all ik are functions and solving for x
#help-49
latent kite
last slate
Ohh ok
night hawk
8 - 10 are all correct.
latent kite
and factoring
😭
latent kite
im taking 2 online classes this summer prey for me
latent kite
i tried taking an online remedial math class last summer and had to withdraw
idk what compelled me to do that
last slate
Ohh
latent kite
omg i just realized this is the help channel 💀
last slate
HELP
latent kite
im so sry for wasting ur time dawg
latent kite
8 - 10 are all correct.
this guy said u did it all correct tho so ur prolly fine
last slate
OH MY GOD
Wait I have question cause I only knew a bit so I also did 12-16 and I’m more lost
Sob sob sob
😿 😭
night hawk
- ✅
night hawk
- ✅
night hawk
- ✅
last slate
YAY
night hawk
- ✅
night hawk
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midnight plankBOT
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west lily
the twisted cubic curve is isomorphic to P^1, but P^1 has degree 1 and the twisted cubic has degree 3. What's the explanation?
midnight plankBOT
the twisted cubic curve is isomorphic to P^1, but P^1 has degree 1 and the twist...
west lily
Also, the degree genus formula gives a relationship between genus and degree. Does that mean isomorphic varieties can have different genuses?
midnight plankBOT
@west lily Has your question been resolved?
twilit field
Quick question, what exactly am I supposed to show here
midnight plankBOT
Quick question, what exactly am I supposed to show here
twilit field
so I consider an arbitrary basis for V and W?
prime hornet
yeah
twilit field
I was wondering if rank -nullity can be appplied here
I think it can
formalising it may take a while though
subtle blaze
i would like to start with a basis of imT then extend that to W
but that wouldn't be all bases of W
prime hornet
like idk how i'd do this
think about the rank of the matrix rep, I think 
twilit field
I was wondering if rank -nullity can be appplied here
so this is wrong?
prime hornet
well it might not be wrong, but it’s not what first jumped to my mind
twilit field
idk, I stupidly avoided matrices like the plague in LA1.
I resorted to linear maps instead
to prove stuff even related to matrices
subtle blaze
i mean matrices are kinda ass
lavish venture
you can let j = dim range T and select j linearly independent vectors from the list Tv_1, … , Tv_n
subtle blaze
everytime you want to right one down you need to pick a basis
twilit field
you can let j = dim range T and select j linearly independent vectors from the l...
yea, something like that
lavish venture
you can let j = dim range T and select j linearly independent vectors from the l...
you’ll get j = dimrange T nonzero columns because of linear independence they obviously are all nonzero
subtle blaze
ah that's clever too
prime hornet
that’s pretty much it, yeah
subtle blaze
instead of pivots
prime hornet
hence my comment on rank
lavish venture
i don’t think axler even covered rank at this point in the book
twilit field
you’ll get j = dimrange T nonzero columns because of linear independence they ob...
hmm, yea, makes sebse
*sense
prime hornet
Axler does all sorts of weird choices imo
twilit field
I'm not sure if I should use FIS or axler atp imo
lavish venture
axler
twilit field
for LA 2
lavish venture
i like axler
twilit field
I love axler too
prime hornet
as long as you stuck to one, it’s fine
twilit field
but we do dets early on in the course , so 
lavish venture
i’ve opened FIS and it felt too computational
twilit field
an axiomatic defn
prime hornet
my preference isn’t with Axler, but I think he’s okay
twilit field
i’ve opened FIS and it felt too computational
yea, my problem too
my course reccs axler AND FIS
subtle blaze
an axiomatic defn
how
prime hornet
det is the unique multilinear alternating blah blah blah
FIS has a section on this, iirc
twilit field
okie, will get to axler during the summer I guess, FIS during the sem
Thanks
Will hopefully finish axler over the summer
prime hornet
idk if it’s really necessary to go over both
prime hornet
just start learning other math and pick up what you need along the way 
twilit field
yeah, but when I study dets I suppose I'll use FIS
other than that, axler
I mean I have like 5 LA books with me
💀
lavish venture
we would all still be doing basic shit if we needed to master everything perfectly
subtle blaze
det is the unique multilinear alternating blah blah blah
$det(v_1, ..., v_n) = \bigwedge dx^i(v_1, ..., v_n)$ this one?
grand pondBOT
frosst
twilit field
Axler, FIS, Kumarasan, pooole and planning to get halmos soon
subtle blaze
you dont need to cover LA in such detail or you will spend all your waking ours doing LA
lavish venture
but why
prime hornet
$det(v_1, ..., v_n) = \bigwedge dx^i(v_1, ..., v_n)$ this one?
this one
lyric charm
I mean I have like 5 LA books with me
that is 4 books too many 💀
lavish venture
😭
prime hornet
I want to do my thesis on LA
why
lyric charm
thesis on linalg is sussy
twilit field
thesis on linalg is sussy
UG thesis
lyric charm
linalg will be a tool not an end in itself no?
