but they are demonstrating the same idea, innit?
#help-38
frail heron
ionic pendant
no because the columns themselves are not a vector space, you have to distinguish between a set S and span(S)
trim lichen
there is more to Col(A) than just the columns of A themselves
frail heron
no because the columns themselves are not a vector space, you have to distinguis...
I see, how about null space of A?
trim lichen
null space of A is the set of all vectors x satisfying Ax=0
this one lives in R^n as opposed to the col space living in R^m
trim joltBOT
trim joltBOT
✅
frail heron
one more question. for this matrix, the column space always consists of (0,0,0). Is it correct?
trim lichen
well claiming that Col(A) contains only the cols of A and nothing else is, as has been discussed before, very wrong
frail heron
The question is asking for the column space of A
trim lichen
The question is asking for the column space of A
yeah so you would have needed to do some computational work
you would need to find a basis for the col space and either specify it as the span of that or write it longhand in parametric form somehow
ionic pendant
the lazy way would be to just write span{the columns} but they probably expect you to do the work to reduce it to a basis
frail heron
lemme go over rq. I'll leave this channel open
Ig this is the answer?
(1,0,0) (2,0,0) (0,1,0) (3, -1/2, 0)
ionic pendant
not quite
the way to get a basis of the column space is:
- find the columns in the RREF with pivots in them
- identify the corresponding columns in the original matrix. those vectors form your basis
frail heron
I don’t quite get it, the columns in the RREF forms the column vector of A…right?
or should I write exactly like this?
trim lichen
the way to get a basis of the column space is:
1. find the columns in the RREF w...
that, or you could column-reduce your matrix instead 
ionic pendant
trim joltBOT
trim joltBOT
Available help channel!
twin olive
Hi
twin olive
I was wondering someone could check my proofs for question 1a and 1b, slightly confused on 1c
For 1c I think it is not an equivalence relation, I think I have proved it is reflexive and symmetric but not sure about transitivity as i don't think it is guaranteed
trim joltBOT
@twin olive Has your question been resolved?
twin olive
<@&286206848099549185>
main sigil
For 1c I think it is not an equivalence relation, I think I have proved it is re...
It's not, but can you construct a counterexample?
twin olive
I will try now, I realize there exists scenarios in which transitivity does not hold
Are the proofs for the first 2 questions correct?
main sigil
havent checked yet, i will check it
main sigil
1a is correct, but perhaps a little too wordy to my taste
twin olive
Oops
main sigil
is it really necessarily to dedicate a whole sentence and ton of symbols to say that 2 sets are equal if they have the same elements? Are you supposed to cite every ZF axiom you use?
twin olive
No I was just writing for my sake, when I submit I will rewrite and cut out all the unnecessary stuff
I'm not sure if I have done 1b correctly think I may have mixed up some deifnition
main sigil
for 1b, just provide a counterexample
counterexample is the easiest and best disproof you can possibly give
and its also what they asked for
twin olive
Is my counter example at the end correct
Sorry I did the disproof to make sure I understood everything correctly and intuitively 😭
main sigil
Is my counter example at the end correct
yes, that works
twin olive
For the counter example in 1c, do you think i should define p?
main sigil
For the counter example in 1c, do you think i should define p?
Definitely, you have to explain what p does
twin olive
Because for 1b I let p be arbitrary as I quickly thought of that example, can't seem to do the same for 1c
main sigil
Because for 1b I let p be arbitrary as I quickly thought of that example, can't...
yeah, in 1b it worked. Im not sure whether you could do sth like that in 1c
its definitely easier to just define p
if you cant figure anything out, try building the smallest possible counterexample
with random elements, not necessariyl the well known sets such as reals or naturals
one neat counterexample i found (and the smallest one there is) has |S| = 4 and |T| = 3
twin olive
Oh yeah that's actually very nice, I was trying to come up with some weird definition for p and some surjective function in R
main sigil
that's not necessary, i really like small counterexamples, because they are simple to construct and they capture the essence of why it doesnt work
twin olive
Yh, it seems much cleaner and easier to understand haha
Thank you very much!
.close
trim joltBOT
trim joltBOT
Available help channel!
velvet plinth
trim joltBOT
velvet plinth
I do no lt understand how to get started
main sigil
I do no lt understand how to get started
Do you know what degree will f be?
velvet plinth
No
main sigil
No
okay, say that f is of degree n.
of what degree is f(3x)?
and of what degree is f'(x)? f''(x)?
the point of this is that we may be able to recover degree of f from the equation
f(3x) = f'(x) * f''(x)
since both sides must be of the same degree
main sigil
that's interesting way to approach it lol
dapper swift
yeah stupid question lmao
velvet plinth
n
velvet plinth
and of what degree is f'(x)? f''(x)?
n
dapper swift
yeah (n - 1) + (n - 2)
velvet plinth
Yea
dapper swift
okay now you know n
velvet plinth
Howw
dapper swift
degrees of both sides equal
main sigil
yeah (n - 1) + (n - 2)
f'(x) * f''(x) is gonna have this degree
and f'(X) * f''(x) = f(3x)
so they should have the same degree
velvet plinth
Yea so they are n-3
lunar stirrup
Do you know what degree will f be?
hey i asked the same q and u only answered lmao
main sigil
hey i asked the same q and u only answered lmao
Yeah, i remember lol
main sigil
Yea so they are n-3
it should be 2n-3
(n-1) + (n-2) = 2n-3
that's the degree of f'(x) * f''(x)
velvet plinth
Ohhh right
main sigil
what about the degree of f(3x)?
velvet plinth
2n-3 =n
Coz it will be the same as f(x)
Its jus that the coef changes dehree samee
Degree
lunar stirrup
f can be the zero function: f(x) = 0
can u take that if f(x) is a polynomial?
dapper swift
can u take that if f(x) is a polynomial?
