how can we assume -v exists in U n W
#help-49
spring latch
midnight plankBOT
how can we assume -v exists in U n W
spring latch
wait
U+W is a subspace so it must include 0
so u+w=0 must appear atleast once
no wait
but how can we assume specifically -v appears
prime hornet
how can we assume -v exists in U n W
the intersection of two linear subspaces is a subspace
spring latch
the intersection of two linear subspaces is a subspace
how do we know this
spring latch
okay well
0 is definitely in both of them by definition
if a is in both of them and b is in both of them
then in each of them individually a+b must be in it
and if
wait why did i decide to use a for this
if a is in both of them then xa is in both o fhtem
so its a subspace
okay
spring latch
how can we assume -v exists in U n W
so if v is in U n W then -1 * v is in there
so -v is in there
twilit field
how can we assume -v exists in U n W
Ayy, Axler.
twilit field
Have fun!
spring latch
why is that an emoji lmfoa
anyway so since v and -v are in the intersection
v must be in U and -v must be in W
alright
.solved
midnight plankBOT
twilit field
why is that an emoji lmfoa
I made it one , lol
spring latch
I made it one , lol
oh cool lmao
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whole dove
why do i need to substitute a derivative back into the original function
midnight plankBOT
why do i need to substitute a derivative back into the original function
gaunt nimbus
Could u elaborate?
midnight plankBOT
@whole dove Has your question been resolved?
whole dove
Could u elaborate?
like for example a local maximum or minimum question
i found when f’(x) is 0
the stationary point
when and why do i need to substitute it back into the original function
whole dove
may you please give an example
runic hamlet
by solving f'(x)=0 you find the x coordinate of the min/max
by plugging that x value into f(x) you find the y coordinate of the min/max
is that what you mean?
midnight plankBOT
whole dove
okay so f of x is y? basically
also, are there any scenarios where you would need to substitute and value into the derivative function?
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winter jewel
midnight plankBOT
frank atlas
Jee advanced 2024??
winter jewel
Yes
frank atlas
I just wrote 2025
winter jewel
Can you help how to solve it
winter jewel
I just wrote 2025
How was it
frank atlas
Ok
frank atlas
Kanpur iit
winter jewel
Ooof
frank kraken
P(G⎹C) = P(G).1/2/(P(G).1/2+(1-P(G))*1)
winter jewel
.close
midnight plankBOT
frank kraken
<@&268886789983436800>
midnight plankBOT
daring dagger
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spring wedge
I'm going with the alternate solution, I tried using solve on my classpad
midnight plankBOT
I'm going with the alternate solution, I tried using solve on my classpad
spring wedge
inputting 12000= 8400cos(pie times t over 48)+10200
but i got some output that had words like constns and so on which is definately not the answer when solving for t
can anyone find out why it may be that way
i used gpt and it managed to solve it
.close
woeful turret
can someone explain why its not transitive
midnight plankBOT
can someone explain why its not transitive
pine wave
recall the definition of transitive
lyric charm
R|2345
-+----
2|++
3|++ +
4| +
5| + +
cursive shoal
can someone explain why its not transitive
R is NOT transitive because:
(2, 3) and (3, 5) are in R, but (2, 5) is NOT in R.
midnight plankBOT
@woeful turret Has your question been resolved?
woeful turret
one second ill be back
woeful turret
recall the definition of transitive
if (a,b) is there and (b,c) is there (a,c) should also be there
pine wave
yeah do you see how you can try to find a counterexample to this?
woeful turret
R is NOT transitive because:
(2, 3) and (3, 5) are in R, but (2, 5) is NOT in R...
ohhhh so it has to be true for each and every element
woeful turret
yeah do you see how you can try to find a counterexample to this?
yeah got it
thank u
.close
twilit field
The intersection consists of those vectors that satisfy $v+U_1=w+U_2$. Let $x \in U_1; y \in U_2$. We then have $v+x=w+y$. So $x = (w-v)+y$. . So any vector in $U_1$ can be written as the translate of $U_2$? I feel like I'm doing something wrong
midnight plankBOT
The intersection consists of those vectors that satisfy $v+U_1=w+U_2$. Let $x \i...
grand pondBOT
wai
grim vector
Where is probability :(
twilit field
Where is probability :(
In a while, sorry Yaku.
toxic crane
Hai
rapid galleon
The intersection consists of those vectors that satisfy $v+U_1=w+U_2$. Let $x \i...
if the intersection is nonempty you're asked to prove A1∩A2 = x + U3 for some subspace U3 of V. do you have any idea what this subspace could be? you gotta have some idea to begin with, can't blindly start this
twilit field
if the intersection is nonempty you're asked to prove A1∩A2 = x + U3 for some su...
U_3=.U_1 \cap U_2?
twilit field
if the intersection is nonempty you're asked to prove A1∩A2 = x + U3 for some su...
hmm, I mean why does what I've done not work
It shows that elements of U+1 can be written as translates of U_2
rapid galleon
It shows that elements of U+1 can be written as translates of U_2
you're going in the wrong direction. let x be a member of the intersection. then we actually wish to show A1 ∩ A2 = x + U1∩U2.
thing about x to note is that x = v + u1 = w + u2 => v - w = u2 - u1 ∈ U1∩U2. pick y from A1 ∩ A2 then show y is in x + U1∩U2. again do it conversely to show the equality.
twilit field
you're going in the wrong direction. let x be a member of the intersection. then...