prime hornet
nobody in their right mind reads 5 books on linear algebra
twilit field
linalg will be a tool not an end in itself no?
fair, something LA adjacent
subtle blaze
this one
this requires you to pick a basis then you need to show this definition of det is well-defined
lavish venture
nobody in their right mind reads 5 books on linear algebra
linear algebra is a chore
twilit field
linear algebra is a chore
idk, axler makes it fun
lavish venture
like you’re supposed to learn it so you can learn fun things
subtle blaze
^^that's what i mean by tool
prime hornet
this requires you to pick a basis then you need to show this definition of det i...
this is the actual theorem given
you prove its existence by showing that it’s given by that formula
lavish venture
idk, axler makes it fun
but you said you’re using FIS?
lavish venture
axler >
prime hornet
does it really matter that much?
twilit field
I just started revising LA 1, for which I used axler
lavish venture
nope
subtle blaze
what the fuck even is LA1 and LA2
twilit field
does it really matter that much?
My worry is dets
twilit field
what the fuck even is LA1 and LA2
2 courses in LA
subtle blaze
yeah but LA is LA
lavish venture
was the first course not proof based?
twilit field
was the first course not proof based?
it was
lavish venture
🤔
twilit field
but we only covered until the basics of spectral theorm
prime hornet
this is like saying you read Munkres for topology, but then are worried about the manifolds perspective, so you're gonna pick up Lee too and relearn topology from a different perspective
subtle blaze
good on you my uni after 3 semester still haven't covered that
twilit field
now we'll cover diganolisation, factorisation etc
subtle blaze
and won't 🙂
prime hornet
this is like saying you read Munkres for topology, but then are worried about th...
I feel like this is just a waste of your time
lavish venture
spectral theorem was like one of the last things we did
twilit field
and end with quadratic forms
twilit field
spectral theorem was like one of the last things we did
we didn't do a proof
of spectral
lavish venture
applied math probably
subtle blaze
nope
lavish venture
🤔
prime hornet
frosst has said quite a few things about his uni's courses in the past
subtle blaze
this is the actual theorem given
is it this
prime hornet
I don't know if I recognize this definition 
my stuff with dets is super rusty though
subtle blaze
we've comandeered wai's channel whoops
lavish venture
aren’t you doing abbott too?
twilit field
Though trying to do it without row reduction now
twilit field
aren’t you doing abbott too?
That's for RA
lavish venture
That's for RA
i know but you’re going through multiple at once
twilit field
I can explain
basically I want to cover everything we will do during the sem by myself first
for multiple reasons
- I can focus more on the details in the sem
- Gives me a head start, allows me to relax more during the sem
lavish venture
school is indeed much easier when you know everything ahead of time
or at least have been exposed to it
twilit field
yea, which is what I'm doing rn
subtle blaze
basically I want to cover everything we will do during the sem by myself first
this is very good
all my courses are very easy now because i spent a lot of time in first year getting the general picture when i audited all the classes
i got a lot more time to spend on the specifics and details of the coursework when i actually took them for credit
twilit field
Thoough going to be a bit hard doing AA, probability , RA1 and LA 2 over the summer
subtle blaze
i just sat in the classes
it gives you the correct path of "general idea" of the content
if you follow a book you'll go down a lot of rabbit holes that the course won't
lavish venture
Thoough going to be a bit hard doing AA, probability , RA1 and LA 2 over the sum...
probability sucks tbh
lavish venture
💀
not enough theory in the intro books
i always got bored trying to read blitzstein
twilit field
Here, I'm going to start off by showing that if all entires are 1, then the rank is 1.
\
If All entries are 1, we can row reduce it. to eliminate all rows but the first , and similalry column reduce to only get the first entry as 1. This clearly has a rank of 1, we're done
lavish venture
the exercises just felt like high school
after measure theory maybe it will be interesting
lavish venture
next fall 🙏🏻
subtle blaze
not enough theory in the intro books
that's not the fault of probability
twilit field
Here, I'm going to start off by showing that if all entires are 1, then the ran...
How does this sound
prime hornet
idk measure theory still
prime hornet
Here, I'm going to start off by showing that if all entires are 1, then the ran...
that probably works...
I think you can honestly do something simpler though? 
if all the entries in the matrix rep of T are 1, then T(v_1) = T(v_2) = ... = T(v_n) = w_1 + ... + w_m; in that case, they're all the same vector w in W, so range T = span{T(v_1), ..., T(v_n)} = span{w}, which has cardinality 1 
so dim range T = 1
I think that would be enough
though somebody can correct me if I'm messing this up
twilit field
that probably works... <:pikathink:879066872513970227>
I think you can honestl...
