flame
Just laying the groundwork for how to solve it.
Fixed the orientation of the triangles.
elder vine
,rotate
ocean sealBOT
lone heartBOT
@abstract salmon Has your question been resolved?
little berry
Hi. I was just wondering why the rational algebraic expression $\frac{2x}{x+2}+\frac{x+2}{x-2}$ is equal to $\frac{3x^2+4}{(x+2)(x-2)}$ and not 3x+2?
ocean sealBOT
Shinutsi
little berry
My solution for 3x+2 is this:
gray isle
you violated fraction laws in your attempt at cancellation
just because an expression appears in the numerator and denominator doesn't my want you can "cancel" it
(x-2) isn't a common factor of the numerator and denominator,
same for (x+2)
little berry
I see. How would you know if you can cancel a factor or not?
gray isle
factorised/product form will explicitly indicate whether there's a common factor
little berry
oh. may I know why (x+2) and (x-2) aren't a common factor?
gray isle
the numerator isn't in the form
(x-2) * something
nor
(x+2) * something
lone heartBOT
lone heartBOT
Available help channel!
lone heartBOT
worn fox
but you're timesing it by 4
it would be -b^2/64 inside the 4 bracket
but they've pulled it outside
lost coral
idk what to do
lone heartBOT
idk what to do
lost coral
lone heartBOT
@lost coral Has your question been resolved?
harsh girder
yes, then you can solve a
worn fox
so m is the number of distinct eigenvalues?
you need to tell us what m represents otherwise how can we know..
its certainly true when m=n, but it can also be true when m < n
novel night
Example: the identity matrix has one eigenvalue... But it's pretty diagonal.
worn fox
the statement you wrote says "is only diagonalisable if m=n"
worn fox
Example: the identity matrix has one eigenvalue... But it's pretty diagonal.
this is an example of that statement being false
iron mulch
sushi is back with the matrices 😭
worn fox
what would it mean if v was an eigenvector of A^2?
come on sushi
the definition of an eigenvector
are you guessing?
i have just asked you for the definition of v being an eigenvector for A^2
you should know what the definition of an eigenvector is
then go and look it up!
you have notes
ive seen them
use google
anything
if you dont know the definition of something, dont just sit around waiting for someone to tell you, go and find out
okay then what does it mean for v to be an eigenvector of A?
worn fox
then go and look it up!
so you ignored me when i asked you to do this?
like come on sushi you gotta give me something
iron mulch
worn fox
you've spent all this time doing questions about eigenvalues, eigenvectors, eigenspaces, algebraic/geometric multiplicity and you're telling me you have no idea what an eigenvector is??
i guarantee you
in your notes
is the definition of an eigenvector
in those slides you have
it will be there
and youll have seen it
it tells you exactly what an eigenvalue and eigenvector is
in the definition
mess around with A^2(v) = A(A(v)) given that v is an eigenvector of A
im not going to tell you the answer
until you actually do the question and/or do what im asking you to do
lone heartBOT
@acoustic plover Has your question been resolved?
worn fox
you have no idea?
like literally zero idea?
at all?
no idea?
stop thinking about the question if you don't know the fundamental things that the question is asking about
what is an eigenvector?
right a vector that gets scaled up when a matrix is applied to it
so if v is an eigenvector for A
worn fox
tacit arch
Do you know how to multiply a matrix with a vector symbolically?
.
That's the definition of what's happening when matrices multiply
you keep trying to skip understanding concepts. should stop that
tacit arch
https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/multiplication-of-...
write down using this notation what it means for
$Av = \lambda v$
ocean sealBOT
riemann
tacit arch
do you have access to a writing utensil and paper @acoustic plover ?
real gazelle
If you don't prove if it's true or not, then you can't figure out if it's true or not
you need to either prove it or find a counterexample
tacit arch
if you don't want to do the most basic thing of writing stuff down, just say so
real gazelle
yeah we can't help you unless you also put in enough effort for you to understand the problem
do you understand the definition for what an eigenvector is?
tacit arch
https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/multiplication-of-...