I feel like there may be some other way to solve this without knowing what the subspace looks like
twilit field
you're going in the wrong direction. let x be a member of the intersection. then...
what is wrong with my proof though
I'll think about this a bit more
thanks
.close
midnight plankBOT
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twilit field
let $x \in U_1, y \in U_2$ we the. have $x+v=y+u$. So $x= (u-v)+y$ so $U_1 = (u-v) +U_2$ . Now from this , I'm first hoping to show that it's possible that there is no soln ,for x,y, resulting in teh empty set
midnight plankBOT
let $x \in U_1, y \in U_2$ we the. have $x+v=y+u$. So $x= (u-v)+y$ so $U_1 = (u-...
grand pondBOT
wai
twilit field
there can be a solution though
there can be
but I first want to show it's possible there isin't
novel sedge
im not sure how that will be relavant to proving this 😄
twilit field
im not sure how that will be relavant to proving this 😄
the last part of the question
novel sedge
its just asking you to show one of the two must happen for any given u,v,U1,U2
i mean, sure: just take two translates of the same subspace. they wont have any intersection and thus there will be no solution
last slate
A hint can be to consider the example of R³ with two planes -- both parallel at first, then both intersecting
twilit field
A hint can be to consider the example of R³ with two planes -- both parallel at ...
Do I have to start off having a rough idea what the intrsection is?
novel sedge
Okay, I'll continue with the other case then
not sure what you mean by this 🤔 can you post your proof here later?
twilit field
not sure what you mean by this 🤔 can you post your proof here later?
I meant that A_1 cap A_2 is a translate of some subspace fo V
novel sedge
but thats not always the case either
twilit field
but thats not always the case either
it's either the translate of some subspace of V or is the empty set no?
novel sedge
what im saying is, nowhere in your proof do you need to show that empty intersection is possible or not
yes, its saying that either empty intersection or translate must happen. but giving one example where it happens does not prove anything
twilit field
got it
novel sedge
i would just assume that the intersection is not empty and show that A1 \cap A2 is a translate of a subspace
twilit field
okie
last slate
Do I have to start off having a rough idea what the intrsection is?
Start with A1 intersection A2 being non-empty
Then prove that y,x in A1 intersection A2 implies y = x + v where v is in U1 intersection U2
And conversely, that if v in U1 intersection U2, then y = x + v is in A1 intersection A2
twilit field
let the intersection not be empty. There then exists vectors, $x \in U_1, y \in U_2$ such that $x+v=w+y$
grand pondBOT
wai
twilit field
i'm lost sorry
novel sedge
what do you want to show?
twilit field
I'm confused, so I have to show it's the translate of some subset of V?
novel sedge
not some subset, some subspace
novel sedge
yes, so do you know any candidates for what the subspace could be?
twilit field
U_1 and U_2
novel sedge
those are... two subspaces
twilit field
yes
novel sedge
im asking you, do you know any candidates for the subspace that A1\cap A2 could be a translate of
twilit field
im asking you, do you know any candidates for the subspace that A1\cap A2 could ...
hmm
A_1 and A_2
novel sedge
im asking you for a single subspace. and the answer you have given me are two sets that are not necessarily even subspaces!
twilit field
im asking you for a _single_ **subspace**. and the answer you have given me are ...
fair
Lemme try thinking a bit more
Well, I think this is best done by investigating the relationshp b.w A_1 and A_2
no?
novel sedge
those are vague words, that each of us have a different interpretation of what their meaning is
. if in your mind you think it is the way forward, you should do it. either way, i dont think i can be of any help in answering this question 😄
last slate
A hint can be to consider the example of R³ with two planes -- both parallel at ...
A1 intersection A2 = x + S where S is a subspace of V
What's S? (reconsider this hint)
twilit field
A1 intersection A2 = x + S where S is a subspace of V
What's S? (reconsider thi...
U_1 \cap U_2 is my best bet
novel sedge
and what element do you think it will be translated by?
twilit field
v+w
novel sedge
you should check to see if it works
twilit field
See, I get this is a valid way, but is there no way to assumme a completely arbitary subspace and show that it exists
I think there might be
novel sedge
i dont know, maybe there is 😄
twilit field
Let $t \in A_1 \cap A_2$. This tell us $t =v+x ; x \in U_1$ we then have $-v+t=x$. so $-t+ A_1 \cap A_2 =U_1$. We can similarly obtain $-t+A_2 \cap A_2 =y; y \in U_2$. so $-t+ A_1 \cap A_2 = U_2$
grand pondBOT
wai
twilit field
This is somehwat correct?
subtle blaze
How come everytime I try to read what you write I never seem to understand
subtle blaze
There’s always just random jumps in logic and reasoning that don’t follow from your previous sentences
You whip out random shit out the wazoo all the time
You just add the word “it then follows” or “this tells us” but the 2 statements aren’t even connected
twilit field
Let $t \in A_1 \cap A_2$. so $t \in A_1 \implies t=v+U_1$. Let $x \in U_1$. Then $t=v+x.$ so $-v+t=x$ so $-v+ A_1 \cap A_2 = U_1$. We similarly have $-w+A_1 \cap A_2 = U_2$
grand pondBOT
wai
twilit field
Is this a better start
subtle blaze
But why do you say let x in U₁
This choice is not yours to make
You already made that choice when you picked t
novel sedge
you could have said instead: let x\in U1 such that t = v+x
twilit field
But why do you say let x in U₁
x stands for an arbitrary element of U_1
though now that I think of it, that line of thinking is flawed
novel sedge
x stands for an arbitrary element of U_1
then it is not true that for any arbitrary element of U1 that t = v + x
subtle blaze
$t \in A_1 \implies \exists x \in U_1 : t = v + x$
grand pondBOT
frosst
novel sedge
you could have said instead: let x\in U1 such that t = v+x
this statement however, is true
twilit field
yea, that makes more sense
subtle blaze
You can’t pick the t arbitrary then also pick the x arbitrarily
This is what I mean by you make shit up lol
twilit field
This is what I mean by you make shit up lol
I think I'm failing at trying to speed run axler( which results in this)
subtle blaze
(I’m not saying this as like an insult, just an observation that you tend to do this a lot)
It’s hard for me to even read what you write because I can’t follow what you’re saying
The next part is I don’t see how -v + A₁ ∩ A₂ = U₁
Maybe that’s true but it doesn’t seem obvious
And if you’re writing a proof it better be obvious what the connection is
novel sedge
Maybe that’s true but it doesn’t seem obvious
it is not true,
subtle blaze
Well it didn’t sound true in my head anyway
novel sedge
just take two nonparallel lines to be U1,U2 in R^2. the intersection of their translates is a singleton, while U1 or U2 is a line
subtle blaze
If A₁ ∩ A₂ is empty then my U₁ is empty which is a contradiction by the assumption that U₁ is a subspace
last slate
You are on the right track but don't jump between statements in between, like t in A1 intersection A2 implies t in A1 implies there exists x in U1 and v in V such that t = v + x implies t-v = x implies t-v in U1
Similarly, t-w in U2
You can't say -t + A1 intersection A2 is U1 for instance because you're saying -t + (any element in A1 intersection A2) makes up the whole subspace U1, which is false
novel sedge
If A₁ ∩ A₂ is empty then my U₁ is empty which is a contradiction by the assumpti...