That's even better
Thanks
Now for the other direction
hhmm
If dim(Range)=1 , the span is 1 D, so a matrix with all entires 1 is one possibility?
midnight plankBOT
@twilit field Has your question been resolved?
twilit field
<@&286206848099549185>
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cursive swan
Is this true?
e = 2 + 1/6 + 1/4! + 1/5! + 1/6! .....?
midnight plankBOT
Is this true?
e = 2 + 1/6 + 1/4! + 1/5! + 1/6! .....?
lilac sage
,w e = 2+ sum from n=3 to infty of 1/n!
grand pondBOT
misty gorge
Is this true?
e = 2 + 1/6 + 1/4! + 1/5! + 1/6! .....?
where are you getting these beginning terms from? you can use the taylor series for e^x to get a true statement that looks similar to what you have.
cursive swan
wait
grand pondBOT
cursive swan
Yes, is it that:
e = 2 + 1/2! + 1/3! + 1/4! + 1/5! + ......
misty gorge
you should have a 1 in the beginning
cursive swan
what
1 + (2 + 1/2! + 1/3! +....)?
Or 1 + 1/1! + 1/2! + 1/3! + .......................?
thanks
y'all
.close
midnight plankBOT
midnight plankBOT
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tidal turret
what would be the difference between a function from like calculus, f(x)=e^x for example, and a linear transformation?
midnight plankBOT
what would be the difference between a function from like calculus, f(x)=e^x for...
nova yoke
a linear transformation is a particular kind of function between vector spaces that satisfies f(ax + by) = af(x) + bf(y)
where a,b are scalars
sharp coral
"transformation" and "function" are essentially synonyms here
so we could call them "linear functions" instead and it would mean the same thing
subtle blaze
what would be the difference between a function from like calculus, f(x)=e^x for...
Functions from calculus are usually analytic, or smooth, or k-times differentiable or at least continuous, with ideas of limits and sequences that have more to do with metric and topological spaces, as opposed to vector spaces which can be independent of a metric or topology
lyric charm
what would be the difference between a function from like calculus, f(x)=e^x for...
linear transformations have in general a vector space as their domain and codomain
rather than R
tidal turret
this is all great answers, i appreciate it
is it true that in calculus all functions are like, endomorphisms? what I mean is like R->R or C->C, or is it possible the domain is restricted but the image not or something? am I making any sense
nova yoke
endomorphism implies "morphism" of some kind (generally homomorphism)
most calculus functions are not homomorphisms
lyric charm
is it true that in calculus all functions are like, endomorphisms? what I mean i...
only if you are extremely category theory pilled
and call a function from a set to itself an "endomorphism in Set"
which is stupid
tidal turret
Take the modulus function
what would be the domain and codomain of modulus function? by modulus function you mean absolute value right? so R -> Z+?
nova yoke
not Z+
Z = integers only
the codomain can be any set containing all the possible values of the function
so any set containing [0,infty)
tidal turret
ok, só in calculus not all functions are like f: K->K
it could also have a restricted codomain/image
tidal turret
and call a function from a set to itself an "endomorphism in Set"
i just dont know how else to call it when f : K -> K
nova yoke
there may be exceptions, but in calculus usually the codomain is just taken to be all of R (to the extent that anyone even talks about codomains in calculus classes)
the domain may or may not be all of R, depending on the function
tidal turret
ok i appreciate it a bunch, sorry for the cat theory jargon
nova yoke
no problem!
lyric charm
i just dont know how else to call it when f : K -> K
function from a set to itself
midnight plankBOT
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twilit field
We can split this into two cases. Case 1: v_1 is in the kernel, in which case the entire first column would be 0( as the vector is mapped to 0). In the Second case, it doesm
midnight plankBOT
We can split this into two cases. Case 1: v_1 is in the kernel, in which case ...
twilit field
.close
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stray drum
Whats wrong about this to get y? Im somehow missing a c?
midnight plankBOT
Whats wrong about this to get y? Im somehow missing a c?
midnight plankBOT
@stray drum Has your question been resolved?
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tidal turret
can someone explain the geometric intuition behing determinants calculating the volume of a paralelelepipid
midnight plankBOT
can someone explain the geometric intuition behing determinants calculating the ...
cursive swan
what is @paralelelepipid@?
trail sphinx
@tidal turret
tidal turret
tidal turret
Try drawing a parallelogram with vectors
runic hamlet
you should first try to understand why the determinant is the area of a parallelogram in R^2
in particular, what do row operations mean geometrically
afterwards this translates nearly immediately into R^3
tidal turret
how is the determinant the area of a parallelogram for vectors in r2
what?
is like all the possible linear combinations of the vectors or what
i dont really get it tbh
runic hamlet
tidal turret
I dont follow, what is ad-bc
where are the vectors
there is something that I am missing here
runic hamlet
the formula ad-bc screams determinant
tidal turret
I think thats for r2
tidal turret
,, \begin{pmatrix} a & b \ c & d \end{pmatrix}
grand pondBOT
renato
runic hamlet
the columns are (a,c) and (b,d)
tidal turret
trace?