0 is by defn a polynomial
velvet plinth
Yeah, so can you figure out what's the degree of f now?
Yea its 3
dapper swift
(otherwise the function space of polynomials wouldn't have an additive identity)
lunar stirrup
these kinds of methods help a lot actually
dapper swift
I was amazed when that worked lol
otherwise it's 2 or 3 pages of work
or some shit, maybe not that much
lunar stirrup
I was amazed when that worked lol
no usually with the kind of exam, theres a trick for every question
main sigil
Yea its 3
now I'd just write down a general 3rd degree polynomial, expand everything and compare the coefficients
velvet plinth
no usually with the kind of exam, theres a trick for every question
Allen kid?
lunar stirrup
Allen kid?
yh
main sigil
i dont think the 0 trick is intentional, that'd be too good of a trick
velvet plinth
now I'd just write down a general 3rd degree polynomial, expand everything and c...
Okayyy , 👍
.close
manic lagoon
trim joltBOT
manic lagoon
Help is needed by moi.
trim lichen
do you know in general how you can extract the sum of the coefficients of a polynomial
manic lagoon
This ain't no polynomial.
rustic marsh
This ain't no polynomial.
it is a polynomial
empty orchid
This ain't no polynomial.
“Sum of coefficients”
manic lagoon
What is its degree?
trim lichen
well deg(f) must be 1999 if the equation is to hold
empty orchid
In general, you can take x=1 to find the sum (I.e., f(1)=sum of coefficients)
trim lichen
hold on actually
yknow what you have a point i think, there's no way any polynomial f(x) exists that satisfies this
conclusion: question is garbage, skip.
trim joltBOT
manic lagoon
I will also ask my teacher.
vagrant imp
bruh just find f(1)
assume f(x) is a polynomial eqn
then f(1) gives sum of coeffs
eg f(x)=ax^2+bx+c
f(1)=a+b+c
manic lagoon
Can you manually compute the coefficients given enough time? If yes, how? What is its degree? What is the coefficient of the term containing a power of x that is one less than its (the function's) degree?
vagrant imp
Can you manually compute the coefficients given enough time? If yes, how? What i...
I'm pretty sure they don't want yout to do all that but lets see
manic lagoon
.reopen
trim joltBOT
✅
lilac flame
f isn't even a polynomial
vagrant imp
f isn't even a polynomial
how do yk?
lilac flame
you can't divide x^2015 by 16x^16 - 11x + 2
so the sum of coefficients doesn't even make sense
vagrant imp
it could be some inf polynomial maybe
lilac flame
power series yes
vagrant imp
power series yes
yeah so its possible to find the sum of coeffs in that sense
manic lagoon
I don't know power series. What would the first few terms be?
empty orchid
Jesus
lilac flame
,w x^2015/(16x^16 - 11x + 2) power series
solid kilnBOT
Requested by mayervietorus
lilac flame
wolfram alpha dumb bruh
dapper swift
,w x^(2015) / (16x^16 - 11x + 2) quotient and remainder
solid kilnBOT
Requested by sooouth.
red loom
Why not ((16-11+2)f(1)=2016)
solid kilnBOT
PajamaMamaLlama
red loom
f(1) is the sum of the coefficients?
manic lagoon
We know dude.
manic lagoon
I don't know power series. What would the first few terms be?
...
red loom
ahh I did not see any of the other msgs my bad 😅
dapper swift
I mean the assumption is that f is a polynomial
lilac flame
,w factorize (16x^16-11x+2)
dapper swift
the questionmaker must be very lazy not to check that
lilac flame
,w (16x^16-11x+2)/(x+1)
vagrant imp
and apply taylor approximation
wouldn't this work(although differentiation seems a bit nasty)
solid kilnBOT
dapper swift
okay let me try on MATLAB
red loom
,w (16x^16-11x+2)/(x+1)
dapper swift
arghhh I'm having software issues
dapper swift
wait so if 16x^16 = 11x - 2, then raise both sides to the power of 125 to get constant * x^2000 = (11x - 2)^125
dapper swift
to be fair I never learnt this
Desmos seems to tell me it's divisible
there's a turning point around there
so no vertical asymptote it seems
trim joltBOT
@manic lagoon Has your question been resolved?
trim joltBOT
@manic lagoon Has your question been resolved?
nova pumice
Why do a lot of people take differential calc and linear algebra at the same time during a semester in college
nova pumice
Isn’t that hard
Since it’s two calc
Is linear algebra even calc
It looks like cross product
empty orchid
Is linear algebra even calc
In some ways but mostly not
Linear Algebra sets you up for the idea of Multivariable, though
(Obviously we can't visualize higher than 3 dimensions, so how come some problems are proved in, say, the 22nd dimension? Linear Algebra answers these kinds of questions)
manic lagoon
<@1362088890080690258> Has your question been resolved?
sad
limpid dawn
It looks like cross product
more like dot product
trim joltBOT
@nova pumice Has your question been resolved?
trim joltBOT
trim lichen
!msgdel
trim joltBOT
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.