we have assumed A1\cap A2 is nonempty.
(read up)
novel sedge
but A1\cap A2 may be empty in which case we are done 😄
twilit field
You are on the right track but don't jump between statements in between, like t ...
thanks
novel sedge
what’s your main account
im not sure how this discussion is relavant to the solution to this problem 😄
subtle blaze
Yeah I’m very eepy I was just thinking that what he wrote didn’t sound right
midnight plankBOT
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rancid iron
midnight plankBOT
novel sedge
,w pi*(120mm)^2 to cm^2
grand pondBOT
rancid iron
oh shit
novel sedge
looks like option C is closet 😄
midnight plankBOT
novel sedge
no problem! glad to help 😄
midnight plankBOT
Available help channel!
wary ferry
Is this right
midnight plankBOT
Is this right
tired osprey
what is happening
night hawk
Is this right
Are you trying to solve for a tangent line?
fallow scarab
my answer looks a little different
Than what
wary ferry
the calculator thing they put it in
wary ferry
is there a easier or faster way to do that
tired osprey
using a calculator
wary ferry
sorry for the messy work
inland patio
Why does f(x,y) = x^2 - y^4 have no local extremum at (x,y) = (0,0) whereas g(x,y) = x^2 + y^4 has?
midnight plankBOT
Why does f(x,y) = x^2 - y^4 have no local extremum at (x,y) = (0,0) whereas g(x,...
pine wave
Recall the definition of local extremum?
fallow scarab
x^2 - c has a global min and c - y^4 has global max
inland patio
Recall the definition of local extremum?
For a local minimum, there exists a d > 0 such that f(p) >= f(x) whenever || x - p|| < d. Analogously for local maximum. I don't see how I can read that off from the definition though.
pine wave
well, checking you have the right definition is important - to show that something is not a local minimum/maximum, you just need to show it doesn't satisfy the definition
negating the local minimum condition, it says
"for all d>0, there exists some p such that ||x-p||<d and f(p)<f(x)"
inland patio
negating the local minimum condition, it says
"for all d>0, there exists some p ...
ah ok 👍 so it's obvious then that p = (0,0) is such a point then for f
pine wave
ah ok 👍 so it's obvious then that p = (0,0) is such a point then for f
wait how is p=(0,0) such a point
inland patio
wait how is p=(0,0) such a point
it is not a local minimum, since there is no extremum for f at p = (0,0)
pine wave
the reasoning doesn't seem very sound - how did you conclude there is no extremum?
pine wave
How I would write it would be "say i'm given d>0, then I can set p=(0,d/2), it is clear ||(0,0)-p||=d/2<d, and f(p)<0=f(0,0)"
that is to expand on how to demonstrate it is not a local minimum
of course, to be extra clear, you can lay out the definition for local minimum and explicitly show your negation step
inland patio
How I would write it would be "say i'm given d>0, then I can set p=(0,d/2), it i...
yes, sounds much better. And then simply repeat that phrase for local maximum with some minor changes.
.close
muted gull
I don't understand where is my mistake? I transformed this triple integral in cylindrical coordinates, but my notes say that f(r,θ,z)=r (due to dV=rdzdrdθ) not r² (I got that because r=sqrt(x²+y²) multiplied by r from the dzdrdθ
midnight plankBOT
I don't understand where is my mistake? I transformed this triple integral in cy...
dawn dagger
I don't understand where is my mistake? I transformed this triple integral in cy...
the other factor r comes from the jacobian
muted gull
Yes, ik, the handwritten solution is my solution
Is it correct?
Because my profs notes say otherwise
dawn dagger
ellinas lol
muted gull
Από μαθηματικό είσαι?
muted gull
yours should be right, i checked in cartesian
Ty
midnight plankBOT
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plucky prairie
forgot how to do this
midnight plankBOT
forgot how to do this
tired osprey
multply everything by x, then solve it.
midnight plankBOT
plucky prairie
.reopen
midnight plankBOT
✅
tired osprey
it's correct
unkempt skiff
how do i find the angle for the resultant, is there anyway i can cross produyct this
midnight plankBOT
how do i find the angle for the resultant, is there anyway i can cross produyct ...
pine wave
You can figure out the components and calculate the cross product from there
midnight plankBOT
@unkempt skiff Has your question been resolved?
midnight plankBOT
grand pondBOT
wai
twilit field
so what we really have is the quotient space $A/U$
grand pondBOT
wai
twilit field
$A / U = {w+U \mid w \in A}$
grand pondBOT
wai
twilit field
But this doesn't show A is a translate, does it
prime hornet
hm, I don't quite understand how you got that we have this quotient space, but I don't think this shows that A is a translate anyways 
A is not a vector space, so the quotient isn't even defined
remember that to get a quotient V/U, U needs to be a subspace of a vector space V
prime hornet
have you done the => direction yet?
twilit field
That direction would be if A is the translate of some subspace of V,, then whatever, right
twilit field
have you done the => direction yet?
no
prime hornet
hm, do you want to give it a go?
prime hornet
That direction would be if A is the translate of some subspace of V,, then whate...
yes
prime hornet
hm, do you want to give it a go?