tidal turret
the columns are (a,c) and (b,d)
let me do the drawing
tidal turret
https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/Area_parallelogram_as_...
yep but I still dont follow how is the area of the parallelogram ad-bc
maybe I am lacking some parallelogram formulas
base times height
runic hamlet
I am not claiming that its obvious that the area of this parallelogram is ad-bc
the key observation is what happens when you do a row operation
geometrically
draw the parallelograms for $\begin{pmatrix} 1 & 3 \ 2 & 8 \end{pmatrix}$ and $\begin{pmatrix} 1 & 3 \ 0 & 2 \end{pmatrix}$
grand pondBOT
Denascite
runic hamlet
what did the row operation do geometrically
tidal turret
**Denascite**
runic hamlet
hmm ok maybe not the best vectors
tidal turret
when we do a row operation, depending on the operation the determinant changes or maybe it doesnt
for example adding a multiple of one row to another row, doesnt change the determinant
but swapping rows, changes the determinant by a factor of -1
midnight plankBOT
@tidal turret Has your question been resolved?
novel sedge
hi, what is your question? 😄
runic hamlet
for example adding a multiple of one row to another row, doesnt change the deter...
and what are those row operations geometrically
why do they not change the area
midnight plankBOT
@tidal turret Has your question been resolved?
midnight plankBOT
tidal turret
.reopen
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twilit field
#1260937723452330155 message Can I have some help formalising this
midnight plankBOT
https://ptb.discord.com/channels/268882317391429632/1260937723452330155/13751021...
flat spire
wai
fresh sparrow
wai
twilit field
hi
midnight plankBOT
@twilit field Has your question been resolved?
twilit field
<@&286206848099549185>
solid iris
为? 
twilit field
Oh
lethal path
为? <:hmmcatfone:849254980812013639>
喂
solid iris
oops
lethal path
solid iris
im tired i just got back to my hotel
lethal path
nice
twilit field
.close
spice scroll
if W1,W2,U are all sub vector spaces of V and i pvoed that W1(+)W2=V, then can i just intersect both LHS and RHS with U and get the statement in the pic?
midnight plankBOT
if W1,W2,U are all sub vector spaces of V and i pvoed that W1(+)W2=V, then can i...
spice scroll
or is it not enough
i dont get why they put the intersection in each () on RHS instead of just once as a distributive thing
midnight plankBOT
@spice scroll Has your question been resolved?
midnight plankBOT
solid iris
.reopen
solid iris
if W1,W2,U are all sub vector spaces of V and i pvoed that W1(+)W2=V, then can i...
r u able to prove the claim?
midnight plankBOT
Available help channel!
tidal turret
if I have a line that is entirely contained in a plane. how would you know that the direction of the line is orthogonal to the normal of the plane?
midnight plankBOT
if I have a line that is entirely contained in a plane. how would you know that ...
blissful pier
if I have a line that is entirely contained in a plane. how would you know that ...
if the dot product of two vectors is zero, the vectors are orthogonal
pure wraith
if I have a line that is entirely contained in a plane. how would you know that ...
You can use the direction cosines of the line and the normal vector of the plane
What was said above
sharp coral
definitionally the normal of the plane is orthogonal to all vectors lying in the plane, including the direction vectors of the line
tidal turret
definitionally the normal of the plane is orthogonal to all vectors lying in the...
yes, but how?
vagrant crest
definitionally the normal of the plane is orthogonal to all vectors lying in the...
yeah the definition of the normal is literally it being orthagonal to the plane, therefore lines within the plane aswell
vagrant crest
yes, but how?
are you asking how the normal is orthogonal to the plane?
pure wraith
Try drawing/imagining one that isn't
tidal turret
like all the vectors present in the plane are orthogonal to the normal of the plane
tidal turret
Try drawing/imagining one that isn't
hmm
lyric charm
how is the normal of the plane orthogonal to all the vectors inside it
thats kinda the point of a normal lmfao
subtle blaze
how is the normal of the plane orthogonal to all the vectors inside it
What does it mean for a vector to be normal to a plane
vagrant crest
all the vectors inside the plane are well make up the plane, they are all parallels to the plane
tidal turret
What does it mean for a vector to be normal to a plane
good question, idk i don't have a good geometric intuition
tidal turret
all the vectors inside the plane are well make up the plane, they are all parall...
wdym?
subtle blaze
We conventionally define a normal vector to be one that is orthogonal to every vector in the plane
Now you’re asking why they are orthogonal
tidal turret
We conventionally define a normal vector to be one that is orthogonal to every v...
why tho?
subtle blaze
Well, by definition…
subtle blaze
why tho?