this this equation exactly on paper
real gazelle
wait you don't need that equation
the proof is very simple
we're trying to answer, if v is an eigenvector of A, is v also an eigenvector of A^2, right?
worn fox
how can you think this AAAAAAA
tacit arch
how can you think this AAAAAAA
who are you, ann?
tacit arch
we're trying to answer, if v is an eigenvector of A, is v also an eigenvector of...
don't you think it's important for her to know what an eigenvector is?
worn fox
im on the slippery slope down the helper pipline
real gazelle
don't you think it's important for her to know what an eigenvector is?
well yeah but how does multiplying two 2x2 matrices help w that lol
worn fox
do you honestly think that, when trying to decide whether a statement about eigenvectors is true or not, the definition of an eigenvector is not going to be useful or relevant?
tacit arch
well yeah but how does multiplying two 2x2 matrices help w that lol
I said use it as notation to multiply a matrix with a vector
real gazelle
@worn fox it's alright we can tagteam, I'll jump in
worn fox
okay but if you dont even know what it is, how can you possibly know its "relations to other things"?
real gazelle
and when I rage quit, you can come back
real gazelle
😭
worn fox
ill jump back in when i wake up
real gazelle
LOL
okay @acoustic plover I know the other helper already answered this for you, but what does it mean if some vector (let's call it v) is an eigenvector of A^2?
sushi are you still here
okay well
if they don't respond, I'll just close the thread lol
I shall set a 10 minute timer for them to come back
aight
awesome
so what does it mean for v to be an eigenvector for A^2
can you write it down as an equation
yep!
well
not 0
the left side is correct though!
let me ask a simpler question
what does it mean for v to be an eigenvector of A?
okay let's work through some basic examples then so you can figure out what it means
let's say $$A = \begin{pmatrix}1&4\2&-1\end{pmatrix}$$ and $$v = \begin{pmatrix}2\1\end{pmatrix},$$ can you tell me what the eigenvalue would be in this situation?
ocean sealBOT
Eric Tao (he/him)
real gazelle
@acoustic plover are you still here
okay
are you trying to figure it out
do you know how to multiply it by hand
I would recommend multiplying it by hand because you get a better intuition for the subject
when you plug things into wolfram alpha, 1) it's kinda annoying and in this case it's probably faster to do it by hand, and 2) wolfram alpha just hands it to you and you have no feel for what's going on
that only makes it true for that matrix
that does not show it is true for any matrix
that is the question that it wants you to figure out
this might give you the idea that it is true
but it doesn't show you that it is true
your job is to figure out when it is true
and we can do that with a really simple and nice proof
how does this show it's false
are you sure
right, but like I'm just asking if you're sure
let me put it this way
is there some number that you can multiply [0 1] by to get [0 0]?
well what about 0
yes 0 works
no that doesn't prove it's always true
okay well the logic works like this
suppose we know that v is an eigenvector of A, with an eigenvalue of lambda; we want to prove that v is an eigenvector of A^2
so if v is an eigenvector of A with an eigenvalue of lambda, what's the equation we can write down
well what's the definition of eigenvector
there's an equation that defines what an eigenvector is
if we know that v is an eigenvector of A with an eigenvalue of lambda, then what does that actually mean
as an equation
we just went over it
it's this equation right?
Av = lambda v
yeah
so now if we want to show that v is an eigenvector of A^2
then our end goal is to show that A^2 v = something * v
right?
awesome
and A^2 v
that is just AAv
right?
subtle mango
so if Av=λv, then A^2v=A(Av)=A(λv) right
real gazelle
@subtle mango I'm gonna explain it
I just don't wanna give the answer right away
lol
okay so we have AAv
a lot of linear algebra proofs are putting "clever choices of parentheses"
so in this case we already know what Av is, right?
yep lambda v
well yeah but how did you get there?
what do you mean by the original thing
yep exactly, it's still a constant times the eigenvector
but first we have to prove that A^2 v = lambda^2 v
we are getting a little ahead of ourselves here
what original formula?
oh
I mean that by itself doesn't show it's lambda squared
you need to manipulate the equation a bit
okay awesome!