I don't think it'll be too bad
I'm still trying to figure out the <= direction 
hard umbra
hint: you need to divide by 2
twilit field
hint: you need to divide by 2
for which direction
prime hornet
oh snow is here, so it's time for me to run away 
twilit field
okay, got =>
willow arrow
Anyone here who knows integration?
twilit field
midnight plankBOT
@twilit field Has your question been resolved?
midnight plankBOT
@twilit field Has your question been resolved?
rapid galleon
<= direction
anything you've tried so far? this one is tricky ngl
twilit field
anything you've tried so far? this one is tricky ngl
I started by writing it as w+ \lambda(v-w)
rapid galleon
lemme think just, the required subspace is something associated with A
gotta get me sum idea
subtle blaze
i started mine by saying Since A is nonempty, let v ∈ A, and let U = −v + A, we will show that U is a subspace.
i just did the problem via showing closure
rapid galleon
ohh hmm yehh that should work. for x in A, we can write x = v + (x - v), thus x is in v + U, again conversely the same shyt
dat was clever
twilit field
wait, but all we know is $\lambda v + (1- \lambda)w \in A$
grand pondBOT
wai
twilit field
i started mine by saying Since A is nonempty, let v ∈ A, and let U = −v + A, we ...
I don't follow
rapid galleon
**wai**
using that fact, show A - v is indeed a subspace.
rest of the problem is done
twilit field
hmm, okay
subtle blaze
it's not that trivial to do
you need to go through the computations to see it
you know that U := ...?
and then pick 2 things from there and add it, see if it's still in U
twilit field
but I neither know what A looks liker, neutehr do I know U
hard umbra
but you know the condition given
why do you need to know what A "looks like"
what does that even mean?
it just has elements
subtle blaze
you do know what A "looks like" via the condition
twilit field
I know if it's of the given form it's in A
subtle blaze
the idea is to find an equation that things in U will satisfy, via what they will satisfy in A
subtle blaze
i started mine by saying Since A is nonempty, let v ∈ A, and let U = −v + A, we ...
if i pick some v_1 \in U = −v + A, this is nothing but v_1 = -v + a_1
for some a_1 \in A
twilit field
yes
subtle blaze
how do you show closure of vector addition
grand pondBOT
wai
subtle blaze
why is there a w
we didn't define a w
it seems like you've just picked a v_2 from some random ass place
remember we need to pick 2 things from U
things in U have the form -v + something in A
twilit field
yes, so v_2=-v+a_2, a_2 \in A
twilit field
add thme
subtle blaze
okay do that
twilit field
so v_1+v_2=-2v +(a_1+a_2)
subtle blaze
(it will turn out later that doing scalar multiplication makes it easier but you "don't know that" to begin with)
subtle blaze
so v_1+v_2=-2v +(a_1+a_2)
well good now we have something about a_1 + a_2
we know something about a_1 and a_2's
twilit field
a_1,a_2 belong to U
subtle blaze
remember, the whole goal is to show that v_1 + v_2 \in U
meaning v_1 + v_2 = -v + something in A
twilit field
so I have v_1+v_2= -v + (-v+a_1+a_2)
subtle blaze
okay so what do we want to show
twilit field
that this is in A, right
subtle blaze
what is "this"
hard umbra
-v + a1 + a2 had better be in A for the question to even be true
but that's not the question
the question is to explain why
twilit field
I'm so confused, I don't even get why we're doing what we're doing atm
subtle blaze
we want to show that v_1 + v_2 \in U
rapid galleon
we're showing closure with addition then with scalar multiplication
subtle blaze
closure under vector addition
rapid galleon
to show that U is subspace
hard umbra
if A is a translate of a subspace, then A - a is a subspace for any a in A
subtle blaze
i think there's also a better way to "get rid" of the 2 in the -2v
twilit field
if A is a translate of a subspace, then A - a is a subspace for any a in A
right
hard umbra
the goal is to take U = A - a and show it's a subspace
you are showing the subspace properties right now
novel sedge
why do you need to know what A "looks like"
well, one could interpret the condition being that every line connecting two points in A is also contained in A 😄
twilit field
so v_1+v_2 = (a_1+a_2) - 2a, we the divide by 2 to get $v_1+v+2 = (a_1+a_2)-a
subtle blaze
im sorry what
novel sedge
that is not needed, to solve the problem, however, like you have rightly pointed out
hard umbra
the lhs must be halved too
subtle blaze
the right hand side too
hard umbra
that yes
subtle blaze
rapid galleon
twilit field
the right hand side too
a_1+a_2 are still in the subspace, so we can stcik with this notation I think
hard umbra
A is not a subspace
novel sedge
if im understanding your problem correctly, it should not be this complicated 🤔 if A is a translate of a subspace U, then A = a + U for any a \in A. so A - a = U, a subspace
rapid galleon
so I have v_1+v_2= -v + (-v+a_1+a_2)
take a step back bruh. you show -v + a1 + a2 is a member of A. you know for members in A, that (affine) combination must be in A. so you have to show that -v + a1 + a2 is some affine combination of members of A to get the closuree
twilit field
so $\frac{v_1+v_2}{2} = \frac{a_1+a_2}{2}-v$
subtle blaze
you want to show that this is true
twilit field
yes
grand pondBOT
wai
subtle blaze
well what do you know about the a_1 and a_2's?
