is this more factorization 😭
#help-49
jaunty canopy
well what does fermat's christmas theorem say about 3 mod 4 primes?
if p | a^2 + b^2 then
viral dagger
well what does fermat's christmas theorem say about 3 mod 4 primes?
nothing that i know of
jaunty canopy
oh ok wait lemme pull up the statement
oh ok maybe it's not bundled together with the statement of fermat's christmas theorem
but yeah a common thing is that if p | a^2 + b^2, then p | a and p | b
this isn't too hard to prove
jaunty canopy
but yeah a common thing is that if p | a^2 + b^2, then p | a and p | b
but basically use this to conclude that ugly numbers are those whose prime factorisation does not contain a 1 mod 4 prime
this completely characterises ugly numbers and finishes both a and b
viral dagger
btw do you mind if we move to dms to not worry about the channel closing? our timezones are so horribly mismatched theres only really a 2 hour overlap between us ralking :(
jaunty canopy
btw do you mind if we move to dms to not worry about the channel closing? our ti...
yh sure
viral dagger
but basically use this to conclude that ugly numbers are those whose prime facto...
ok so basically since k isnt divisible by a 1 mod 4 then neither does a and b
jaunty canopy
but yeah a common thing is that if p | a^2 + b^2, then p | a and p | b
as in try to use this
let's say $k = 2^t p_1 p_2 \cdots p_l$ where all the $p_i$s are 3 mod 4 primes
grand pondBOT
LY
jaunty canopy
then we have $2k^2 = a^2 + b^2$
grand pondBOT
LY
jaunty canopy
so $p_t$ divides $a$ and $b$
grand pondBOT
LY
jaunty canopy
can you carry on like this to finish?
jaunty canopy
ok nw!
midnight plankBOT
visual ingot
Uff
midnight plankBOT
Available help channel!
granite cipher
midnight plankBOT
granite cipher
How do I go about this
tired vapor
$\frac{df^{-1}(a)}{dx}=\frac{1}{f'(f^{-1}(a))}$. also, you're given that $f(4)=6$, so $f^{-1}(6)=4$, and $f'(4)=\frac{1}{5}$
grand pondBOT
fish
midnight plankBOT
@granite cipher Has your question been resolved?
slate anvil
nyone know the answer to this?
if P(t) is a matrix of continuous functions, then the wronskians of any two fundemental sets of solutions of x' = P(t) x are exactly the same. True or False
i got true but gpt is telling me its false
midnight plankBOT
nyone know the answer to this?
if P(t) is a matrix of continuous functions, the...
@slate anvil Has your question been resolved?
midnight plankBOT
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Available help channel!
tidal turret
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placid spoke
!status
midnight plankBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
placid spoke
well all primes past 2 are odd, so if there are any triplets like that, they would all be odd numbers. so write down some potential candidates.
for example: 7, 9, 11
what's wrong with that triplet? 9 isn't prime because it's divisible by 3
okay now try 11, 13, 15
notice a pattern starting to emerge?
tidal turret
notice a pattern starting to emerge?
No
tidal turret
okay now try 11, 13, 15
15 is not prime
pine thicket
what number seems to always mess the sequence up?
lavish venture
15 is not prime
in each of these triplets there is a number that isn’t prime because it’s a multiple of what?
tidal turret
3
sage olive
suppose you have such a prime triplet, then the first number of the triplet must be either 1 or 2 mod 3
what do you know about one of the other two numbers after it mod 3?
lavish venture
@tidal turret do you know what mod 3 means
sage olive
it’s the remainder after dividing by 3
so a number that is 1 mod 3 is a number of the form 3n + 1, where n is an integer
your first number of your prime triplet, assuming there is one, has to be either 3n + 1 or 3n + 2
if it was 3n + 0 or 3n + 3 it would be divisible by 3
contradicting the fact that it’s prime
tidal turret
Ye
sage olive
then each number after that is 2 more than the previous number
so suppose your first number is 3n + 1
the next number is 3n + 1 + 2
which can’t be prime because …
tidal turret
Div3 remainder0
sage olive
right
then suppose your first number is 3n + 2
the next number is 3n + 4, nothing wrong so far
the third number in the triplet would be 3n + 6
tidal turret
Id there any proof
sage olive
is there any proof of what?
tidal turret
Only triplet 357
sage olive
well yeah it’s the proof i just gave you lol
suppose there’s a prime triplet with this construction, then you’ll always find that one of the numbers is divisible by 3 and therefore not prime
the one exception is the triplet (3,5,7) in which case yes the first number is divisible by 3, but is clearly prime
3 is the only multiple of 3 that is prime
midnight plankBOT
@tidal turret Has your question been resolved?
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edgy knot
midnight plankBOT
edgy knot
the red 2y at the bottom doesnt exist in the solution im wondering why?
how do we get rid of it?
lethal path
the red 2y at the bottom doesnt exist in the solution im wondering why?
You should have du/dy = 2y
u is a function of y so chain rule
Which implies du = 2y dy instead
So that gets replaced: you dropped the dy
In the integral
edgy knot
so the 2y just gets replaced by du/dy?
midnight plankBOT
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Available help channel!
edgy knot
how do you solve the integral of 1/sqrt(x)?
midnight plankBOT
how do you solve the integral of 1/sqrt(x)?
lethal path
Power rule for integrals
terse timber
If x²+y²+z²=20
and x+y+z=0
Is there a solution?
midnight plankBOT
If x²+y²+z²=20
and x+y+z=0
Is there a solution?
brazen shuttle
If x²+y²+z²=20
and x+y+z=0
Is there a solution?
yes
as long as when they are added without the sqrt they = 0
like you could have x=1 y=-3 and z=2 and when they are squared they will equal a pos number
rain wasp
If x²+y²+z²=20
and x+y+z=0
Is there a solution?
yes. one is a sphere with the radius of 2sqrt5, while one is a plane cutting through said sphere
brazen shuttle
so according to what fungus said, then any solution is where they will intersect eachother
im not too sure whether this is abstract thought or you are doing 3 dimensions, but either work
gusty falcon
maybe not really a math question but do you guys learn math better with lecture notes or actual books?
midnight plankBOT
maybe not really a math question but do you guys learn math better with lecture ...
gusty falcon
gray moat
I only use notes that I make by myself from books tbh
sullen jungle
memory from explanation 👍
gray moat
memory from explanation 👍
how do you refresh your memory then?
gray moat
not yet uni though
aahh then yeah
rain wasp
i tend to actually internalize the concepts better during the lecture if i don't take notes
i can't say the same to everyone else
delicate sage
maybe not really a math question but do you guys learn math better with lecture ...