empty orchid
.reopen
burnt mulch
(answered in another server)
late laurel
.close
trim lichen
imploding channels are non-reopenable
trim joltBOT
Available help channel!
urban grotto
trim joltBOT
urban grotto
long shot, but does anyone wanna take a crack on trying this integral with me or seeing if theres an easier way to go about it? I really dont think the problem im doing is supposed to be this bad
bright oracle
Oh no it's the monster integral again
cobalt drift
$\int \frac{\dd{\theta}}{k\sin \theta + 1}$
solid kilnBOT
jewels!
cobalt drift
This is it right?
bright oracle
It looks like you made it even worse
bright oracle
Yeah it's tan(θ/2) time
You know it's serious when that comes out
(He asked about this question yesterday but went to have another look at it to see if the monster integral could be avoided)
cobalt drift
can't avoid monsters in physics
bright oracle
Sadly
The real world doesn't lie down into a convenient setup, even if we make our cows spherical
urban grotto
im gonna cry and throw up in real life
cobalt drift
$\sin \theta = \frac{2\tan (\theta /2)}{1 + \tan^2(\theta /2)}$
solid kilnBOT
jewels!
bright oracle
Looks fine
cobalt drift
Do you need help proceeding @urban grotto
urban grotto
no im good, thaks for the help. ill give it a shot with this sub
i figured this was the case
bright oracle
Bottom is sec^2(θ/2), which turns it into 2 sin cos
urban grotto
becuse i saw someone mention itin a similar problem on stack exchange
mmm okay
thanks again, ill go give this a shot now while i wait for my students to show up
bright oracle
You're a tutor?
urban grotto
im a physics TA lol, if you can believe it
I TA engineering physics
anyways ttyl
.close
trim joltBOT
cobalt drift
I TA engineering physics
coffee overdose
empty orchid
It's been a while, I don't remember if this is right
It's good
That integral still looks shitty though
trim joltBOT
Available help channel!
marsh forum
Doesn't (b) imply (c)
trim joltBOT
Doesn't (b) imply (c)
marsh forum
I mean trivially
grim sparrow
yeah
marsh forum
Cool
Thanks!
Here I just show it's not closed under addition, right
oh wait
The 0 transformation isn't invertible
we're done on those ground alone
amber python
it says noninvertible
queen beacon
I mean trivially
Yes
amber python
in particular the claim is true for dimV = 1
marsh forum
it says noninvertible
yea, and every subpace needs a. 0 vector
amber python
so it contains 0
queen beacon
Well, you want it to not be closed
marsh forum
yea, so I have to find some other LT
queen beacon
Instead you need A+B invertible and A, B aren’t on they own
amber python
sharp on the helper grind
marsh forum
$T_1( \sum_{i=1}^{n} \alpha_i e_i) = \sum_{i=1}^{n-1} \alpha_i e_i; T_2(\sum_{i=1}^{n} \alpha_i e_i )= a_n e_n$
solid kilnBOT
wai
marsh forum
Now to prove every LT from V -> V when V is 1 D works
queen beacon
It just says to show it doesn’t work for n>1
queen beacon
Now to prove every LT from V -> V when V is 1 D works
But you can just list all of those yes
marsh forum
But you can just list all of those yes
I mean here I think rank-nullity will help?
queen beacon
I mean here I think rank-nullity will help?
Yeah
broken cave
Can somebody help me with 52?
trim joltBOT
Can somebody help me with 52?
left oriole
hint: you know an eigenvalue of A
broken cave
what's the context then?
It's matrices
left oriole
all is see on this page is linear algebra
cobalt drift
Well do you know what a characteristic polynomial is
broken cave
Well do you know what a characteristic polynomial is
Yes
trim lichen
you know charpoly but you do not know eigenvalues?
cobalt drift
The roots of this polynomial are called the eigenvalues
trim lichen
broken cave
you know charpoly but you do not know eigenvalues?
Well , Because it has been taught to us but they didn't tell us anything more than that 🥲
Even the characteristic equation was out of the syllabus , but I still studied it
Well
|A-yI|=0
So this matrix will satisfy
A²-(a+b)A-|A|I=0
Right
How can I proceed from here
left oriole
rearrange AB = B
trim lichen
A²-(a+b)A-|A|I=0
a+d not a+b
broken cave
Oh sorry
left oriole
not like that, i meant more like AB - B = 0, i.e. (A - I)B = 0
broken cave
Oh okay
Because I can't really Say |A|≠0 so I can't say that A^-1 even exists so I shouldn't use it
left oriole
you can say something about |A - I|
cobalt drift
I mean lowkey if you know the shortcut for finding the inverse of a 2x2 matrix this isn't that bad
broken cave
|A-I|=(a-1)(d-1)-bc
broken cave
I mean lowkey if you know the shortcut for finding the inverse of a 2x2 matrix t...
I know
cobalt drift
Because by symmetry A^-1 = A
broken cave
Well reverse the diagonal values and take the negative of non diagonal values divide it by |A|
broken cave
We know the det is non-zero so it's invertible
Ha
cobalt drift
Because by symmetry A^-1 = A
now use this
broken cave
Oh shit yes I can use that A^(-1)=A
cobalt drift
$AB = B \implies A^{-1}AB = A^{-1}B \implies A^{-1}B = B$
solid kilnBOT
jewels!
left oriole
that certainly does not imply A = A^-1
broken cave
|A-I|=(a-1)(d-1)-bc
I got this
broken cave
(A-I)B=0
But I can't take determinant both sides
Because det of B is not defined eventually
Well what should I do
left oriole
But I can't take determinant both sides
if that equation holds for some nonzero B, what does that tell you about A - I?