twilit field
they belong to A
subtle blaze
and what does that mean
twilit field
they are closed under addition
subtle blaze
(you have 2 arbitrary vectors in A...what property do they have?)
subtle blaze
they are closed under addition
A is once again, not a subspace
rapid galleon
they are closed under addition
no their affine combination is in A
twilit field
A is once again, not a subspace
right, I keep forgetting that
subtle blaze
A is special though
twilit field
so their combination is in A because of the form we have here
A is a of certain form
novel sedge
hmm, why do you say so 🤔
subtle blaze
what's special about A
rapid galleon
I'm completely lost
is lin algebra your first exposure to writing proofs?
twilit field
is lin algebra your first exposure to writing proofs?
no, but I went a sem without proof writing
so trying to get back in the gooove
I've already taken an LA course tbh
rapid galleon
i see well get back into our question
subtle blaze
if i just gave you any random A the iff from the problem won't hold
so at some point we must need to use what makes A special
twilit field
The A should be such that \lambda v + (1- \lambda)w \in A
novel sedge
ah okay, i was confused by the notation 😄
hard umbra
specifically (a1 + a2)/2
subtle blaze
remember, we want to show that $\frac{v_1 + v_2}{2}$ can be written in the form $-v + a$ where $a \in A$
grand pondBOT
frosst
subtle blaze
and we just said that $\frac{v_1 + v_2}{2} = \frac{a_1 + a_2}{2} - v$
grand pondBOT
frosst
twilit field
so we wish to show that a_+a+1/2 \in A, right....
subtle blaze
yes, but you have a lot of typos there
subtle blaze
The A should be such that \lambda v + (1- \lambda)w \in A
and we have this
this holds for any lambda, so it will also hold for one in particular
twilit field
yes
subtle blaze
we want to show (a_1 + a_2)/2 \in A using the fact that λa_1 + (1 - λ)a_2 \in A for all λ
hard umbra
frosst switches to greek keyboard
twilit field
we can set lambda to 0.5
subtle blaze
i have text shortcuts set up on my computer but it doesn't autoreplace in discord :/
subtle blaze
we can set lambda to 0.5
why?
subtle blaze
so the full statement is...?
hard umbra
mfw $a_{1/2}$
grand pondBOT
zenith storm
I agree
hard umbra
should've done scalar mult first
zenith storm
That’s correct but what’s the relation?
twilit field
so the full statement is...?
As lambda is arbitrary and lambda a_1+ (1-lambda)a_2 \in A, we have 1/2 a_1 + 1/2 a_2 \in A?
twilit field
let this a_1/2+a_2/a = b \in A
zenith storm
why do we care that 1/2 a_1 + 1/2 a_2 \in A?
That’s not really the core but it’s something you can’t leave out if that makes sense
twilit field
why do we care that 1/2 a_1 + 1/2 a_2 \in A?
this proves closure
zenith storm
Exactly
hard umbra
how
subtle blaze
this proves closure
does it?
hard umbra
where
subtle blaze
where does it say that v_1 + v_2 \in U
twilit field
We can then multiply by 2 to get that, I suppose
subtle blaze
hold up
zenith storm
It’s an option
subtle blaze
why do we care that 1/2 a_1 + 1/2 a_2 \in A?
zenith storm
Closure
hard umbra
not asking you
twilit field
why do we care that 1/2 a_1 + 1/2 a_2 \in A?
closure , so that we can conclude that U is a vector space?
hard shard
subtle blaze
now we know something about (a_1 + a_2)/2
this guy is in A
what does that mean
twilit field
yes
twilit field
(v_1+v_2)/2 can be written as -v+a; a \in A
subtle blaze
which means...?
zenith storm
<@&268886789983436800> this guy is being really annoying
What
twilit field
which means...?
v_1+v_2/2 \in U
subtle blaze
right
which DOES NOT prove closure
it says the sum of 2 vectors multiplied by 1/2 is in the set
not that the sum of 2 vectors is in the set
twilit field
I'm sorry but atp I think I should just crack this bymyself, almost everything is just going over my head
thanks a lot for all the help, but for some reason I'm lost
zenith storm
Please stop posting random images in this channel
Yes señor I’m sorry I was just trying to help at first
subtle blaze
let $v \in A$, and $U = -v + A$. suppose $v_1,v_2 \in U$, then $(v_1 + v_2)/2 = (a_1 + a_2)/2 - v$ which meant that $(v_1 + v_2)/2 \in U$
grand pondBOT
frosst
subtle blaze
at this point you can't really do anything about v_1 + v_2, you'd want to start working on closure under scalar multiplication
if U is closed under scalar multiplication then (v_1 + v_2)/2 \in U ==> v_1 + v_2 \in U which then proves closure under vector addition
that's why we said you should do scalar multiplication first, but prior to doing the problem you don't know that
twilit field
Frosst once again. I'm lost. Can I just try this bymyself, I think I need to self introspect to get this
I'll close this, and post a proof by end of day
subtle blaze
- it's not that easy of a problem
- maybe take a break and come back to it, you say you're lost but you don't specify where. the best thing about maths is that you can identify at which step you've become lost
hard umbra
frosst you should do the question when 2 = 0 in F 
subtle blaze
i was thinking about that tbh
rapid galleon
yep i was thinking about that as well
rapid galleon
at some point we ended up assuming the field is real numbers
hard umbra
spoiler: it's false
twilit field
at some point we ended up assuming the field is real numbers
that's what axler does
.close
midnight plankBOT
twilit field
thanks a lot everyone!