My advice would be to, just write formulas and create a formulae list, and look to proofs from books whenever you are unable to recall them.
sullen jungle
if i understand the concept it will remain engrained in my brain
idk i have something like a logical memory
something i made up
gray moat
idk i have something like a logical memory
that stops working once you get to incredibly dumb stuff to memorise
sullen jungle
probably
gray moat
I just remember how to derive them instead 🥲
sullen jungle
I just remember how to derive them instead 🥲
exactly
gray moat
to this day I still don't remember what's the derivative of cot(x) 💀
misty gorge
maybe not really a math question but do you guys learn math better with lecture ...
it depends, i learned calculus really well from lectures and i learned geometry almost solely from books, but i think you kind of need both
remote spade
maybe not really a math question but do you guys learn math better with lecture ...
What are math notes
misty gorge
lecture notes eventually just become books so i don’t distinguish them too much
remote spade
I only make sst notes!!
gray moat
literally all my notes = the book but keeping on the necessary and non trival stuff
misty gorge
here's an example of a set of lecture notes my professor decided to use in one of my topics classes, this set of lecture notes is what we used but it really reads more like a book with exercises and everything https://arxiv.org/pdf/1011.1690
so is there really gonna be that much of a difference later on? not really
i think the best thing that you can do is go talk to people about what you've learned, and then it won't matter whether you got it from lecture notes or from a book
what kind of subject are you trying to learn? we also have #book-recommendations if you are trying to choose something to read
midnight plankBOT
@gusty falcon Has your question been resolved?
midnight plankBOT
gusty falcon
.reopen
midnight plankBOT
✅
gusty falcon
Actual books
Aren’t they too 😭exhaustive most of the time. Like imagine going over 500 pages
twilit field
Aren’t they too 😭exhaustive most of the time. Like imagine going over 500 pages
I have time
I'm only doing maths
and one other subject evry sem
Well, 2
but yeah
I have time
just work 9-10 hours a day( assuming you only have 3 hours of classes a day)
gusty falcon
here's an example of a set of lecture notes my professor decided to use in one o...
Ohh i see, interesting. So you’ve lost been using lecture notes most of the time and then if necessary you find exercises from relevant textbooks?
or is this only relevant when you’re at a moderately high level of mathematical maturity
rain wasp
Aren’t they too 😭exhaustive most of the time. Like imagine going over 500 pages
reading math books isn't like reading a novel
you are supposed to spend months on it
gusty falcon
Yes uh but well eventually it’ll take longer than just a course on it? Idk unless i’m doing smthing wrong
For example, i’m guessing you guys are aware of the mainstream calc book: stewart
u have like 1000-2000 pages on that book
it’ll probably take ages to go over the entire thing
dreamy lichen
I also don't see how are lecture notes less time expensive than books. Either they cover a thinner topic, which could be solved by only reading the relevant topic in the book or they are written more densely, which makes you spend more time on understanding the material. (A book would arguably be a better option in this case, because if someone explains it to you it's easier than thinking about it on your own)
twilit field
yeah, stewart is bad, like very bad
get a better book imo
I used it for one sem, I hated it
formal palm
took me two months to do stewart, thomas is better though
heavy falcon
it’ll probably take ages to go over the entire thing
dont take everything by the number of pages
the content matters alot
gusty falcon
but most books are still pretty big right? Idk for me it takes over 5-6 months to go over a book
This is longer than me just working through an entire course on the subject/topic
rain wasp
who knows if the actual content is in the first 100 pages and the remaining 900 are acknowledgements
heavy falcon
depends on the book
gusty falcon
who knows if the actual content is in the first 100 pages and the remaining 900 ...
Lol
heavy falcon
for example stewart also covers multivariable calculus
twilit field
some books will take 6 months, It took me 4 months just to do a third of LADR
gusty falcon
some books will take 6 months, It took me 4 months just to do a third of LADR
yeah see so that's what i mean
heavy falcon
not only single variable
formal palm
courses are often longer, often miss some topics and also don't bring as much clarity as a book does, having a reference book with a course is always good
heavy falcon
which is like 2-3 courses in uni
gusty falcon
I don't have an year to spend on LADR for example if i need to get it over with in 1 semester or less
😭
formal palm
i'd rather read 500 pages than attend all lectures of a course
gusty falcon
no don't universities offer lectures notes online
twilit field
Just do the relvant topics from ladr
gusty falcon
like you can google "lecture notes ...."
twilit field
no don't universities offer lectures notes online
depends on the uni
gusty falcon
so i was curious if those are better
more concise and maybe use the books for the problems?
But also that raises the question
formal palm
LADR + strang + david c lay is doable in 4-5 months
gusty falcon
on what problems should i be doing
gusty falcon
LADR + strang + david c lay is doable in 4-5 months
strang took ages for me just for the first few chapters
twilit field
LADR + strang + david c lay is doable in 4-5 months
including dual spaces?
gusty falcon
maybe cuz i was doing all the problems
formal palm
strang took ages for me just for the first few chapters
are you doing all the problems
gusty falcon
yes
and strang's book doesn't have repetitive problems like stewart's for example
so i thought it'd be instructive to do all of them?
heavy falcon
more concise and maybe use the books for the problems?
honestly you should decide yourself
gusty falcon
but like okay we all have time constraints so i'm looking for a practical alternative
twilit field
Hmm, maybe just do the first 5 problems from each exercise in LADR
heavy falcon
you should find what you prefer and choose it
gusty falcon
i can't invest years of my life, for example, just learning linear algebra when i need it solely for a foundation for another overarching goal
twilit field
depending on how deep you want to go
gusty falcon
maybe that makes sense?