Is it invertible?
broken cave
Is it invertible?
Well it is not invertible
Because if it was
Then B=0
Which is not possible
broken cave
=0
left oriole
yep good
so putting it a bit more suggestively, det(A - lambda I) = 0 when lambda = 1
broken cave
|A-I|=(a-1)(d-1)-bc
So this is equal to 0
ad-1-(a+d)-bc=0
(a+d)=2021
Given in the question
So ad-bc=2022
Am I right?
fierce lake
Yeah
dusky thunder
So ad-bc=2022
huh
little glen
what's the context then?
It's jee
broken cave
But again , you know you're not supposed to solve this question with linear algebra
But fine
native shuttle
Is it 2022
dusky thunder
ad-bc-(a+d)+1=0
left oriole
ad-1-(a+d)-bc=0
(a+d)=2021
this isn't quite right
fierce lake
ad-bc-(a+d)+1=0
Oh shi right
left oriole
should be +1
dusky thunder
dont make those silly mistakes
broken cave
And:2020
dusky thunder
costs a lot
broken cave
Ans*
left oriole
yep that is right
native shuttle
Without using any of that
broken cave
dont make those silly mistakes
Yes it cost 5 marks and 5000 rank 🥲
dusky thunder
Without using any of that
what did u use
left oriole
Without using any of that
probably using eigenvalues, trace and determinant?
native shuttle
multiply and use it
left oriole
but OP doesn't know that stuff
broken cave
multiply and use it
Yeah jee expects us to do this way
native shuttle
a+c=1 , b+d=1
broken cave
💀
fierce lake
Well yeah time pressure is a very influencing factor
native shuttle
and solve
broken cave
But ofc it's a computer based test so I can use any method
broken cave
Well yeah time pressure is a very influencing factor
Yeah
Because I have no knowledge of linear algebra and jee is asking me problems from linear algebra and expects me to do it with , basic knowledge of matrices
native shuttle
I did it with basic knowledge lol
broken cave
Yep
fierce lake
I did it with basic knowledge lol
Exactly by guessing right?
Which is what op tryna say
dusky thunder
having computational lin alg in your toolkit is very powerful for this type of exams
broken cave
Yeah but the thing is I also have to prepare it for chemistry and physics too , so don't really have enough time to do so many things 🥲
dusky thunder
idk how hard the questions are but u can always make time for some lin alg 
fierce lake
broken cave
2 years is sufficient
I've only 1 now
broken cave
idk how hard the questions are but u can always make time for some lin alg <:awo...
Ogey
Thanks guys 🫂
.close
left oriole
idk how hard the questions are but u can always make time for some lin alg <:awo...
an eternal truth
trim joltBOT
fierce lake
I've only 1 now
Assuming you haven't wasted the 1 year
broken cave
Assuming you haven't wasted the 1 year
I didn't
fierce lake
Then good
trim joltBOT
✅
broken cave
Well I have another problem
🥲
53
I got this
If I put K=1 then this whole thing satisfies
As I can see
So
Cant I just say K=1
Yes I'm right the solution agrees with me yayv
Thanks anyways
.close
trim joltBOT
trim joltBOT
Available help channel!
jagged wharf
I'm reading a proof about a certain property relating to bilinear forms $H$, but I have a more basic question concerning the existence of a basis. In the proof, we have some ordered basis $\beta={v_1,v_2,\ldots,v_n}$ for $V$ and an invertible matrix $Q$. Then the authors define $\gamma={w_1,w_2,\ldots,w_n}$, where $$w_j=\sum_{i=1}^nQ_{ij}v_i\quad\text{for }1\leq j\leq n,$$and say that "...since $Q$ is invertible, $\gamma$ is an ordered basis for $V$." Why is this excerpt true?
solid kilnBOT
psie
trim lichen
you can reexpress each v_i in terms of the w's using the entries in Q^-1 as coeffs
and since each v_i is in the span of gamma, by linearity so is all of V
trim joltBOT
@jagged wharf Has your question been resolved?
jagged wharf
you can reexpress each v_i in terms of the w's using the entries in Q^-1 as coef...
Ok, thank you. I think I understand now.
frank rain
I need help with part iii)
just to confirm, is the answer to part i) 8! and the answer to part ii) 8P4 = 1680
frank rain
is part threes answer 12P4
trim lichen
nope it's 12C4
the question says "in no particular order"
youd be hard pressed to find a clearer indication than this that the selection is unordered and thus a combination
trim lichen
yeah they are
frank rain
oh ok
part iv is 8!/(2! x 3!) = 3360, and part v) is 2x6!/3! = 240?
or for part v), is there any need to divide by 3!?
frank rain
part iv is 8!/(2! x 3!) = 3360, and part v) is 2x6!/3! = 240?
<@&286206848099549185>
wraith hinge
part iv is 8!/(2! x 3!) = 3360, and part v) is 2x6!/3! = 240?
Yes you do need to divide it by 3! and it will ultimately be 720/6 = 120 and that is because there are 6 tiles with 3 I's repeating
frank rain
so part 4 is 3360
frank rain
but for part 5, i thought dividing was unnescary coz it didnt ask for distinct ways
frank rain
Yes you do need to divide it by 3! and it will ultimately be 720/6 = 120 and tha...