midnight plankBOT
Available help channel!
late violet
Can I find a=1?
midnight plankBOT
Can I find a=1?
late violet
Yes u can
Any help I am stuck , I did like terms but didn’t get anywhere
Probably I should put g(x)=
And solve it with auxiliary function but I’m not sure
slender kernel
Do you know about lhospital
late violet
Do you know about lhospital
Yeah
slender kernel
U can solve with that since it is a indeterminate form of infinite/infinite
lyric charm
you don't need l'hop here dear lord
at all
you can do it much easier algebraically
late violet
I can’t apply lhospital if don’t know if the lim exists right ?
slender kernel
you don't need l'hop here dear lord
No one asked for ur opinion
lyric charm
$\frac{ax+2+\frac{a}{x+2}}{x}$ can juust simplify into $a + \frac{2}{x} + \frac{a}{x(x+2)}$
grand pondBOT
Ann
late violet
Oh yeah that’s easier
lyric charm
No one asked for ur opinion
are you serious rn
late violet
Thanks both 🙌🏽
slender kernel
Yea derivative of X is to tuff huh?
slender kernel
Thanks both 🙌🏽
❤️
lyric charm
Yea derivative of X is to tuff huh?
i'm not saying it's too tough.
i'm saying it's overkill.
slender kernel
👍
slender kernel
Whatt???
lyric charm
which again not tough but it is just unnecessary and gives you more places to screw up
so just why
did i make my point clear? @slender kernel
slender kernel
you can be less condescending when u r making ur points, seems like u r aggressive all the time
midnight plankBOT
@late violet Has your question been resolved?
lofty chasm
help
lyric charm
rewrite both x and dx/dy as trig functions of 8y
lofty chasm
ion get it
lethal path
ion get it
you can use the double angle identity on 8 sin 4y cos 4y
lofty chasm
ok i got sin8y
lethal path
ok i got sin8y
you should get 4 sin 8y
lofty chasm
where is the 4 coming form
lethal path
8 sin u cos u = 4 * 2 sin u cos u = 4 sin(2u)
lofty chasm
erm
lofty chasm
how
lethal path
8 sin u cos u = 4 * 2 sin u cos u = 4 sin(2u)
you just sub in u = 4y into here
lofty chasm
how yk its 4y
OK I GOT IT
ok i got 8sin4y
no i mwan
4sin8y 😭
ok then what do i do
midnight plankBOT
@lofty chasm Has your question been resolved?
lethal path
ok then what do i do
after thinking about it I was wrong; that's not the right thing to do here
lethal path
dx/dy is 8sin4ycos4y
instead go back to here and you have $\frac{dx}{dy} = 8 \sqrt{x} \sqrt{1 - x}$
grand pondBOT
south
lethal path
see if you can figure out how
lethal path
i understand thi
great so now $\frac{dy}{dx} = \frac{1}{dx/dy}$
grand pondBOT
south
lethal path
if you can now complete the square of x(1 - x), you're good
you can get x(1 - x) into that q + r(x + s)^2 form
oh you do have to do $\frac{1}{8 \sqrt{x(1 - x)}} = \frac{1}{\sqrt{64x(1 - x)}}$
grand pondBOT
south
grand pondBOT
south
lethal path
then combine sqrt(64) with sqrt(x (1 - x))
lethal path
did you complete the square of x(1 - x) yet?
midnight plankBOT
midnight plankBOT
Available help channel!
solemn solar
midnight plankBOT
solemn solar
can sum1 provide me a brief direction to go from here?
ive found length BC using basic pythag
frank atlas
can sum1 provide me a brief direction to go from here?
Find AX
frank atlas
Can u find the area of the triangle?
solemn solar
ohh
frank atlas
So u got AX?
solemn solar
hollyp
frank atlas
?
solemn solar
hollup
frank atlas
What is hollup??
solemn solar
means holdup
frank atlas
Kk
solemn solar
wait so im finding area of ABC?
frank atlas
Yea
solemn solar
how do i rearrange for h tho
frank atlas
Tan^-1
midnight plankBOT
frank atlas
Wlcm
midnight plankBOT
Available help channel!
vocal inlet
Let $f$ be a continuous $T-$periodic function with $T>0$. Prove that for all $a \in \mathbb{R}$ we have : $\int_{a}^{a+T}f(x)dx = \int_{0}^{T}f(x)dx$
midnight plankBOT
Let $f$ be a continuous $T-$periodic function with $T>0$. Prove that for all $a ...
grand pondBOT
Da Brilliant Boi
vocal inlet
I tried a substitution (u = x-a)
but then i have to prove $\int_{0}^{T}f(u+a)dx = \int_{0}^{T}f(u)dx$
grand pondBOT
Da Brilliant Boi
vocal inlet
and i'm not sure how to do that
lyric charm
then for the case that it's not, there exists n ∈ Z such that nT ∈ (a, a+T) and you can split the integral at nT and move either piece forward or backward by a period
vocal inlet
so smth like $\int_{a}^{nT}f(x)dx + \int_{nT}^{a+T}f(x)dx$
f(x)* sorry
wait i'm not sure where this is going, i'll try
lyric charm
f(x) not f(x+a)
vocal inlet
yeah but i don't see what to do next
lyric charm
in both
grand pondBOT
Da Brilliant Boi
lyric charm
then you substitute u := x+T in the first
vocal inlet
alright...