formal palm
including dual spaces?
my bad i didnt consider that i already had a first course in LA before starting LADR so it was quicker for me to finish all three books
twilit field
LADR is a 2 sem book if you're a beginner IMO
heavy falcon
i can't invest years of my life, for example, just learning linear algebra when ...
it makes sense but at the same time there is no method that leads to this
whether using a book or lectures or whatever
you wont take years in learning linear algebra
formal palm
how many LA courses does your university offer?
gusty falcon
LADR is a 2 sem book if you're a beginner IMO
is that axler?
gusty falcon
how many LA courses does your university offer?
i'm not in uni
twilit field
is that axler?
yes
gusty falcon
okay so my idea was strang -> axler
twilit field
I'd prefer axler+ strang as axler avoids dets
gusty falcon
strang also avoids dets
formal palm
it's generally two so you can use strang for the first course and start LADR towards the end of the course
twilit field
dets are really important
gusty falcon
so okay how many problems do you guys do in general? Or rather how do you choose which problems to work on for a given book?
twilit field
it's generally two so you can use strang for the first course and start LADR tow...
I used axler against my prof's reccomendation for the first course 
twilit field
so okay how many problems do you guys do in general? Or rather how do you choose...
Around 6 a day if the topic is new
like I'm currently doing RA
formal palm
I used axler against my prof's reccomendation for the first course <:opencry:586...
based
twilit field
hardly get 3-4 problems done a day from abbott
gusty falcon
does LADR have any prereqs? u need to know proof writing?
twilit field
Yes
gusty falcon
hardly get 3-4 problems done a day from abbott
so wouldn't the book take ages?
😭
even in your case
twilit field
so wouldn't the book take ages?
I'm doing it a semseter early
so I have time
I practically have a year before I take RA
well, 8 months
gusty falcon
what all did you take so far?
twilit field
I've taken calc 1, LA 1 and proof writing
twilit field
yes
gusty falcon
technically not even integration is covered
twilit field
and epsilon-delta limits
gusty falcon
cuz in the US they call it calc 2
gusty falcon
and epsilon-delta limits
yeah fair
twilit field
I tried to use spivak for calc 1, but that didn't go well
gusty falcon
that's rough
i was halfway through the strang book
and half way through multivariable calculus
but then i dropped all of it and just decided to do a bit of comp math so i'm revising precalc and stuff
and i constantly get discouraged by how long a book takes
😭 i guess it's normal
whoops
formal palm
serge lang's LA
gusty falcon
LADR + strang + david c lay is doable in 4-5 months
anyway so for this
what do you do that allows you to maybe work thru all that in 4-5 months?
u don't do all the problems?
formal palm
u don't do all the problems?
yeah i skip problems that look doable or similar to the ones i have done before, problem overlap over all three books is huge of course since all three books have the same primary focus that is linear algebra
formal palm
definitely
gusty falcon
I;ve spent 8+ hours on single LADR problems
that sounds scary
okay well i just gotta go with the flow i guess
thanks guys
formal palm
i really haven't used strang tbh, only used it for transformation, but im sure it's a really good beginners book judging from the way he teaches
gusty falcon
It's a bit verbose
But if you do all of his questions u have a good handle on that topic
generally
formal palm
i see
hollow oyster
serge lang's LA
langs LA is a bit too much... well FIS is a good one
formal palm
langs LA is a bit too much... well FIS is a good one
yeah i think i agree with that, haven't read FIS till now
hollow oyster
FIS is a big book ,I'm currently using H&K's book
midnight plankBOT
@gusty falcon Has your question been resolved?
misty gorge
that sounds scary
ideally one would ask for help before you’ve spent 8 hours on a single linear algebra problem
gusty falcon
ideally one would ask for help before you’ve spent 8 hours on a single linear al...
😭
what did you do?
You'd work through books/lecture notes?
misty gorge
You'd work through books/lecture notes?
yes, working through books is good
being unable to do exercises is usually a sign that you didn’t understand the material
so they’re good litmus tests
i have spent a long time on some problems but in things like LADR (or LADW if you prefer) you really should be pairing material from the book and exercises, and maybe a lecture series, and ask for help when you need it or feel like you don’t understand a problem or are stuck on an exercise
again the exercises in LADR aren’t as scary as some would make it out to seem
gusty falcon
I see
Can you do calculus and linear algebra in tandem?
Like together?
Or do both courses need some kinda proof writing experience
misty gorge
Can you do calculus and linear algebra in tandem?
you can
what kind of calculus do you wanna do
like intro real analysis?
or just like straight up calculus
gusty falcon
like intro real analysis?
midway between the two
misty gorge
ladr will have proofs in it for the record
gusty falcon
not intro real analysis cuz i've heard you need proof and a lot of other things to get started
misty gorge
you’ll pick up on how to prove things as you go
gusty falcon
but also not high school calculus because i know most of the algorithms lol
and hs calc is just mostly algorithms
misty gorge
not intro real analysis cuz i've heard you need proof and a lot of other things ...
you don’t need so much extra stuff, knowing what a proof by contradiction (and a few other things) is and a willingness to spend time internalizing definitions is probably enough
misty gorge
is it a good idea to jump straight into intro ra?
eh
i’ve gone back and forth on my opinions on it
gusty falcon
everyone i asked said it was a bad idea
hollow oyster
is it a good idea to jump straight into intro ra?
maybe it depends on what book do you take for RA honestly
gusty falcon
and at the very least try to get a solid mastery over calc (so i thought of just revising calc again)
abbott?
pearl idol
I think Abbott is pretty friendly to go into even if you don't have any proof experience
gusty falcon
or is there a lecture series i can follow 😭
gusty falcon
I think Abbott is pretty friendly to go into even if you don't have any proof ex...
oh so no prereqs?
misty gorge
everyone i asked said it was a bad idea
there is no harm in shrimply trying, and if you really dislike it or feel stuck or frustrated try to gain a better foundation as needed. if you don’t like that idea then you can try an intro proofs style thing or philosophical logic (<- underrated!) style thing
gusty falcon
there is no harm in shrimply trying, and if you really dislike it or feel stuck ...