no problem; often in math you don't know what you are looking for ahead of time though, so you have to get used to not knowing ahead of time
(this is not just applicable in math, but in life in general obviously)
😭 what happens when you come to a point where you're doing something original
you won't always have people who can tell you the answer
hopefully :)
tacit arch
we're really not supposed to just give away answers even though that's all you want
real gazelle
yeah we're not allowed to give away answers
and it isn't the purpose of the help channels; we're only allowed to give you hints and suggestions for what to try, really
nudges in the right direction, if you will
tacit arch
but that doesn't mean we can just break rules for you
noble sinew
There also is a lot of difference in understanding a proof and making one yourself
real gazelle
well you learn the answer to that question because we told you the answer lol but you won't learn more about the subject in general without grappling with the question
noble sinew
And you aren’t doing yourself any favors by just trying to avoid doing it on your own
real gazelle
I know for me personally, there are problem sets that I struggle with for days, or just never figure out
but it's good mental exercise to think through them
and i learn a lot, even if I never solve them
tacit arch
I wonder if sushi can figure out if v is an eigenvector of A with eigenvalue k, what
$A^{200} v=?v$
ocean sealBOT
riemann
real gazelle
yeah it is
real gazelle
I wonder if sushi can figure out if v is an eigenvector of A with eigenvalue k, ...
I mean that question is kinda simple
anyways is anyone against me closing the thread lol
or is there more
lone heartBOT
lone heartBOT
Available help channel!
native blaze
I would like to know how I am obligated to prove a statement P(x) considering the quantifier using induction
native blaze
So if I wanted to prove statement $(x^{y} > xy) = P(x)$, how does the structure of what I want to prove contained outside of P(x) affect how I conduct my proof by induction?
ocean sealBOT
muppetman
lone heartBOT
@native blaze Has your question been resolved?
lone heartBOT
@native blaze Has your question been resolved?
vague mirage
Prove that a³+b³+c³-3abc =
1/2(a+b+c){(a-b)²+(b-c)²+(c-a)²}
lone heartBOT
Prove that a³+b³+c³-3abc =
1/2(a+b+c){(a-b)²+(b-c)²+(c-a)²}
vague mirage
How to do it how should I approach
pliant dune
expanding the brackets and seeing if they're equal seems like a start
vague mirage
expand stuff on the right?
1/2(a+b+c){(a-b)²+(b-c)²+(c-a)²}```
mellow tusk
Prove that a³+b³+c³-3abc =
1/2(a+b+c){(a-b)²+(b-c)²+(c-a)²}
just prove asq+bsq+csq-ab-bc-ca = (a-b)²+(b-c)²+(c-a)² into 1/2
and then use (a3 + b3 + c3 - 3abc) = (a + b + c)*(a^2 + b^2 + c^2 - ab - bc - ac).
vale wigeon
just prove asq+bsq+csq-ab-bc-ca = (a-b)²+(b-c)²+(c-a)² into 1/2
this symbol ^ exists
vale wigeon
your algebra seems ok so far
mellow tusk
grt
native blaze
I think I did it
remove the first implication arrow in your proof; if your teacher is very strict he will take off a small mark
Also remember to quantify a,b,c in some set if your teacher cares about that
like rationals
vague mirage
remove the first implication arrow in your proof; if your teacher is very strict...
So what should I use?
native blaze
nothing
for the very first line
you have an equal sign
so you dont need an implication
but honestly if ur teacher isnt strict about that it doesnt matter, i doubt they will care
hot lantern
leaden hound
I would like to understand how wolfram gets this result
have I typed something in a odd way?
marsh rapids
Is there a step you don't understand in the step by step solution ?