2x6! is 1440
wraith hinge
2x6! is 1440
Well to be honest 2 Ms at start and at end (and they would be the same even if they were interchanged so don't do x2)
frank rain
so if it were distinct ways, then we would not multiply by 2 and instead divide by 3!
but if its total possible ways, we would multiply by 2 and not divide by 3!
right?
wraith hinge
right?
Well I know it might seem hard for you to understand since you are just starting permutations and combinations but you must account for repeated arrangements
We don't multiply by 2 - you only do that if there are two different ways to do something, which isn’t the case here
We divide by 3! - every time you have three I’s, because they are identical and the words would change because there are 3 Is
trim joltBOT
@frank rain Has your question been resolved?
trim joltBOT
trim joltBOT
Available help channel!
rustic marsh
$\int (1-x^2)^{100} , dx \[0.3cm]
= \int \sum_{k=0}^{100} {100\choose k} \cdot (-1)^k \cdot x^{2k} \[0.3cm]
= \sum_{k=0}^{100} {100\choose k} \cdot (-1)^k \cdot \frac{x^{2k+1}}{2k+1} + C$
solid kilnBOT
Flatus
rustic marsh
need to double check if this is right
cuz when graphed in desmos, doesnt look right
cobalt drift
looks alright
cobalt drift
cuz when graphed in desmos, doesnt look right
What do you mean by this?
rustic marsh
i dont think the constant changes it
cobalt drift
Can you send a screenshot
rustic marsh
but also
if i wanted bounds from
-1 to 1
on the integral
i subbed into sigma and also let desmos just calculate the integral
and got super diff answers
cobalt drift
Uh
rustic marsh
Can you send a screenshot
rustic marsh
I think desmos is just freaking out
is it f(1)-f(-1) to solve int from -1 to 1?
cuz um
it doesnt match up at all
cobalt drift
it might be a desmos issue lol
rustic marsh
it might be a desmos issue lol
ill test on other software and websites later when i get back to pc
trim joltBOT
@rustic marsh Has your question been resolved?
rustic marsh
🥀
trim joltBOT
@rustic marsh Has your question been resolved?
trim joltBOT
Available help channel!
summer crypt
Find the side lenght of cube ABCDA1B1C1D1 if distance between BD and AC1 is d
round imp
rustic marsh
Find the side lenght of cube ABCDA1B1C1D1 if distance between BD and AC1 is d
diagram?
summer crypt
i dont have one sorry
rustic marsh
i dont have one sorry
where did u get the q from
heavy lion
your question itself doenst make sense
rustic marsh
can u js send the q
heavy lion
if BD = d, then AC1 cant be = d, in a normal cube
or you can just find the side length with BD = d alone
rustic marsh
wtf is AC1?
is it just AC?
how can you have a distance between BD and AC?
distance between 2 lines?
perpendicular distance?
summer crypt
Its q10
A1 , B1, C1, D1 are the upper points
AC1 is the diagonal
Yea the distance is perpendicular to both
rustic marsh
oh wait AC1 is the line between A and C1?
rustic marsh
why does translation not mention anything u named
u asked to find side length but it's asking to find volume?
summer crypt
i will just cube it it reckoned it doesnt matter
summer crypt
why does translation not mention anything u named
its weird, distance from diagonal to side not intersecting it
is d
rustic marsh
i will just cube it it reckoned it doesnt matter
i see
summer crypt
its perpendicular to both
rustic marsh
its perpendicular to both
yeah idk what the question is asking
im just assuming it's probably this?
oh wait
$AC \ne CH$
solid kilnBOT
Flatus
rustic marsh
eh ignore the solution but
is the idea that AH = 1 right? cuz i can't wrap my head around the translation
summer crypt
its d not 1
forest needle
Just consider a cube in the 3d cartesian coordinates having side length l, then use midpoint and distance formula and find l in terms of 1
summer crypt
it says it's a unit
i told you the translation is weird it is d
we are supposed to express a in terms of d
summer crypt
Just consider a cube in the 3d cartesian coordinates having side length l, then ...
i dont want to use cordinates nor vectors
forest needle
Any specific reason for not wanting to use coordinates or vectors?
summer crypt
because we havent gone over them yet
forest needle
So you wanna do this using basic geometry?
summer crypt
So you wanna do this using basic geometry?
yes
forest needle
Oof
Coordinates is far simpler
I just tried it using geometry it's just pythagorous used once
Try drawing a diagram @summer crypt
trim joltBOT
@summer crypt Has your question been resolved?
summer crypt
Try drawing a diagram <@755415569163092068>
ok wait i will do it
trim joltBOT
@summer crypt Has your question been resolved?
trim joltBOT
trim joltBOT
Available help channel!
round minnow
trim joltBOT
round minnow
why don't we write +C after everything?
I solved and got it all right but my answer was different in that after I got 3 * x^6 / 6 I added +c
round minnow
and for the rest I added +c also
round minnow
But all additive constants sum up to a single constant
oh
trim joltBOT
trim joltBOT
Available help channel!
tiny elm
trim joltBOT
tiny elm
2 values of a?
no 1st option has <, >, =>, =<
cobalt drift
What is the gradient at that point?
tiny elm
What is the gradient at that point?
its 2xy+a, x^2
cobalt drift
Okay
cobalt drift
You know that $D_{\vb u} f(x, y) = \grad f(x, y) \cdot \vb u$ right?
solid kilnBOT
jewels!
tiny elm
yess
cobalt drift
You can use the definition of the dot product in terms of the cosine
On the right
to establish a bound
since |cos x| <= 1 for real x
Bot?