omggg philosophy again 😭
misty gorge
yeah or you can read abbott and see how you like it
misty gorge
omggg philosophy again 😭
have you done any philosophy?
gusty falcon
no actually
philosophy just reminded me of that probability question we both worked on
😭
misty gorge
it’s good to know some logic, the idea of fallacies, and deduction using formal logical rules, and some history behind it (personal opinion, i think it should be required instead of an intro proof class, though maybe some people would get mad at me for saying this)
gusty falcon
as long as you don't take Rudin as your first intro to RA it's not that bad
no i don't really want to jump into actual RA cuz i guess i can't handle that yet
but at the same time i don't want a calc book either
smth in between
the only in between one i know is spivak
but ... i guess that's very hard in itself 😭
hollow oyster
the only in between one i know is spivak
Apostol has some good proof based clac book
misty gorge
you could try reading a non-proof based linear algebra book
non proof-based*
sorry
hollow oyster
its a two volume long book and the second volume contains some multi var calc stuff +LA
misty gorge
i always put the hyphen in the wrong place
hollow oyster
you could try reading a non-proof based linear algebra book
Strang?
gusty falcon
Yeah strang has a bit of proof
misty gorge
which were great
gusty falcon
the first linear algebra book i read was schaums notes
Oh
gusty falcon
Apostol has some good proof based clac book
I see
Is that the "in between"
also is spivak the "In between" too?
hollow oyster
Yes
gusty falcon
yes to both questions?
hollow oyster
also is spivak the "In between" too?
there are some books which are more friendly intro to RA but their difficulty can be considered as in between
hollow oyster
yes to both questions?
yes
gusty falcon
there are some books which are more friendly intro to RA but their difficulty ca...
that’s intro to RA?
i mean spivak
or is it a bit lower level than that
don’t really want an intro to RA book cuz no proof writing experience but yeah
hollow oyster
i get 2 courses in 1 book 😭
It's a great book
hollow oyster
don’t really want an intro to RA book cuz no proof writing experience but yeah
that's fair then again some people have their first intro to proofs by RA or LA
gusty falcon
that's fair then again some people have their first intro to proofs by RA or LA
what makes spivak hard?
gusty falcon
It's a great book
yeah i think i’ll work through that 
obtuse wing
A triangle ( \triangle ABC ) is inscribed in a circle with a radius of 4. The length of side ( AB ) is 6. Side ( BC ) has a length of ( 4\sqrt{3} ) and is the longest side of the triangle.
Calculate the length of side ( AC ) of triangle ( \triangle ABC ). Why can't I use the law of cosines right away?
midnight plankBOT
A triangle \( \triangle ABC \) is inscribed in a circle with a radius of 4. The ...
grand pondBOT
john3515
obtuse wing
although i think it would only help if Δ was 0
midnight plankBOT
@obtuse wing Has your question been resolved?
obtuse wing
.close
paper whale
midnight plankBOT
spring wave
Do you know stars and bars?
distant moat
What did I do wrong?
midnight plankBOT
What did I do wrong?
distant moat
grim vector
Why would it be wrong ?
grim vector
,w dy/dx = (x+1)/y
grand pondBOT
grim vector
Well wolfram agree also
leaden hemlock
,w dy/dx = (x+1)/y
yo when ur done can u pls help
grim vector
You're welcome
distant moat
.close
midnight plankBOT
leaden hemlock
sorry
grim vector
You opened a channel
leaden hemlock
i didn’t see the channel
grim vector
Its ok
distant moat
.close
midnight plankBOT
Available help channel!
sullen wasp
just a question I thought of: If an event happens 10 times every day (at the same times) what is the average time a person would have to wait for the next event? is it just 24/10 hours? or could it be more complicated cause a person could have to wait for the next day for the next event
midnight plankBOT
just a question I thought of: If an event happens 10 times every day (at the sam...
radiant roost
bleak berry
just a question I thought of: If an event happens 10 times every day (at the sam...
it would be more complicited
imagine all 10 events happened right after each other
then the expected wait time would be 24/2
bleak berry
imagine all 10 events happened right after each other
as in basically the same time
midnight plankBOT
@sullen wasp Has your question been resolved?
midnight plankBOT
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Available help channel!
tidal turret
midnight plankBOT
@tidal turret Has your question been resolved?
midnight plankBOT
@tidal turret Has your question been resolved?
warped horizon
can you translate?
is it find all vectors such that the image of these vectors is 1,3,2?
and find a first?
tidal turret
Let $B = {(-1, 2, 1), (1, 1, 0), (1, 0, 0)}$ be a basis of $\mathbb{R}^3$, and let $f : \mathbb{R}^3 \to \mathbb{R}^3$ be the linear transformation such that
[
M_{BE}(f) =
\begin{pmatrix}
a & 2 & 1 \
1 & 2 & 1 \
2 & 2 & 1
\end{pmatrix}.
]
Find $a \in \mathbb{R}$ such that $f(1, 2, -1) = (1, 3, 2)$, and for the value of $a$ obtained, find all vectors $\mathbf{v} \in \mathbb{R}^3$ such that $f(\mathbf{v}) = (1, 3, 2)$.
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
tidal turret
can you translate?
Here is the translation sir
grand pondBOT
tidal turret
,w {{a,2,1},{1,2,1},{2,2,1}} * {{0,0,1},{0,1,-2},{1,-1,3}} * {{1},{2},{-1}}
grand pondBOT
warped horizon
Okay yeah so a = 3
tidal turret
Ye
tidal turret
Okay yeah so a = 3
Look, if i multiply M_BE(f) with M_EB(id) i get M_EE(f) and then i multiply by (1,2,-1)
You see what i mean
warped horizon
Yes yes I understand
M_BE(f) is the matrix representation for f in the basis B
so for the normal basis, first multiply by the inverse of the basis B to get your vector in terms of B's basis vectors
then multiply by M_BE
as your final transformation
so now you need to solve this matrix equation
tidal turret
Look, if i multiply M_BE(f) with M_EB(id) i get M_EE(f) and then i multiply by (...
And what is M_BE(id)?? That's the vectors of B placed as columns in a matrix!!!