(the free version goes deep enough here to show pretty much everything)
leaden hound
I see, I think it would be very much easier to get (1-cos(t/2)) as u and the use substitution
would you agree?
marsh rapids
It could work as well
leaden hound
thing is, I don't get that absurd result
I will show what I did if you don't mind
one sec
found my problem
thanks anyway!
marsh rapids
.close
alpine sable
I dont understand this
lone heartBOT
I dont understand this
echo socket
Have you tried anything to solve the inequality so far?
alpine sable
Ye
mellow tusk
I dont understand this
case 1 when x>=5/2 case 2 when x<5/2
alpine sable
?
worn iris
\lvert f(x) \rvert > a &\iff\left[\begin{matrix} f(x)<-a \ f(x)>a\end{matrix}\right.\
ocean sealBOT
blueberry faygo
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
lone heartBOT
alpine sable
.reopen
lone heartBOT
✅
alpine sable
So 3?
high rapids
What?
chrome plank
Go to sleep
If you want to solve this, you have to divide it into two cases:
First case ) [Absolute value] > [Other side]
Secon case ) - [Absolute value] > [Other side] Notice the - sign!
This is because when you have something like
|x| = 5, x can be both 5 and -5, because |-5| = 5
alpine sable
Ohh
chrome plank
Of course you have to keep the same inequality sign (<, >, <= or >=), in this case we had > and I just went with that because it's pretty messy to always write them all
Oh you are right, my bad! (i'm fixing the old message)
keen pasture
Ok
lone heartBOT
keen pasture
Thanks for the help
What are results?
lone heartBOT
Available help channel!
sand light
Can someone tell me the answer to this question and explain
lone heartBOT
Can someone tell me the answer to this question and explain
mellow tusk
ratio of areas?
compact cedar
is FG line parallel to BC line
worn iris
you cant determine specifically if F and G are merely points that satisfy FG//BC
compact cedar
they cant be similar
cus if FG isn't parallel to BC the angles could be wild and it totally wouldnt be similar to ABC
lone heartBOT
@sand light Has your question been resolved?
terse cosmos
17
terse cosmos
17
vale wigeon
do you have any progress or are you stuck at the beginning?
there is a trick that greatly simplifies this
||let u := 3^(x+y) and v := 2^(x-y). you get a linear system in u and v. solve for the values of u and v, hence arrive at the system x+y=__, x-y=__ which is also easy to solve||
@terse cosmos
lone heartBOT
@terse cosmos Has your question been resolved?
vale wigeon
alpine sable
How would I modify this curve so that it exponentially goes up, stays mostly flat for a settable amount and then continues exponentially rising?
0.5 - 0.5Cos(pi * x)
alpine sable
I only care about the graph in 0 < x , y < 1
ocean sealBOT
mortal trellis
not a lot of exponential growth going on here
mortal trellis
paint
alpine sable
alpine sable
How would I modify this curve so that it exponentially goes up, stays mostly fla...
depends
what's ur defintion of modify
what are things u can modify
Well I want it to keep that shape for the beginning and end parts, just change the middle
The equation can be completely different
alpine sable
Well I want it to keep that shape for the beginning and end parts, just change t...
oh ok
can't u use piecewise functions?
I havent heard of it, Ill look into it
mortal trellis
my first idea would be to put two logistic curves next to each other. but that's not really exponential
alpine sable
Yeah I could combine functions
Maybe exponential is the wrong word, just a cos curve sort of shape
alpine sable
Maybe exponential is the wrong word, just a cos curve sort of shape
can u draw it?
alpine sable
Here
alpine sable
Here
u can sample let's say 10 points and get a 9th degree polynomial passing through those points
mortal trellis
that would probably look like shit
polynomials of high degree are not good for interpolation
mortal trellis
or well, if you want the graph to look good
alpine sable
oh
pliant dune
depending on how strong your restrictions are you could just try multiplying by some sort of gaussian function
alpine sable
So I guess it would have 3 variables
StartPoint, Duration, Slope
Then while x < StartPoint => Cos curve
while StartPoint < x < StartPoint + Duration => Slope
while x > StartPoint + Duration => Cos curve
alpine sable
So I guess it would have 3 variables
StartPoint, Duration, Slope
Then while x <...
lemme guess u are coding something
Yep lol
well then just use peice wise functions
Hence my lack of math expertise
in certain bounds give exponential function
in certain bounds give constant flat
after that give exponential
mortal trellis
only problem with that would be if they want smooth transitions
alpine sable
Im worried about how to go from one curve to the next cleanly
lemme code it
well it won't be continious but eye soothing
mortal trellis
well it probably should be at least continuous
alpine sable
import numpy as np
def f(x) :
if 0.25>x>0:
return np.exp(x)-1 #first exponential
if 0.75>x>=0.25:
return np.exp(0.25) - 1#flat
else:
return np.exp(x) - np.exp(0.75)+np.exp(0.25) - 1# exponential
Whats np.exp(x) ? x^2?