lyric charm
get $\int_{a+T}^{(n+1)T} f(u-T) \dd{u} + \int_{nT}^{a+T} f(x) \dd{x}$
grand pondBOT
Ann
lyric charm
see if you can figure out how to stitch these
you will need to explicitly use the periodicity of f
vocal inlet
oh
i see it now
that substitution was a nice move
just to make sure...f(u-T) is the same as f(u) by periodicity, reuniting the two integrals i get integral from nT to (n+1)T of f(x), which can easily become from 0 to T with another substitution
anyway i'm satisfied
thx
.close
tranquil turtle
midnight plankBOT
tranquil turtle
help please
lyric charm
lyric charm
,w (1,-3,1)×(1,0,-1)
grand pondBOT
lyric charm
er
ok i screwed up the input but (6,4,6) is correct actually
that is the cross product
do you see what to do from here?
tranquil turtle
do you see what to do from here?
oh half it?
lyric charm
well your answer has to be a number not a vector so just halving your vector ain't it
you need to also do something else
tranquil turtle
well your answer has to be a number not a vector so just halving your vector ain...
sqrt 22
find the magnitude
forgot
lyric charm
yes
tranquil turtle
yes
ok thanks. i also wanna check this
|a +b| = |a|^2 + |b|^2 - 2|a|b| * cos (180 - 120)
angle between a + b and a = sin 60 / |a + b| = sin theta / |b|
is that right
midnight plankBOT
It is suggested that you limit yourself to one question per help channel, opening a new one once your original question is answered and your original channel has been closed. This is to make your channel easier to follow for potential helpers and can bring attention to the fact that your question has changed.
tranquil turtle
sorry
lyric charm
ok thanks. i also wanna check this
|a +b| = |a|^2 + |b|^2 - 2|a|b| * cos (180 -...
the first one of these is kind of needlessly overcomplicated cause the third term could have simply been +2|a| |b| cos(120°)
and the other one uh...
are you trying to apply the sine law to some triangle
tranquil turtle
yeah
lyric charm
that's gonna be a bit overkill
just find (a+b)•b normally and do the standard formula for angle between two vectors
tranquil turtle
just find (a+b)•b normally and do the standard formula for angle between two vec...
oh ok, is the answer 46.1 for the angle
so i know if its correct
lyric charm
... don't know off the top of my head
but it feels like it's at least in the right ballpark.
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
I'm trying to axiomatically prove $P(E \cup F) = P(E) + P(F)- P( E \cap F)$
midnight plankBOT
I'm trying to axiomatically prove $P(E \cup F) = P(E) + P(F)- P( E \cap F)$
grand pondBOT
wai
lyric charm
ok, progress?
twilit field
So firstly, I want to write EuF as the union of two disjoint events
so $P(E \cup (E^{C} \cap F)$
grand pondBOT
wai
lyric charm
missing closing bracket at the end but yes
twilit field
I can then write it as $P(E) + P(E^{C} \cap F)$, by setting every other element in the series to be the null event
grand pondBOT
wai
twilit field
I then have to write E^C \cap F as the union of two disjoint events
maybe $( (E \cap F )\cup ( E^{C} \cap F)$
grand pondBOT
wai
chrome rain
instead, try writing F as disjoint union of two things
twilit field
instead, try writing F as disjoint union of two things
I have, have I not?
chrome rain
I mean
F = (E complement int F) union ?
grand pondBOT
wai
chrome rain
👍
twilit field
so $P(F) = P(EF) + P(E^{C} F)$
grand pondBOT
wai
midnight plankBOT
midnight plankBOT
Available help channel!
woeful turret
midnight plankBOT
woeful turret
i used the f(x1) = f(x2) method
and i got 2 conditions x1=x2 or x1x2=1
how do i comment about one-one or many one for an interval based on this?
frank atlas
x1x2=1 is impossible if x1,x2 >1
frank atlas
how do i comment about one-one or many one for an interval based on this?
So x1=x2 is the only way
So it is one one in [1, ♾️)
frank atlas
Wlcm
woeful turret
could u also help me do this graphically?
the eqn ive got is
(y-1)(x^2+1)/2x = 1
can i use this in any way?
frank atlas
(y-1)(x^2+1)/2x = 1
Make that y=... Form
frank atlas
And idk how the graph of that function looks like
frank atlas
Maybe u can search that in some graph plotter site
woeful turret
yeah i know but i wanted to plot it myself
cause in the exam we wont be able to use graph plotters right
frank atlas
cause in the exam we wont be able to use graph plotters right
Tbh I don't know
woeful turret
oh ok sure
modern sapphire
well not exactly two squares, but close enough
woeful turret
how will that help us?
modern sapphire
you should know how a plot of the square looks
woeful turret
parabola?
modern sapphire
yea
woeful turret
but what about that term on the bottom
modern sapphire
close to infinity, the ratio would get close to 1
and you only need to analyze at -1, 0 and 1 for more details
at intermediate intervals, its gonna be monotonic
woeful turret
and you only need to analyze at -1, 0 and 1 for more details
why at those points tho?
modern sapphire
why at those points tho?
well, you can see the values are significant for the individual functions
like x^2 + 1 has minimum at 0
(x+1)^2 has min at -1
woeful turret
oh ok right and the 1?
because its a square we are assuming it will be kind of symmetric?
or like because theres a -1 its a good idea to check 1 also
modern sapphire
well, its not gonna be symmetric
but I called out that point coz you have that condition x1x2 = 1
and 1 is important in that context
woeful turret
but I called out that point coz you have that condition x1x2 = 1
oh no we cant use that because if im doing it graphically then i wouldnt have gotten that condition
modern sapphire
you already know the function has a critical value there
modern sapphire
uhh sure
woeful turret
using quotient rule?
modern sapphire
yea
midnight plankBOT
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Available help channel!