Problem what is M_EB(id)?? That's the inverse of M_BE(Id)
,w {{3,2,1},{1,2,1},{2,2,1}} * {{0,0,1},{0,1,-2},{1,-1,3}} * {{x},{y},{z}}
grand pondBOT
warped horizon
$$ \begin{pmatrix}
3 & 2 & 1
1 & 2 & 1
2 & 2 & 1
\end{pmatrix} \cdot
\begin{pmatrix}
0 & 0 & 1
0 & 1 & -2
1 &-1& 3
\end{pmatrix} v =
\begin{pmatrix}
1
3
2
\end{pmatrix} $$
grand pondBOT
vehnil
tidal turret
$$ \begin{pmatrix}
3 & 2 & 1 \
1 & 2 & 1 \
2 & 2 & 1
\end{pmatrix} \cdot
\begin{pmatrix}
0 & 0 & 1 \
0 & 1 & -2 \
1 &-1& 3
\end{pmatrix} v =
\begin{pmatrix}
1 \
3 \
2
\end{pmatrix} $$
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
warped horizon
ah do they just not show up in text
weird
this is what you need to solve right?
tidal turret
Ye
tidal turret
this is what you need to solve right?
v is a generic vector in R3
,, \begin{cases} x + y + 2z = 1 \ x + y = 3 \ x + y + z = 2 \end{cases}
warped horizon
okay yeah
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
warped horizon
best bet is to multiply the two matrices together
and then do rref with an augmented matrix to find the entries of v
and seems like you already have the product of the mtrices together
tidal turret
,w rref {{1,1,2,1},{1,1,0,3},{1,1,1,2}}
grand pondBOT
warped horizon
set either x_1 or x_2 as the free variable and solve
we know that x_3 = -1
x_1 + x_2 = 3
tidal turret
- x1 + x2 = 3
- x3 = -1
(x1,x2,x3)=(3-x2, x2, -1) = x2(-1,1,0)+(3,0,-1)
warped horizon
perfect
but you probably want to parametrize with a different variable like t
(x1,x2,x3) = (-1,1,0)t +(3,0,-1) for t \in R
warped horizon
its good work!!
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viral dagger
where did i go wrong?
midnight plankBOT
where did i go wrong?
viral dagger
(shut up i did not just write A^2C^2 LMAO)
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@viral dagger Has your question been resolved?
modern sapphire
Uhh, how did you get AC*BC = 72?
modern sapphire
where did i go wrong?
36 is the area of ABD, not ABC.
modern sapphire
Yea, I kinda expected that youd get negative since you got the hypotenus AB as 6.
Like, AB has to be longer than 12
viral dagger
i mean is it not
modern sapphire
no?
Due to the area being 36, theres a hard limit that BC<6
since AD gotta be larger than 12
Since AB is the hypotenuse, it has to be larger than 24
and that skews the ratio of AB/BC
Lets say AD = 18. So, we have BC = 4, and the angle bisector thm makes AB = 6, which is way less than AC.
So, AD gotta be even longer. Lets say AD = 36. That makes BC = 2, and angle bisector thm makes AB = 6 (still)
So no matter how long you make AD, the hypotenus is fixed to be 6, its not gonna grow any longer
Maybe, you can try to find a lowest possible area for ABD such that the problem has a solution
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gusty birch
Hi! $2^n$ plays a fundamental role in discrete calculus, similar to $e^x$ in infinitesimal calculus, it is it's own derivative.\
When solving linear homogeneous \textit{difference} equations do we use $2^{rn}$ instead of $e^{rx}$?
midnight plankBOT
Hi! $2^n$ plays a fundamental role in discrete calculus, similar to $e^x$ in inf...
grand pondBOT
woomy
gusty birch
i.e, are the solutions to linear homogeneous difference equations of the form:
$$C_1 2^{r_1 n} + C_2 2^{r_2 n } +... + C_k 2^{r_k n}$$
Where $r_k$ are the roots of the characteristic equation for the DE
grand pondBOT
woomy
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@gusty birch Has your question been resolved?
gusty birch
<@&286206848099549185>
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@gusty birch Has your question been resolved?
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@gusty birch Has your question been resolved?
gusty birch
Specifically I need help solving this:
I know the answer and if my above assumption is correct then its pretty much solved.
(of course the solution being the Binet's Formula and the sequence in question being the fibonacci sequence)
floral dune
assuming deltas mean differentiation.
the characteristic equation you look for is r^2 + r - 1 = 0
the two roots of this will give you the solution such that f = c_1e^r_1x + c_2e^r_2x
which leads to c_1cos(x) + c_2sin(x)
with the initial conditions you will find the coefficients c_1 and c_2 and you are done
gusty birch
The deltas are differences not derivatives
Which are the discrete version of derivatives
floral dune
correction cos(r_1x) and sin(r_2x)
if the function is continuous and differentiable you can take its laplace transform that gives the characteristic equation
gusty birch
Thats what my question is about, if i can treat differences the same as derivatives when solving ODEs
floral dune
all I know is this
glass dome
i.e, are the solutions to linear homogeneous difference equations of the form:
$...
no, it's still e^rx, right?
wait no i'm dumb
gusty birch
My thought process for it becoming 2^rx is that since 2^x is its own difference, the reasoning we go through to build the solution for linear homogeneous ordinary differential equations should be the same, but instead of derivatives we use differences
glass dome
ok ok ok
so if you have uhhhh
ok so let's take the fibonacci sequence
$f_n = f_{n-1} + f_{n-2}$
grand pondBOT
Kaisheng21
glass dome
then that's m^2 = m + 1, so m^2 - m - 1 = 0
let m_1 and m_2 be the roots
then the general solution turns out to be $Am_1^n + Bm_2^n$ where $A$ and $B$ are arbitrary constants
grand pondBOT
Kaisheng21
gusty birch
oh so i dont exponentiate anything?
glass dome
yeah it just works out
gusty birch
no i mean, the roots get exponentiated but they arent in the exponents
glass dome
sure
gusty birch
unlike for ODEs
grand pondBOT
Kaisheng21
gusty birch
yea makes sense but its just weird that it kinda just works out lol
wait so
And now i just use the initial conditions to find out C1 and C2
glass dome
yes
gusty birch
**Kaisheng21**
this seems different though
cause i have the first and second difference, you have different terms of the sequence
gusty birch
f_n-1 is the n-1th term of the sequence no?
glass dome
can't you just turn the differences into the different terms of the sequence?
it'll end up as some recurrence relation in the end
gusty birch
okay yes sorry i was typing to write down my though process
yess thankyouuu!!