mortal trellis
,w plot 0.5/(1+e^(-50(x-0.25))), 0<x<1
ocean sealBOT
mortal trellis
I would maybe choose something like this for the first part
and then piecewise the same kind of thing for the other half of the interval
alpine sable
One thing is the whole graph needs to fit between 0 and 1 for both x and y
mortal trellis
,w plot piecewise[{{0.5/(1+e^(-50(x-0.25))), 0<x<0.5}, {0.5/(1+e^(-50(x-0.75)))+0.5, 0.5<x<1}}], 0<x<1
ocean sealBOT
mortal trellis
something like this. although I guess that's not actually continuous at 0.5
you would have to play around with the parameters a bit
mortal trellis
yeah you can move the plot around by changing the parameters
the 0.25 and 0.75 are the inflection points
alpine sable
It works if I set 0.25 to 0.1 and 0.75 to 0.9 mostly
mortal trellis
,w plot (1/(1+e^(-50(x-0.25))) + 1/(1+e^(-50(x-0.75))))/2, 0<x<1
ocean sealBOT
mortal trellis
you can also just add the two curves
that probably also generalizes easier
(and is now smooth everywhere)
alpine sable
The graph is a little off when I do that for some reason
It shrinks by 0.25 in the y axis
mortal trellis
,w plot (1/(1+e^(-50(x-1/6))) + 1/(1+e^(-50(x-3/6))) + 1/(1+e^(-50(x-5/6))))/3, 0<x<1
ocean sealBOT
mortal trellis
It shrinks by 0.25 in the y axis
you probably need to change the factors
alpine sable
It works if I do f(x) + g(x) - 0.5
Where f(x) and g(x) are the two equations
mortal trellis
note that the second function in my example does not have the +0.5 anymore
alpine sable
Ah that was it
mortal trellis
but anyway I think this works now. you can change the height of the flat part, you can change how long that part is, you can change how steep the growth part is etc
mellow tusk
which ?
mellow tusk
alright ill try them
alpine sable
alright ill try them
tysm!!! :)))
lone heartBOT
Available help channel!
alpine sable
mellow tusk
ok so
mellow tusk
i proved first 3 parts, except d [ idk what is asked in d part ]
mellow tusk
im yet to solve q3
alpine sable
i proved first 3 parts, except d [ idk what is asked in d part ]
thank you so much!!! for d it says 'find the value of m so that the shortest vertical distance coincides with the y-axis'
thank yo so much!! i absent so i didnt understand much of it before
mellow tusk
thank you so much!!! for d it says 'find the value of m so that the shortest ver...
i couldnt understand that part, sorry
mellow tusk
thank yo so much!! i absent so i didnt understand much of it before
no problem
alpine sable
i couldnt understand that part, sorry
its alright :))
mellow tusk
ill send q3 too give me 5-10 mins
alpine sable
ill send q3 too give me 5-10 mins
okay 🙂
mellow tusk
in b part i didnt calculate the area, but i marked the coordinates of the right angled triangle
alpine sable
omg thank you so much!!!!!
mellow tusk
np
hollow lark
lone heartBOT
hollow lark
How to do c and d
lone heartBOT
@hollow lark Has your question been resolved?
hollow lark
<@&286206848099549185>
lone heartBOT
@hollow lark Has your question been resolved?
vale wigeon
it feels as if parts c and d may not supply enough information
nothing is known of the relationship between B and C, so how could we possibly calculate P(A|C)?
yeah, no, P(A|C) can LITERALLY have any value between 0 and 1 with the given data.