woeful turret
midnight plankBOT
hoary drift
Substitute gif = x - fpf
Bring {x} to one side and use {x} ε [0, 1) to get an inequality in x
Split the solution of this inequality into intervals between integers
woeful turret
okay then this
hoary drift
Yes
woeful turret
4x^2 - 20x+9 = -20{x}
woeful turret
oh ok i think i got what ur saying
but im kinda getting something ugly on the lhs
no worries let me try and solve
i am getting
0<=-4x^2+20x-9<20
so -4x^2+20x-29<0
hoary drift
0<=-4x^2+20x-9<20
For this x ε [1.5, 4.5]
So the cases that we need to consider are
1.5 to 2
2 to 3
3 to 4
4 to 4.5
woeful turret
For this x ε [1.5, 4.5]
i think we got it no? we need to take all integers
only solving that shitty inequality is tough
hoary drift
i think we got it no? we need to take all integers
Why
woeful turret
So the cases that we need to consider are
1.5 to 2
2 to 3
3 to 4
4 to 4.5
how do we proceed then?
hoary drift
how do we proceed then?
We know [x] for each separate interval
And we can write {x} = x - [x]
hoary drift
We know [x] for each separate interval
So well solve each case separately
But you might need to verify all your solutions by substituting them into the original equation
woeful turret
huh
isnt it getting too lengthy?
is there a better way? i already spent a lot of time solving the inequality
hoary drift
You could use the discriminant of the original equation to eliminate some cases
I don't think it will eliminate that many cases
woeful turret
ok ykw im done with this question
thanks for helping
ill post it again when im ready to do it
.close
hoary drift
👍
midnight plankBOT
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Available help channel!
dry cove
midnight plankBOT
dry cove
this is the question
this is the formula
i just dont understand how did they get the -2 as an exponent
nimble leaf
Can you send them as images
dry cove
wait ill try to screenshot
question
formula
i dont know how they got the -2 as an exponent
midnight snow
When you move the decimal point towards the right, you reduce 1 from the power of 10
dry cove
When you move the decimal point towards the right, you reduce 1 from the power o...
im sorry, i dont get it, which part or which number was it reduce there, the exponent 9? cause thats the only given exponent im seeing
midnight snow
Let us say you have a number 0.04 × 10⁵
Now when you change the 0.04 to a 4
Which is basically 0.0400 to 004.00
You are moving the decimal point towards right by two places
So you reduce the power by 2
Here, 5. Reduce 5 by 2
So
0.04 × 10⁵ = 4 × 10³
midnight plankBOT
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Available help channel!
woeful turret
midnight plankBOT
woeful turret
is there a better way to do this than differentiating 4 times
quartz hornet
oh wow
the question off the bat screams trignometry
but i cant think of nything except differentiating 4 times
woeful turret
trignometry ? 💀
woeful turret
oh right
woeful turret
my teacher gave me these shitty ass questions bro im not able to solve anything
and the ones i am able to solve are taking too long
woeful turret
interesting but
whered u go
midnight plankBOT
@woeful turret Has your question been resolved?
woeful turret
.close
safe oriole
$\int_{-a}^a\sqrt{a^2-x^2}\dd x=\int_0^{2\pi}-a\sin\theta\sqrt{a^2-a^2\cos^2\theta}\dd\theta=\int_0^{2\pi}-a^2\sin\theta\abs{\sin\theta}\dd\theta=\int_0^{\pi}-a^2\sin\theta\abs{\sin\theta}\dd\theta+\int_{\pi}^{2\pi}a^2\sin\theta\abs{\sin\theta}\dd\theta=0$ where $a>0$
midnight plankBOT
$\int_{-a}^a\sqrt{a^2-x^2}\dd x=\int_0^{2\pi}-a\sin\theta\sqrt{a^2-a^2\cos^2\the...
grand pondBOT
pirateking0723
sharp coral
you shouldn't substitute a cos(theta) for this integral because cos(theta) is not invertible between -pi and pi, and your substitutions must be invertible
safe oriole
you shouldn't substitute a cos(theta) for this integral because cos(theta) is no...
ohhh i see
sharp coral
instead substitute a sin(theta) because it is invertible between -pi/2 and pi/2
safe oriole
this was driving me insane lol
sharp coral
but also your bounds are not correct here, you need to substitute bounds on theta which match the bounds being from -a to a
safe oriole
cant i replace the bounds by any bounds with length 2pi because of the periodicity of sin and cos ?
sharp coral
now that i think about it you could probably make the sin substitution work if you had the correct bounds
no, the bounds definitely should not be length 2pi
they should be length pi because you need it to go from -a to a, not a to a
safe oriole
now that i think about it you could probably make the sin substitution work if y...
for sin the bounds will be -pi/2 to pi/2 right?
safe oriole
they should be length pi because you need it to go from -a to a, not a to a
ah yes
length 2pi will make the integral 0 (?)
safe oriole
for sin the bounds will be -pi/2 to pi/2 right?
or 3pi/2 to pi/2
sharp coral
for sin the bounds will be -pi/2 to pi/2 right?
yes
safe oriole
**pirateking0723**
so it is also possible to do the substitution $x=a\cos\theta$ but after rewriting the integral as $2\int_0^a\sqrt{a^2-x^2}\dd x$ right because $\cos$ is bijective between $0$ and $\pi$
grand pondBOT
pirateking0723
somber folio
Prove that (-cosx + 1 + sinx)/(cosx - 1 + sinx) = sec (A) + tan(A)
midnight plankBOT
Prove that (-cosx + 1 + sinx)/(cosx - 1 + sinx) = sec (A) + tan(A)
somber folio
This makes me doubt my abilities.
fallow scarab
what's A
fresh sparrow