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tidal turret
The measures of angles $A$ and $B$ are both positive, integer numbers of degrees. The measure of angle $A$ is a multiple of the measure of angle $B$, and angles $A$ and $B$ are complementary angles. How many measures are possible for angle $A$?
midnight plankBOT
The measures of angles $A$ and $B$ are both positive, integer numbers of degrees...
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
glass dome
The measures of angles $A$ and $B$ are both positive, integer numbers of degrees...
what have you tried
tidal turret
don't know where to start
tidal turret
can you help me
glass dome
can you help me
write what's in the question as equations
tidal turret
A = 180-B
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
A= k(B -180)
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
<@&286206848099549185>
spring wave
(K+1)B=180
tidal turret
(K+1)B=180
how
spring wave
A+B=180
tidal turret
<@&286206848099549185>
midnight plankBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
midnight plankBOT
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
tidal turret
As a helper, please do not give out answers that could be copied as a homework s...
Have the student work THROUGH the problem and guide them ALONG the wae
bleak berry
B(k+1)=180
from here use the fact that k is an integer
tidal turret
from here use the fact that k is an integer
king, can you guide me through
bleak berry
it might benifit you to write that k = (180/b) - 1
since k is an integer, the right hand side must also be an integer
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
it might benifit you to write that k = (180/b) - 1
why? how did you arrived to that
bleak berry
why? how did you arrived to that
i divided both sides by b and then subtracted 1
bleak berry
B(k+1)=180
from here
tidal turret
oh ye
tidal turret
since k is an integer, the right hand side must also be an integer
so?
b has to be a factor of 180? or what
bleak berry
b has to be a factor of 180? or what
just make sure to subtract 1 because b =180 gives k = 0
tidal turret
,w divisors of 180
grand pondBOT
tidal turret
you can now just count the factors of 180
18 divisors
tidal turret
just make sure to subtract 1 because b =180 gives k = 0
wdym
B(k+1)=180
(k+1)=180/B
k = 180/B - 1
k = (180/b) - 1
tidal turret
just make sure to subtract 1 because b =180 gives k = 0
so 17 divisors?
bleak berry
yeah
tidal turret
answer is 17 then
bleak berry
wdym
i meant that literally plugging in b = 180 would give A = 0 which ig works but probably not counted in the actual problem
bleak berry
answer is 17 then
yes
tidal turret
i meant that literally plugging in b = 180 would give A = 0 which ig works but p...
measures are positive integers
bleak berry
oh ok
bleak berry
ohh noo
uhm
oh i see now
a and b complementary means a + b = 90
not 180
you can do the excact same steps with that now
whoops mb i should have noticed before
tidal turret
, w divisors 90
grand pondBOT
tidal turret
k = (90/b) - 1
bleak berry
great
mint ravine
How would I solve this question:
midnight plankBOT
How would I solve this question:
fallow scarab
mint ravine
ok
drifting root
midnight plankBOT
drifting root
from part a to part b, im just wondering how they relate to each other for c
cause these are two different graphs, normally when doing trapezoidal dont you compare the same graph to its exact value?
also im wondering why im differentiating y=xlnx
i dont see what relevance is has to b
nvm i understand it know, i hate this question
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last slate
midnight plankBOT
last slate
Happy new year!
last slate
For c
Idempotent mean M^2 =M
So
M^2 - M =0
Which gives us M= 0 matrix
And M= identity matrix
sharp coral
matrix algebra is not quite the same as real number algebra in this respect
when you solve polynomials like that, you are implicitly using the fact that if ab = 0, then a = 0 or b = 0. but that's not true of matrices
last slate
when you solve polynomials like that, you are implicitly using the fact that if ...
Yeah
They gave an example on that
But I thought the matrix is similar to the real number algebra in case of polynomial solving
:p
Found an example
1 1
0 0
sharp coral
well it gives you some solutions (since it's still true that any matrix * 0 = 0) but not necessarily all of them
last slate
Is there a way to find an example without trial and error?
twilit field
Is there a way to find an example without trial and error?
solve a bunch of equations
like $\begin{bmatrix} a&b\c&d \end{bmatrix} \cdot \begin{bmatrix} a&b\c&d \end{bmatrix} = \begin{bmatrix} a&b\c&d \end{bmatrix}$
grand pondBOT
math_rocks
last slate
That's gonna take a lot of effort 
twilit field
so solve $\begin{bmatrix} a^2+bc& ab+ bd \ ca+dc & cb + d^2 \end{bmatrix}= \begin{bmatrix} a&b\c&d \end{bmatrix}$
grand pondBOT
math_rocks
twilit field
you can also exploit the fact that $det(A) = 0$ or $1$
grand pondBOT
math_rocks
last slate
Ah
sharp coral
a notable example of idempotent matrices are projections (which you will probably learn later)
last slate
Det polynomials follow real number algebra
last slate
a notable example of idempotent matrices are projections (which you will probabl...
Okay
Thanks cloud,wai
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grand pondBOT
last slate
Can someone check it please
Not sure about h
It is true for a 2x2 matrix
But how to prove it for an nxn matrix?
sharp coral
there is a property connecting transpose and inverse
grand pondBOT
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last slate
Are the rest of the right ?
sharp coral
the last one is sus
last slate
Lop
Lol
$(A+B)^{-1}= A^{-1} + B^{-1}$
And since A inverse and B inverse exist ,A+B whole inverse also exists?
sharp coral
where did you get that formula?
grand pondBOT
sharp coral
it's not in the book for a reason
grand pondBOT
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twilit field
Could someone explain the proof of the bolzaon-weistrass theorm
midnight plankBOT
Could someone explain the proof of the bolzaon-weistrass theorm
twilit field
specifically this part
Let $(a_n)$ be a bounded seqeunce so that there exists $M>0$ satisfying $\abs{a_n} \leq M , \forall n \in \N$ bisect the closed interval $-[M,M]$ into two half intervals at 0. Now it must be that atleast one of thse intervals constains an infinite number of terms in the sequence $(a_n)$
why is this
grand pondBOT
math_rocks
crude galleon
OK
twilit field
$-M \leq a_n \leq M$
grand pondBOT
math_rocks
crude galleon
If you got it why are you still here?
twilit field
for all $n \in \N$
grand pondBOT
math_rocks
steel tendon
If you got it why are you still here?