@hollow lark parts c and d are unsolvable as stated.
mellow drift
hello
mellow drift
cam soneone help me?
lone heartBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
iron mulch
!15min
lone heartBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
iron mulch
READ THE RULES
@mellow drift are you going to ask a question or are you going to ping us and then disappear?
mellow drift
so my question is: i have two complex numbers (5+23i) and (2+3i) and i have to multiply them. the result should be -59+61i. but i wanted to try to solve this problem with the polar forms of the two complex numbers: sqrt of (5^2+23^2) e^iarctan of 23/5 and sqrt of (2^2+3^2)e^i*arctan of 3/2 but that gives me a completely different result i think i completely fckked up something
abstract fractal
so my question is: i have two complex numbers (5+23i) and (2+3i) and i have to ...
Looks correct to me
Why do you think it's wrong?
mellow drift
because i calculated that in a complex numbers calculator and the result wasnt -59+61i but
abstract fractal
I think it's the same number written extremely weirdly
,w sqrt(554*13)e^(i(arctan(23/5)+arctan(3/2)))
ocean sealBOT
abstract fractal
,w √(554*13)*e^(2.339529366i)
ocean sealBOT
mellow drift
how can i see it
abstract fractal
See what
mellow drift
the result of the bot
abstract fractal
Is it not showing up for you?
mellow drift
no just either more or trash
abstract fractal
mellow drift
but thanks for your help
vale sapphire
You have probably disabled embeds in your settings (Text & Images)
abstract fractal
Now why did I have to go such a roundabout way for wolfram to display the exponential as a complex number?
terse cosmos
lone heartBOT
median oar
have u done similar question before?
lone heartBOT
@terse cosmos Has your question been resolved?
alpine sable
ok
alpine sable
last question before i go to bed
f(x)=2x³-16x
the tangent line on (a,f(a)) : g(x)
g'(x)=t
number of different roots of |f(x)|=g(x) : h(t)
when lim h(t) =/= h(k), the min value of k=-1
t->k
h(-10)+h(5)=?
vale wigeon
do we even need that info about the discontinuities of h
also... wait
i think your g may not be well-defined
mellow tusk
what does this part mean
vale wigeon
$\lim_{t \to k} h(t) \neq h(k)$
ocean sealBOT
Ann
mellow tusk
okay
alpine sable
i think your g may not be well-defined
wdym
alpine sable
idk
vale wigeon
y=g(x) is the equation of the tangent line to y=f(x) at the point where f'(x) = t, yes?
except there need not be just one such point...
alpine sable
(a,f(a))
mellow tusk
y=g(x) is the equation of the tangent line to y=f(x) at the point where f'(x) = ...
its tangent at x=a
vale wigeon
ok then what's the relationship between a and t
if t = f'(a) then how can we say the number of solutions to |f(x)|=g(x) is a function of t
alpine sable
idfk
vale wigeon
idk
what do you mean you don't know
it's your question
are you translating it from another language?
alpine sable
if t = f'(a) then how can we say the number of solutions to |f(x)|=g(x) is a fun...
whats the purpose of this question
mellow tusk
if t = f'(a) then how can we say the number of solutions to |f(x)|=g(x) is a fun...
t is a constnat
vale wigeon
whats the purpose of this question
i'm trying to understand your definition of h.
alpine sable
yeah i know
mellow tusk
cause gx is linear, g dash x is constant , g dash x = t
alpine sable
since the slope of g(x) is t
alpine sable
the number of different roots changes as t changes
vale wigeon
what if there are two different tangent lines with the same slope but different amounts of intersection points with the graph of y=|f(x)|
alpine sable
thats why theres (a,f(a))
vale wigeon
hold on. let me try to illustrate my point.
mellow tusk
f(x)=2x³-16x
the tangent line on (a,f(a)) : g(x)
g'(x)=t
number of different roo...
can someone change this ques into latex, pls
vale wigeon
ok so @alpine sable
let's say i want to find h(0) based on your definition
this will have to be the number of solutions of |f(x)|=g(x), where y=g(x) is the equation of the tangent line to y=f(x) whose slope is 0.
do i understand the setup correctly so far?
alpine sable
yes
vale wigeon
ok now look at this graph.
drawn in red and black are the graphs of y=f(x) and y=|f(x)| respectively.
drawn in purple and green are the tangent lines at x=a for a = 4/sqrt(6) [purple] and a = -4/sqrt(6) [green]
the purple line intersects the black graph zero times while the green line intersects it four times.