Oh I lagged
solid kilnBOT
jewels!
tiny elm
ohh the definition you mean cosx= u.v/their magnetidues
solid kilnBOT
jewels!
tiny elm
right right
that makes soo much sense, i wouldve never thought of that
thank youu
.close
round minnow
right?
dusky thunder
yes
round minnow
trim joltBOT
round minnow
what are critical points for this?
manic lagoon
Differentiate it.
trim lichen
why do you think t=0 should not be included
trim lichen
btw may i suggest you write your t's with a little hook on the bottom so they are not just tall plus signs
trim lichen
I wrote it like this and t = 0 makes 1/0 which is undefined
where is 1/0
round minnow
oh my bad
trim lichen
$f'(t) = 2t - \frac{4t}{t^2+1}$
solid kilnBOT
Ann
trim lichen
when you plug in t=0, where is the 1/0?
round minnow
my bad, I didn't mean 1/0
basically I solved t^2 + 1 = 0
t = sqrt(-1) which is imaginary
trim lichen
t^2 + 1 = 0 has no real solutions
round minnow
so I thought u just exclude 0
round minnow
ohh
trim lichen
don't go finger-pointing at t=0...
round minnow
lol
regarding x values that make something undefined, is there a situation where I would exclude it?
I recall a situation where the x value we found was undefined and it got excluded
trim lichen
you exclude things that actually cause an 'undefined' to appear somewhere when plugged in.
round minnow
you exclude things that _actually_ cause an 'undefined' to appear somewhere when...
okay bet, noted
thanks!
.close
lunar stirrup
trim joltBOT
lunar stirrup
not able to find out my mistake..answer is 18 im getting 180
im messing up the units somehow
ionic pendant
can you show your work?
ionic pendant
why is it 10^(-8) now
lunar stirrup
centimetre to metre right
ionic pendant
no that's coulomb * meter
lunar stirrup
oh shit 😭
ionic pendant
centimeter is with lowercase c, coulomb is uppercase C
trim joltBOT
runic trail
well, it cant be centimeter because its a dipole moment
the units wouldnt match up 🤔
lunar stirrup
well, it cant be centimeter because its a dipole moment
yeah my bad i dont focus on units that much and just kind of read it as centimetre
trim joltBOT
Available help channel!
gray brook
f(x) = 2x - 1
what is the distance between the point M and the function given M(-3/2,0)?
spiral ocean
grand light
TOCT!???
trim joltBOT
trim joltBOT
Available help channel!
coarse meadow
trim joltBOT
nimble stone
what are ones thoughts
trim joltBOT
@coarse meadow Has your question been resolved?
coarse meadow
.close
trim joltBOT
trim joltBOT
Available help channel!
swift grove
trim joltBOT
swift grove
inequation
bright quarry
(x - 3)(x - 2)
but tbh here you don’t need that
it’s a concave up quadratic
clearly not A or C
swift grove
so.. D?
dense grove
bright quarry
the problem with D is that it seems to have open circles around the equality points
bright quarry
the problem with D is that it seems to have open circles around the equality poi...
oh wait i was looking at E
yea D is good
swift grove
nice one
i thought its A, but someone said to me: "Because the roots are -2 and -3, adding them together forms -5 and multiplying them forms 6, but you have to invert the signs, then 2 and 3 become positive, so its D"
solid kilnBOT
trim joltBOT
trim joltBOT
Available help channel!
smoky sun
Good day, or evening! I have done an exercise for my math exam, and I in fact am not 100% sure have I done it right. This is the exercise in question:
Given the function f(x) = 2/(|x|-1) sketch the graph of the function g(x) = -f(x), and then:
a) Determine the domain and range of the function g
b) Identify the intervals of monotonicity of the function f
c) Check the parity (whether it is even or odd) of the function g
So I would say
a. domain is R{-1,1} and range is R{0}
b. the interval will be growing at (-∞, -1) and (0, 1), it will be getting smaller at (-1, 0) and (1, ∞)
c. the function is even, because f(−x)=f(x) is true
Also here is the picture of the function g(x) (yes it is form an online function drawer program just because I don’t feel like making one in MS Paint)
I am sorry if you can’t understand what I am saying, I have in fact translated this exercise from another language, and I am not really that used to do math in English, therefor mistakes are bound to happened, but I will be more than happy to resolve any misunderstandings.
trim joltBOT
Good day, or evening! I have done an exercise for my math exam, and I in fact am...
@smoky sun Has your question been resolved?
tardy warren
Good day, or evening! I have done an exercise for my math exam, and I in fact am...
does R{-1, 1} = R - {-1, 1}
?
looking at the graph it seems the image is R - ]0, 2] for f (or for g it would be R - [-2, 0[ I guess)
and the domain is R - {-1, 1} for f and g
does monotonicity mean when the function is either strictly increasing or decreasing? The function decreases in ]-infty, -1[ U ]-1, 0] and increases in [0, 1[ U ]1, infty[
to find this out you could inspect the sign of the derivative of f(x)
I have no idea what "parity" means here
limpid dawn
does R{-1, 1} = R - {-1, 1}
probably yes, discord likes to make the backslash invisible
\
R\\{-1,1}
tardy warren
does monotonicity mean when the function is either strictly increasing or decrea...
actually this could be more simply put
function decreases when x<0
function increases when x>0
sign(f'(x)) = sign(x)
smoky sun
probably yes, discord likes to make the backslash invisible
or I have just simply forgotten to add it xd
tardy warren
Good day, or evening! I have done an exercise for my math exam, and I in fact am...
ah parity means the function is symmetric about the y-axis?
smoky sun
looking at the graph it seems the image is R - ]0, 2] for f (or for g it would b...