Well because it’s something we want to talk about a lot
A normal vector to a plane is used a lot so we want to name it so we have a shortcut to say “a vector with so and so properties”
vagrant crest
wdym?
u can visualize it like this
if u draw a normal to the plane the lines will naturally have the same normal
tidal turret
How do you define it then?
good point hahaha
tidal turret
A normal vector to a plane is used a lot so we want to name it so we have a shor...
wdym "normal vector to a plane"?
vagrant crest
if u draw a normal to the plane the lines will naturally have the same normal
u have to be sure of the definition of a normal tho lol..
subtle blaze
I mean a vector with the property that it is orthogonal to every vector in the plane
Is a vector we like to use often
tidal turret
the normal of the plane
tidal turret
well i just call it normal vector of the plane
subtle blaze
It’s just a name
subtle blaze
A name just means it is used often enough we want a shortcut to say it
tidal turret
ok
subtle blaze
Being “parallel” is not a good idea to think about
tidal turret
How do you define it then?
is just hard to find an definition about it online
subtle blaze
It’s not
A normal vector to a plane is one that is orthogonal to every vector on the plane
tidal turret
https://en.m.wikipedia.org/wiki/Normal_(geometry)
it doesn't mention it's orthogonal to all the vectors lying in the plane
subtle blaze
In three-dimensional space, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P.
A plane is its own tangent plane
Well, not really but that’s not important here
That’s technically not true by some wacky ass weird thing you don’t have to worry about at all
subtle blaze
A plane is its own tangent plane
But just pretend its true
whole dove
Hey anyone here does MUN’s
midnight plankBOT
Hey anyone here does MUN’s
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
tidal turret
what i don't really understand is why you guys say "definationally the normal of the plane is orthogonal to all the vectors lying in the plane"
like WHAT definition wtf or is it geometric intuition
graceful flume
what i don't really understand is why you guys say "definationally the normal of...
Imagine a plane and all the vectors on it
pure wraith
what i don't really understand is why you guys say "definationally the normal of...
Do you know the vector equation that represents a plane
grand pondBOT
jewels!
graceful flume
Look at this normal to the plane and image the vectors on this plane
Like the ones already drawn
tidal turret
i only know the cartesian equation of the plane, vector eq for the plane hasn't been covered in my class
sharp coral
i really don't know how you could define "this vector/direction is perpendicular to a plane" other than "it's perpendicular to every line in the plane"
since angles are by default defined between two lines or vectors
tidal turret
yeah i see what you guys mean now
if the normal is orthogonal to the plane is orthogonal to every line or vector inside it, otherwise it wouldn't be a normal vector of the plane because is not orthogonal to the plane
tidal turret
This diagram illustrates it as well
yeah i appreciate the geometric intuition i think i get it now, well idk i think it's more of an geometric intuition or rather a consequence of the definition of normal of the plane (vector that is orthogonal to the plane)
whatever, i didnt attend class when planes where covered i think i was ill
but i will recheck the definitions in the reader, i appreciate everyone who contributed
tidal turret
i really don't know how you could define "this vector/direction is perpendicular...
thanks, i appreciate it
tidal turret
Do you know the vector equation that represents a plane
well wasn't covered in my class but this looks like an interesting explanation to it
i appreciate it
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tidal turret
midnight plankBOT
pure wraith
Can you translate it
steel crest
Let
L₁: X = λ(0, −1, 1) + (4, 2, 1)
L₂: X = λ(0, 1, 2) + (4, −2, −4)
and A = (4, 2, 1).
Find a point B on L₁ and a point C on L₂ such that triangle ABC is right-angled at A.
pure wraith
Gave up? 
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
Gave up? <:KEK:586240877358350341>
why would you?
tidal turret
I would start by shifting coordinates so that A is the origin
wdym?
tidal turret
Let
L₁: X = λ(0, −1, 1) + (4, 2, 1)
L₂: X = λ(0, 1, 2) + (4, −2, −4)
and A = (4,...
I appreciate the translation, ty
steel crest
wdym?
I mean change the coordinate system so that instead of looking for a right triangle at A, you look for a right triangle at O=(0,0,0)
you need a coordinate system change that doesn't change angles, of course
pure wraith
why would you?
He was LaTeXing and eventually settled on plain text, that's what I meant
steel crest
oh lol
steel crest
I mean change the coordinate system so that instead of looking for a right trian...
think of it like
if you start with a circle:
$$(x-3)^2+(y+1)^2=11$$
then you can define new coordinates
$$x'=x-3$$
$$y'=y+1$$
$$x'^2+y'^2=11$$
grand pondBOT
gfauxpas
steel crest
now you have a circle in a cooridnate system such that it's centered at 0, it makes the math easier
and once you find the solution in the changed coordinate system, shift iti back to the original one using
$$x=x'+3$$
$$y=y'-1$$
grand pondBOT
gfauxpas
steel crest
does that make sense
tidal turret
but we havent covered circles, there is a simpler path that we are missing
steel crest
I'm just illustrating the principle, doesnt matter that it's a circle
the point is, by a change of variables, you can make the problem into an easier one
you want a coordinate system such that A becomes the origin, so
$$T(A)=O'$$
grand pondBOT
gfauxpas
tidal turret
you want a coordinate system such that A becomes the origin, so
first we need to find the intersection of this two lines L1 and L2 that is exactly B
steel crest
first we need to find the intersection of this two lines L1 and L2 that is exact...