so according to you, h(0) should be both 0 and 4 at the same time.
what gives?
and i don't really want to hear any excuses along the lines of "we don't care about h(0) for the problem"
alpine sable
figure it out
thats all i have for this question
vale wigeon
what do you mean, "figure it out"
mellow tusk
lmao
vale wigeon
as far as i'm concerned, the problem is ill-defined pending further clarification
and to insist otherwise is just asinine
hell
this same sort of thing happens even for values that we do care about
for h(-10), here are the two tangent lines of slope -10:
see how there's an ambiguity in the intersection count again?
should h(-10) be 2 or 3?
tacit arch
Poor Ann
alpine sable
no
vale wigeon
then where do you get your question from
did it come to you in a message directly from the heavens
alpine sable
book
vale wigeon
cool, can you take a picture of the book
tacit arch
Do you have a smartphone
vale wigeon
at the relevant page
mellow tusk
its not tangent
bruh
lone heartBOT
@alpine sable Has your question been resolved?
peak island
Hello everyone !
I'm completely stuck currently due to my lack of math skills ( still working on improving and trying to understand trigo but that's irrelevant here ^^ )
https://www.desmos.com/calculator/i3mmcqw2rt
This is a set of function given a x value give me a C value
Now let's admit I want to reverse that system and start from a C value how could I find the B value that would have generated C ?
Context it will be used for : A shader in a video game, I currently have a shader drawing a grid on top of a sphere with various parameters that use the result of C to draw or not the lines.
Now it don't give me much flexibility on how to balance the grid & i'm looking to improve the shader with custom values of C , but I need to reverse the process to get the B value equivalent so the rest of the shader don't break.
Thanks for your help 🙂
peak island
A little bit more of context is that i'm still learning how to read mathematical equation and all ( i'm more used to the programming function usage rather than reading mathematics anotations so sorry if it complicate your explanation for you 😢 )
pliant dune
so given the value of C you want to find what B was?
fluid eagle
right...
pliant dune
the problem I can see is the floor function, so you won't get an exact value
fluid eagle
expressing B in terms of C (so an equation B = (something with C)) is hard because of the floor
yea
pliant dune
though you could probably get a range of values B could have been
fluid eagle
note how the function looks btw
pliant dune
there's no way to reverse the floor function
fluid eagle
if you have a specific c (vertical line), it will cross the graph infinitely many times (or zero times) because the graph repeats
pliant dune
there's no way to reverse the floor function
you lose some information when it's applied
peak island
that's what I was afraid of :/
fluid eagle
since you're programming and all, you could try to make a table of all possible c-values (there are finitely many because of the floor) and link it back to a range/approximation for b
or you could ignore the floor to be honest
it'll give you an approximation for b anyway
peak island
Ok maybe i'm taking the problem in the wrong angle.
let me give you more context.
I ended up with this shader with the help of someone :
https://www.shadertoy.com/view/sdKBD3
what i'm trying to do is getting more control over how the sphere is split
pliant dune
the problem for this is when there are multiple oscillations in the range you can run into problems trying to reverse sine
peak island
here it's just using the lattitude of the sphere to plug into the equations to find the angles theta to apply and deduce when to draw a line or not
Like : at that lattitude split into X segments
Given that i already have a way to provide the X value given the lattitude
fluid eagle
yea, if you know that some b-value gives you a specific c-value, then b+2pi is going to give you the same c-value
but that doesn't matter i think because you want to take b in [0,2pi] probably
you can also work from c = 200 sin(b)
peak island
Hummmm
fluid eagle
then once you get a b-value using arcsin(c/200), you have the starting value of the interval
peak island
that could potentially work I guess , let me try real quick
Huuuum I see where the problem is ...
Trying to match W & B values
the complete flow is : x -> A -> B -> D -> C
when I try to input C -> Z -> W it's working for the first half ( kinda )
but then come the other side of the function
Any ideas ?
because ultimately the floor of the D & C function are there to give an integer , the value C that I already have is already an integer so technically I don't need the floor in the reverse system