Be nice lol
twilit field
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slim eagle
why is it wrong
midnight plankBOT
why is it wrong
slim eagle
to calculate no of toss needed for n on a dice like this
vagrant flume
you forgot $- \left ( \frac{5}{6} \right )^n}$ on the left
grand pondBOT
Slayer05
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
midnight plankBOT
@slim eagle Has your question been resolved?
slim eagle
you forgot $- \left ( \frac{5}{6} \right )^n}$ on the left
y?
dreamy lichen
to calculate no of toss needed for n on a dice like this
like n would be the solution to that equation?
dreamy lichen
why is it wrong
Oh i see what you're trying to do
the RHS is the probability that you get a given number after at most n tosses
slim eagle
yea
dreamy lichen
yea
well, is that probability ever gonna be 1?
You could e.g. keep tossing 1s and never toss a single 6
no matter how many times you toss the dice, you cant be certain that the given number appeared
the expression on RHS will only approach 1
slim eagle
so how do i recitfy the approach
dreamy lichen
wdym?
What exactly are you trying to calculate?
You could e.g. modify your question to be
"After how many tosses can I be x% certain that the number appeared on the dice"
or "How many tosses will I need on average to get the given number"
midnight plankBOT
@slim eagle Has your question been resolved?
slim eagle
or "How many tosses will I need on average to get the given number"
this
midnight plankBOT
@slim eagle Has your question been resolved?
pastel tree
"Determine the value d such that the plane
$x+y+z=d$ tangents hyperbolid $x^2+y^2-z^2=1$"
midnight plankBOT
"Determine the value d such that the plane
$x+y+z=d$ tangents hyperbolid $x^2+y...
grand pondBOT
Merineth 🇸🇪
pastel tree
Could someone help me with this one ?
When doing exercises such as these, should i be thinking in 2d to make it easier for me to understand what to do?
pastel tree
So it would be looking something like this? Where the red line is the plane and the blue circle is the paraboloid
twilit field
yes
pastel tree
And to my understanding, to solve this exercise it has something to do with gradients, normalvalues and them being parallel?
So we start with getting the gradient for the paraboloid like you mentioned
$\grad f(x,y,z) = (2x,2y,-2z)$
grand pondBOT
Merineth 🇸🇪
twilit field
cool
pastel tree
So the gradient of the paraboloid gives vectors in the entirety of the space such as the purple arrows, and they point in the direction of the fastest rate of change. And someone said it's always away from the paraboloiid.
But we are only interested in the green arrow, right=
pastel tree
So we need to find the corresponding x,y,z values that match the intersection between the plane and the paraboloid, right?
Are my concepts correct so far? Like in my understanding?
twilit field
I'd instead say, consider a point (a,b,c) on the paraboloid, The equation of the plane is then $z - \sqrt{a^2+b^2-1} = 2a(x-a)+2b(y-b)$
grand pondBOT
math_rocks
twilit field
Are my concepts correct so far? Like in my understanding?
I think so, yes
pastel tree
**math\_rocks**
I don't understand how you got this
That seems to be wrong
Considering that isn't mentioned in the answer sheet
twilit field
given a curve, do you remember how to find the tangent plane?
I'm using that method
pastel tree
My understanding here is that i'm trying to find a vector that is paralell to the normal vector of the plane
$\grad f || (1,1,1)$
grand pondBOT
Merineth 🇸🇪
twilit field
sure, that works too
grand pondBOT
Merineth 🇸🇪
pastel tree
The book says it's like this
I don't understand why the k?
Could this be the reason?
That when we solve for x,y and z giving us :
$x = 2/k\y=2/k\z = -2/k$
grand pondBOT
Merineth 🇸🇪
pastel tree
That it could be any of the green dashed lines but we are only interested in the blue dashed line?
twilit field
yes, only in the blue dashed line
twilit field
To be honest, I'm not comfortable with this method, best wait for someone else
humble torrent
This is pretty straightforward if you’ve aware of how to express a tangent to a surface
In your case the surface is described by x^2 + y^2 - z^2 = 1
humble torrent
My understanding here is that i'm trying to find a vector that is paralell to th...
You pretty much already said it here
That’s essentially what I was gonna say
But phrased like this
Let F(x,y,z) = x^2 + y^2 - z^2
Then F(x,y,z) = 1 describes ur surface
And gradF(a,b,c) • (x-a, y-b, z-c) = 0
Is the tangent at the point (a,b,c)
pastel tree
Let F(x,y,z) = x^2 + y^2 - z^2
^So this describes the paraboloid but it doesn't have a fixed location in space
Then F(x,y,z) = 1 describes ur surface
This describes both the paraboloid in space and also it's location?
Would that be right?
humble torrent
Yea sorta, think of it in a dimension lower, when you have say f(x,y) = x^2 + y^2 then we can describe certain curves through the level curves of f
The same thing goes for our F but instead F = 1 is a specific level surface
pastel tree
I honestly can't remember too much from Linear Algebra. But if i took the dot product between two vectors and they = 0, that meant that they are paralell to each other?
humble torrent
That they are orthogonal to eachother
humble torrent
Yeah
pastel tree
Okok so we take the vector from the gradient
dot it with the normal vector of the plane
and should be = 0
humble torrent
It’s alot easier to visualize in a lower dimension imo, but yeah basically that
pastel tree
And gradF(a,b,c) • (x-a, y-b, z-c) = 0
I'm not entirely sure i understand this tho
humble torrent
Since our F can only be graphed in 4D it can be quite tricky to make that jump
Well let’s think about it in a dimension lower
Say we had f(x,y) = x^2 + y^2
And we were looking at the curve f = 1
pastel tree
ok
humble torrent
Now what does grad f represent with respect to its level curves?
pastel tree
Well the gradient of that function would represent every point in space with a vector which points towards the fastest rate of change?