easier to find the intersection of the two lines L1' and L2' that is exactly B' in the new coordinate system
how about you try it my way, then try doing it the other way, and that will help you see why my way is easier
or is your question, how do you know that the intersection in the new cooridnate system will correspond to the intersection in the original coordinate system?
pure wraith
affine 🔥
If you don't want to, you could just bash through in the current coordinate system
steel crest
If you don't want to, you could just bash through in the current coordinate syst...
sure
pure wraith
But the idea of moving the origin will help you
steel crest
it's just ... harder
pure wraith
I like it
tidal turret
why will it help? changing the coordinates system?
tidal turret
how about you try it my way, then try doing it the other way, and that will help...
sure, I just dont follow
pure wraith
Ease
steel crest
sure, I just dont follow
I could explain it, but, I think moreh elpful than an explanation will be doing it my way, and trying to do it the other way, and seeing what complications there will be
feeling more than reading will give you an intuition of WHY I'm giving this advice
spice scroll
r u able to prove the claim?
I mean it's pretty immediate isn't it? Since intersection of W1,W2 is only the zero vector, so you can intersect the stuff on RHS like shown with U. If you have a proof/contradiction then show me
steel crest
WLOG is why you CAN choose the origin. The reason to choose the origin is because things being 0 simplifies math
tidal turret
ok so A = (0,0,0) then?
steel crest
T(A)=(0,0,0)
tidal turret
T?
steel crest
$$A = \left( \begin{array}{c}
4 \
2 \
1
\end{array} \right)
\quad \Rightarrow \quad
T(A) = O' = \left( \begin{array}{c}
0 \
0 \
0
\end{array} \right)$$
grand pondBOT
gfauxpas
steel crest
T?
a change of coordinates
a linear map
$$\begin{aligned}
x' &= x - 4 \
y' &= y - 2 \
z' &= z - 1
\end{aligned}$$
grand pondBOT
gfauxpas
steel crest
this is like the circle example I gave, i'm just using the symbolism of linear algebra
the change of a vector into a new coordinate system is a linear map (why?)
wait no
it's AFFINE not linear
durp
so maybe misleading to use the symbol "T"
since T is usually used for linear mappings, my apologies
tidal turret
but
steel crest
tidal turret
what you are saying is that for B i will subtract 4 in x1 subtract 2 in x2 and subtract 1 in x3
for B = (x1,x2,x3)
pure wraith
It's just a substitution that makes the job easier for us
steel crest
and you have to do it to every point that you're analyzing
did you take basic physics yet? did you learn about choosing a frame of reference?
pure wraith
You don't have to worry about the geometric intuition, which is shifting the origin
tidal turret
did you take basic physics yet? did you learn about choosing a frame of referenc...
i did
but i didnt passed
steel crest
yeah it took me 3 tries to pass physics, I had to drop out twice, I had horrible profs , but anyway
Imagine I tell a physics student
tidal turret
but I know what FoR is
steel crest
to choose a coordinate system to drop a ball off a building
if I'm going to ask them questions about the process of dropping the ball, it might be natural to put the origin at the top of the building
if im going to ask about what happens when it hits the ground, it might make sense to put the origin at the bottom of the building
subtle blaze
Do you know how to find the angle between 2 lines
steel crest
but it's the same process, just a question of, what should I call the starting point?
and here, the point of interest is A, so I want that to be the starting point
subtle blaze
Because I would do that and travel 1 unit along 1 line from the intersection, and cos(θ) along the other direction
And those would be my 2 points
pure wraith
"travelling" would be hard computationally
subtle blaze
It’s not
tidal turret
guys I swear maybe u guys are overcomplicating it
steel crest
guys I swear maybe u guys are overcomplicating it
I still think my idea is the easiest way to do it 
tidal turret
yes but idk physics
subtle blaze
If you start at a point on the line eg the intersection then you can just add some copy of (0,-1,1) or (0,1,2)
steel crest
i was just giving an illustration of why you can choose a frame of reference based on convenience
steel crest
If you start at a point on the line eg the intersection then you can just add so...
if the intersection is at the origin!!!
steel crest
it doesnt matter in principle,. but in computation it does
otherwise it becomes thorny
subtle blaze
It’s the same as doing an affine shift to re-centre but the exact formulation is not particularly insightful
steel crest
anyway, I'mg oing to play skyrim, gl
tidal turret
anyway, I'mg oing to play skyrim, gl
gl dragonborn!!
tidal turret
It’s the same as doing an affine shift to re-centre but the exact formulation is...
guys using affine knowledge is using advanced machinery for this simple problem
pure wraith
he made a substitution lol
thats all it is
dont get bogged down in what its called
but sure, you can go ahead and work as is
tidal turret
but if we translate A to the origin why is it easier?