The graph is made for the g(x), and yeah I think it would make sense for the range to be R{0, 2} there is a gap there
smoky sun
ah parity means the function is symmetric about the y-axis?
by some searching, the exercise wanted me to check is the function even or odd
tardy warren
by some searching, the exercise wanted me to check is the function even or odd
what does that even mean?
I've never heard that term to refer to functions before
for me, even is 2k and odd is 2k+1 lol
smoky sun
does monotonicity mean when the function is either strictly increasing or decrea...
i guess they wanted me to say both, when it is increasing and decreasing
tardy warren
alright. Although the function is growing when x>0 and decreasing when x<0
limpid dawn
so it is even
i was at your channel, wanted also to confirm 36
tardy warren
so I suppose your b is incorrect?
tardy warren
i was at your channel, wanted also to confirm 36
very good
means I get an additional decimal on my grade 😋
smoky sun
Good day, or evening! I have done an exercise for my math exam, and I in fact am...
so if i get it right:
a. domain is R{-1,1} and range is R{0, 2}
b. function is growing when x>0 and decreasing when x<0
c. its just fine
limpid dawn
the range is wrong
limpid dawn
you dont want to only exclude 0 or 2 but the interval
tardy warren
yeah he used a set instead of an interval didn't he
limpid dawn
it will never reach 0 but it reaches 2
tardy warren
he put {0, 2} instead of ]0, 2]
limpid dawn
so it's half open
empty orchid
so if i get it right:
a. domain is R{-1,1} and range is R{0, 2}
b. function is g...
I think you should say $\bR\setminus[a, b]$ instead (or something of the like)
solid kilnBOT
;(
tardy warren
Good day, or evening! I have done an exercise for my math exam, and I in fact am...
a. domain is R - {-1, 1} and range is R - ]0, 2]
b. the function increases when x > 0 and decreases when x < 0
c. [your answer is already correct]
empty orchid
smoky sun
he put {0, 2} instead of ]0, 2]
okey okey, i shall do it like that
tardy warren
a. domain is R - {-1, 1} and range is R - ]0, 2]
b. the function increases when ...
is this correct?
tardy warren
**;(**
why not just use a minus symbol, you aren't even saving any characters now lol
limpid dawn
why not just use a minus symbol, you aren't even saving any characters now lol
it's a slanted minus
tardy warren
isn't set difference just a negative sign
empty orchid
Using R-(a, b) feels...off
I don't like the notion of subtraction on sets in this manner, it makes sense to use \ for me
tardy warren
well A - B is the set of the elements that belong to A but don't belong to B, right?
empty orchid
I don't like the notion of subtraction on sets in this manner, it makes sense to...
^
tardy warren
a. domain is R - {-1, 1} and range is R - ]0, 2]
b. the function increases when ...
well anyways, aside from the minor notation disagreements, this is correct?
accidentally wrote R{-1, 1} but meant R - {-1, 1} -- edited to fix that
limpid dawn
well increasing/decreasing at x=1 or x=-1 is difficult
tardy warren
well increasing/decreasing at x=1 or x=-1 is difficult
well yeah indeed since the function isn't defined there, neither is its derivative lol
still an improvement?
smoky sun
I think you should say $\bR\setminus[a, b]$ instead (or something of the like)
sooo, you suggest to write it down more like this? Or to use the [ ] not { }?
tardy warren
sooo, you suggest to write it down more like this? Or to use the [ ] not { }?
R\]0,2]
limpid dawn
i mean technically, you can leave it that way since you defined the domain without -1 and 1 so x > 0 or x < 0 means in this context without 1 or -1
ionic pendant
just add spaces:
R\{1} produces R{1}
R \ {1} produces R \ {1}
empty orchid
sooo, you suggest to write it down more like this? Or to use the [ ] not { }?
R\(a, b) or [a, b] depending on the context
tardy warren
i mean technically, you can leave it that way since you defined the domain witho...
yee
empty orchid
{} means the set! (which of course is not necessarily true when it comes to problems like these)
tardy warren
R\\(a, b) or [a, b] depending on the context
shouldn't the left side be excludent though since 0 is only obtained at the limits?
empty orchid
shouldn't the left side be excludent though since 0 is only obtained at the limi...
wut
tardy warren
instead of [0, 2] it should be ]0, 2] or (0, 2] (depends on your notation preference)
empty orchid
No this is like a general form
limpid dawn
shouldn't the left side be excludent though since 0 is only obtained at the limi...
yes
empty orchid
Not for this specific problem
tardy warren
aight
tardy warren
{} means the set! (which of course is not necessarily true when it comes to prob...
pedant's notice: [a, b] is also a set: [a, b] = {x / real x such that a <= x <= b}
although [a, b] contains way more elements than just a and b
an uncontable amount or something I believe cuz it's using the real set
wait
no but the interval is bounded
so that means it's countable???