I still have no idea what level curves are
Someone tried explaining it to me yesterday for lik 6 hours
humble torrent
Are you Swedish?
humble torrent
Har du hört talas om nivå kurvor?
pastel tree
Nej, allt måste man lära sig själv.
Det enda jag vet är att om vi har en karta och "kapar" en figur horisontellt då är linjerna som skapar figuren nivå kurvorna?
humble torrent
Japp
f = 1 t.ex är ju snittet som görs med planet z = 1 och ytan som f utgör i grafen
pastel tree
Could this be the reason?
Det är lite så jag försöker tänka när jag arbetade med denna figuren
Ja, så vi kapar "kartan" när z = 1
humble torrent
Yes
Detta kallas då för en nivå kurva till f
Speciellt så säger då gradienten av f något mer
Att för varje punkt på nivå kurvan så är gradienten vid denna punkt vinkelrät/ortogonal mot kurvan
Kan det kännas rimligt?
pastel tree
Ja det känns nog okej.
För då menar du dom blå streckade pilarna, tex?
om då den blå circkeln var våra nivåkurvor till x^2+y^2
humble torrent
Precis
pastel tree
Okej det låter 100% rimligt
humble torrent
And gradF(a,b,c) • (x-a, y-b, z-c) = 0
Härligt, då om vi tittar tillbaka till detta
På motsvarande sätt här kan vi säga att
grad f(a,b) • (x-a, y-b) = 0 är elevationen för tangenten till kurvan vid punkten (a,b)
pastel tree
punkten a,b. Är det punkten där dom tangerar?
humble torrent
Om vi t.ex använder samma f som innan och tittar på kurvan f = 1,
så kan vi t.ex kolla på punkten (0,1), detta är en punkt på kurvan.
Tangenten vid den punkt till kurvan blir då
(2*0, 2*1) • (x - 0, y - 1) = 0
Dvs.
2(y-1) = 0,
Dvs.
y = 1, vilket är ju rimligt!
pastel tree
Mm men jag förstår fortfarande inte varför du tar x-a och y-b i den andra vektorn
humble torrent
Vi kan tänka det som att vi förskjuter koordinat systemet så att vårt origo är punkten (a,b)
Eftersom vektorer ju alltid börjar från origo
pastel tree
Um det visste jag inte..
Det blir lite för mycket nya koncept
Jag lär mig främst genom visuellt, jag kan inte bara anta att saker och ting börjar vid origo därav (x-a, y-b) därav osv...
Jag förstår att den gröna pilen är den vi söker. Den är parallell med planets normalvektor. Så jag måste ju börja med att hitta den vektorn
humble torrent
Okej låt oss säga så här, tänkt att vi var intresserade av att hitta tangenten till en kurva vid (0,0)
Låt oss säga att kurvan faktiskt går igenom här
humble torrent
Nej?
pastel tree
Va jo?
Ganska säker på att någon förklarade samma sak igår
Om den har mer än 1 point of intersection
då går ju planet igenom hyperboloiden vilket betyder att den inte är tangent emd den
humble torrent
Jo visst det är ett sätt att göra det på, men det kan väldigt lätt bli knepigt
pastel tree
Jo kanske det men det är väl den enda sättet jag förstår
humble torrent
Låt oss säga att kurvan faktiskt går igenom här
Det var därför jag försökte förklara det på annat här
humble torrent
Detta låter ju jätte rimligt?
Ingen stoppar dig ifrån att försöka på detta dock
Då det verkar vara väldigt rimligt för dig
pastel tree
ja men problermet är att jag har suttit med denna uppgfiten i 15 h
och ingen kan förklara på ett sätt som är förståeligt
alla ska försöka med sitt egna sätt
humble torrent
Så hur vill du ha det? Lära dig eller sitta och sucka?
pastel tree
Vad menar du sitta och sucka?
humble torrent
Vi försöker vårt bästa
pastel tree
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tidal turret
1.- Let $S = \langle (1, -1, -1, 0), (-1, 1, 2, 1) \rangle$, $T = { x \in \mathbb{R}^4 \mid x_1 + x_2 - 2x_3 + x_4 = 0 }$, and
$W = { x \in \mathbb{R}^4 \mid x_1 - 3x_3 + x_4 = 0 }$.
Find, if possible, a scalar $a \in \mathbb{R}$ and a linear transformation $f : \mathbb{R}^4 \to \mathbb{R}^4$ such that
$f(1, 0, 0,-1) = (a, 1, 1, -3)$ and $f(S) \oplus f(T) = W$.
grand pondBOT
938c2cc0dcc05f2b68c4287040cfcf71
twilit field
start by finding a basis of T
tidal turret
x1 = -x2+2x3-x4
(x1,x2,x3,x4)=(-x2+2x3-x4,x2,x3,x4)
x2(-1,1,0,0)+x3(2,0,1,0)+x4(-1,0,0,1)
T=<(-1,1,0,0),(2,0,1,0),(-1,0,0,1)>
tidal turret
start by finding a basis of T
x1= 3x3 -x4
(x1,x2,x3,x4)=(3x3-x4,x2,x3,x4)
x3(3,0,1,0)+x2(-1,1,0,0)+x4(-1,0,0,1)
W=<(3,0,1,0),(-1,1,0,0),(-1,0,0,1)>
midnight plankBOT
@tidal turret Has your question been resolved?
midnight plankBOT
@tidal turret Has your question been resolved?
grim vector
<@&268886789983436800>
loud dragon
Oops wrong channel.
midnight plankBOT
worthy wing
<@&268886789983436800>
I am curious now 
grim vector
I am curious now <:uponthewitnessing:1204923968096706581>
It was a random roblox gif xd
midnight plankBOT
Available help channel!
worthy wing
It was a random roblox gif xd
Thx i needed to know for some reason 