calculations are simpler, but why?
tidal turret
you're gonna have to solve linear equations
i mean only for finding the intersection of the lines L1 and L2, only there
or maybe I am missing something 🤔
pure wraith
Assume B is governed by a parameter t, and C is governed by a parameter s
You're gonna need to solve BA.AC = 0
tidal turret
You're gonna need to solve BA.AC = 0
oh that part too
tidal turret
Assume B is governed by a parameter t, and C is governed by a parameter s
no
B is an exact point, the L1 n L2
C is parametrized yeah
but B is NOT
tidal turret
You're gonna need to solve BA.AC = 0
yeah but A and B we know
tidal turret
we only lack C
pure wraith
Fixed
tidal turret
.
Let L1 : X = λ(0,-1,1)+(4,2,1) , L2 : X = λ(0,1,2)+(4,-2,-4) and A = (4,2,1)
Find a point B that is in L1 and in L2, and a point C ∈ L2 such that ABC is a right triangle at A
tidal turret
"Hallar un punto B que pertenezca a L1 y a L2"
pure wraith
And apply the dot product condition on C insead
pure wraith
Yeah my bad
tidal turret
Yeah my bad
I think maybe this is why lad proposed a change of coordinates, the translation was incorrect
novel sedge
hi, first find the interesection of the lines. that point is your point B. then use the fact that prependicular vectors have 0 dot product to find point C 😄
tidal turret
because the problem is very simple
tidal turret
hi, first find the interesection of the lines. that point is your point B. then ...
yeah
tidal turret
hi, first find the interesection of the lines. that point is your point B. then ...
ok, let's do the number crunching shall we?
novel sedge
yep!
pure wraith
good luck
novel sedge
yes, that looks correct 😄
tidal turret
yes, that looks correct 😄
novel sedge
the method is correct, and the answer is also looks good !
tidal turret
the method is correct, and the answer is also looks good !
ty mate, exercise looked complicated but after you dismantle by parts is not bad mate
tidal turret
the method is correct, and the answer is also looks good !
where you from mate?
novel sedge
glad to help 😄 make sure you can do the dismantling on your own
<@&268886789983436800> this seems to be a scam (fake top level domain, steamcommunity spelled incorrectly) 
tidal turret
<@&268886789983436800> this seems to be a scam (fake top level domain, steamcomm...
i appreciate the preciseness
midnight plankBOT
@tidal turret Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
slow thorn
is the lower statement really always true?
midnight plankBOT
is the lower statement really always true?
midnight plankBOT
midnight plankBOT
Available help channel!
gray zenith
How do I show the realtion between them? I was able to prove that they are dependent but I couldn't show u in terms of v or vice vesa.
midnight plankBOT
How do I show the realtion between them? I was able to prove that they are depen...
gray zenith
If someone could just give me a hint, that would be great.
hard umbra
check out the tangent angle sum formula
gray zenith
check out the tangent angle sum formula
thanks, lemme try that.
tan u = tan (x+y/1-xy)
but I'm not sure how to go any further
this does look very much like what I have but
gray zenith
check out the tangent angle sum formula
how would I get all those x, and y with tan?
gray zenith
check out the tangent angle sum formula
I also found this formula. I'm not sure how this was derived but by any chance did you mean for me to use this one instead of the one I mentioned earlier?
hard umbra
well theyre consequences of each other
gray zenith
well theyre consequences of each other
I see. I haven't come this far in my trig lessons so I didn't know that.
hard umbra
in [ \tan(\alpha + \beta) = \f {\tan\alpha + \tan\beta} {1 - \tan\alpha\tan\beta} ] simply $\tan^{-1}$ both sides and use $\alpha = \tan^{-1} x$, $\beta = \tan^{-1}y$
grand pondBOT
gray zenith
ohh. yes that makes sensee.
wait dam how did you get the latex to render white
is that in the settings?
hard umbra
yea
gray zenith
I should be able to solve it now. let me get another crack at it
midnight plankBOT
@gray zenith Has your question been resolved?
gray zenith
<@1089928213871993054> Has your question been resolved?
alomost there
need another min. my math slow :-0
yep got it.
thanks @hard umbra
.close
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
if Null(S)=Null(T), then if S=ET, E has to be injective, right
midnight plankBOT
if Null(S)=Null(T), then if S=ET, E has to be injective, right
twilit field
Here's my proof for that
S(v)=ET(v). Let v not in Null(S), but T(v)\in Null E, we then have S(v)=0, which is a contradiction
nah, this "proof" is wrong, is it not
oh, existance
E is the identity matrix
Does that work
.close
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
Prove $\sum_{i=1}^{n} (-1)^{i} \binom{n}{i}= 0$
\
Proof :We can split this sum into $\sum_{i=1}^{n} \binom{n}{2i} - \sum_{i=1}^{n} \binom{n}{2i+1}$
\
Now I wonder if shifting the index of the second sum will quickly allow me to conclude it's zero
midnight plankBOT
Prove $\sum_{i=1}^{n} (-1)^{i} \binom{n}{i}= 0$
\\
Proof :We can split this sum ...
grand pondBOT
wai
pure wraith
I mean
dusty portal
Damn 😔
pure wraith
its the binomial expansion of (1 - 1)^n = 0
dusty portal
Prove $\sum_{i=1}^{n} (-1)^{i} \binom{n}{i}= 0$
\\
Proof :We can split this sum ...