I have no idea
ionic pendant
every interval of real numbers is uncountable
tardy warren
a. domain is R - {-1, 1} and range is R - ]0, 2]
b. the function increases when ...
anyways, this should be the correct answer I reckon
might have (a) wrong since idk if it's asking regarding f(x) or g(x)
limpid dawn
every interval of real numbers is uncountable
i counted them all
tardy warren
might have (a) wrong since idk if it's asking regarding f(x) or g(x)
might have to switch-a-roo ]0, 2] for [-2, 0[
smoky sun
anyways, this should be the correct answer I reckon
yay, so thanks
smoky sun
might have (a) wrong since idk if it's asking regarding f(x) or g(x)
oh, a. is asking for g(x)
smoky sun
anyways I'm off now
yeah it is "rather late" would be fair to take a nap, so thanks again for help
trim joltBOT
@smoky sun Has your question been resolved?
rain pulsar
can someone help me prove that this set is linearly independent in the vector space of polynomials of degree n?
rain pulsar
<@&286206848099549185>
supple copper
can someone help me prove that this set is linearly independent in the vector sp...
What do you need to prove independence?
trim joltBOT
@rain pulsar Has your question been resolved?
rain pulsar
What do you need to prove independence?
that the trivial solution is the only solution for the linear combination to equal 0
supple copper
that the trivial solution is the only solution for the linear combination to equ...
Also what do you mean by equal to 0? What zero?
(There are many different things in math that are all called zero but are different kinds of zeros)
rain pulsar
i guess for the power series to equal 0
supple copper
We’re in a vector space
There are no power series in a bare vector space
You have no notion of convergence meaning no limits
supple copper
i guess for the power series to equal 0
But close
You’re looking for the zero polynomial
The polynomial p(x) = 0 for all x
rain pulsar
where i would start for showing that the linear combination equals the 0 polynomial?
rain pulsar
if the only solution to the 0 vector is when all the coefficients are 0?
or if no vector in the set is a linear combination of the other vectors
supple copper
Let $V$ be a vector space. A subset $B = {v_1, \ldots, v_n} \subseteq V$ is linearly independent if…?
solid kilnBOT
frosst
supple copper
if the only solution to the 0 vector is when all the coefficients are 0?
The words you’re using are exactly correct but we need it in terms of mathematical notation so we can start doing some solving
rain pulsar
sorry im not too familiar with latex
supple copper
That’s okay just type it out as best you can (even without latex) I’ll try to interpret
supple copper
The only thing you are missing is an implication sign
$\sum_{i=1}^n t_iv_i = 0 \implies t_1 = \ldots = t_n = 0$
solid kilnBOT
frosst
supple copper
Does that make sense?
rain pulsar
so far yes
supple copper
Okay so sub stuff in
supple copper
$\sum_{i=0}^n t_i(x-a)^i = 0$
solid kilnBOT
frosst
supple copper
Show that this equation already implies that all the t_i’s must be 0
For polynomials the usual way is to equate coefficients
But this problem if you try to equate coefficients it looks like very nasty binomial theorem
It might be easier to first prove that {1, x, x², …, xⁿ} is linearly independent
rain pulsar
It might be easier to first prove that {1, x, x², …, xⁿ} is linearly independent
isnt this the standard basis for Pn
so it should be known to be linearly independent
supple copper
Well you don’t know that until you prove it to be so
And if you already have that as a result you can use then its simple
rain pulsar
And if you already have that as a result you can use then its simple
i think i do in my course
supple copper
p(x) = 0 for all x iff p(x-a) = 0 for all x and any a
supple copper
It might be easier to first prove that {1, x, x², …, xⁿ} is linearly independent
If you can show this then you can show that they are really the same question and quote the result here
rain pulsar
p(x) = 0 for all x iff p(x-a) = 0 for all x and any a
where does this come from
supple copper
That’s what I’m saying you should prove
trim joltBOT
@rain pulsar Has your question been resolved?
trim joltBOT
trim joltBOT
Available help channel!
broken cave
For question number 58 can somebody suggest any methods which is short , I can find P inverse and multiply with A and finally I can take out the values but it's too lengthy
broken cave
I don't wanna take that lengthy approach
Any ideas which will not make this that lengthy
trim lichen
$P^{-1}AP - I = P^{-1}(A-I)P$
solid kilnBOT
Ann
trim lichen
$P^{-1}AP - I = P^{-1}AP - P^{-1}IP$
solid kilnBOT
Ann
trim lichen
and factor out P^-1 on the left and P on the right
you can take for granted that P has an inverse
trim joltBOT
trim joltBOT
Available help channel!
mystic pilot
How can i find asintheta from acostheta = 28/√41 in the picture
fierce lake
How can i find asintheta from acostheta = 28/√41 in the picture
Cross product
mystic pilot
I know cross product
fierce lake
So?
mystic pilot
But I am want to be familiar with dot product in this problem
mystic pilot
Yes
mystic pilot
I have a formula that is acc_net = √[(a_c)² + (a_t)²]
mystic pilot
What is a ? Magnitude of acceleration?
Yes the acc vector given in probelm
In this formula acc net is a vector that is given
If I can use it to find?
fierce lake
Yes the acc vector given in probelm
Then shouldn't you divide with magnitude of v to get a cos theta
mystic pilot
Then what do.i do
fierce lake
First of all
mystic pilot
Cuz
fierce lake
Find cos theta
mystic pilot
Ac = asin theta and At = acos theta