Can we not use binomial?
dusty portal
Now you're damned 😔
fiery plover
Now you're damned 😔
fr
twilit field
its the binomial expansion of (1 - 1)^n = 0
thanks
polar star
what happens when n->inf
fiery plover
I genuinely find Axler's LA easier
sheldon?
fiery plover
That's miles easier than this
if you like proofs
LA is gonna be easier for you
but
twilit field
if you like proofs
I LOVE proofs
fiery plover
if you dont
pure wraith
isn't that an indeterminate form
No 0^n is always going to be 0
twilit field
0^{\infty} is indeterminate
dusty portal
Wut
fiery plover
0^{\infty} is indeterminate
said who
polar star
0^{\infty} is indeterminate
sums of partial sums is 0
dusty portal
1^{\infty} is as well
fiery plover
0^x as you take x large will be 0
dusty portal
Ugh what is the compiled list of indeterminate forms
polar star
,w indeterminate forms list
grand pondBOT
Requested by _mia_34
fiery plover
L
dusty portal
Oh well
twilit field
pure wraith
calculus for ants, volume 1
twilit field
twilit field
okay, i'll eat and then get back to combi
Can't wait to do RA / LA tomorrow
Thanks a lot again everyone!
fiery plover
Can't wait to do RA / LA tomorrow
what do you abreviate as RA
twilit field
what do you abreviate as RA
Real analysis
twilit field
Easier than this fr
fiery plover
Easier than this fr
ok then
search up combinatorial proofs
youll love
actually
just do graph theory for real
twilit field
just do graph theory for real
3rd year elective
fiery plover
also where did you even get such a question from
twilit field
also where did you even get such a question from
This? A probability book
Sheldon ross
fiery plover
3rd year elective
well you can do olympiad graph theory problems until then
midnight plankBOT
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Available help channel!
jade jackal
i thought the final result would be
$$(arcsin(x^2))^2 * 1/4$$
midnight plankBOT
i thought the final result would be
$$(arcsin(x^2))^2 * 1/4$$
grand pondBOT
Snek
slender walrus
yes, thats what it should be. (and don't forget the +C)
jade jackal
yes, thats what it should be. (and don't forget the +C)
oh so its the writers mistake
slender walrus
yes
pure wraith
it looks like a plus in the denominator lol
fallen sparrow
it looks like a plus in the denominator lol
Yes exactly
jade jackal
it looks like a plus in the denominator lol
wait shit youre right lmao
slender walrus
the first line of work implies its intended to be -
jade jackal
Yes exactly
im sending a complaint to the proff telling him to compencate for the time i wasted solving this
hopefully he raises my grades
or expells me
either way im happy
midnight plankBOT
@jade jackal Has your question been resolved?
solemn silo
How do I get k<=9 from this
midnight plankBOT
How do I get k<=9 from this
solemn silo
?
pure wraith
nvm, rational root theorem
gaunt nimbus
solemn silo
Then why did alexheinsis say that at the finale
pure wraith
is k only natural numbers
pure wraith
okay makes sense then
You can lowkey guess and check
with the IVT
since its only naturals
solemn silo
Nah I want correct proof
pure wraith
IVT is valid
pure wraith
Ah this is calculus
I mean
Ehehe
Basically if you have a continuous function f : R --> R, such that f(x) < 0 and f(y) > 0, then there must exist a root of f between x and y
So you show how it's positive for k = 10
And negative for k = 9
midnight plankBOT
@solemn silo Has your question been resolved?
midnight plankBOT
@solemn silo Has your question been resolved?
solemn silo
<@&286206848099549185>
fallow scarab
Then why did alexheinsis say that at the finale
if you're gonna ask about someone's answer, you should just provide the link
shrewd tusk
How do I get k<=9 from this
by induction?
spring wave
||for x>= 10 (8x^3 + 16x^2 +16x)/x^4 = 8/x + 16/x^2 + 16/x^3 <= 0.8 + 0.16+0.016 = 0.976 < 1
hence (8x^3+16x^2+16x) < x^4 for x > 10, hence the polynomial is positive||
idk how to give hints for this
dont open spoiler
midnight plankBOT
@solemn silo Has your question been resolved?
novel sedge
How do I get k<=9 from this
rewrite this as $$k^4 + 1\leq 8k^3 + 16k^2 + 16k$$ The right hand side is a cubic, while the left is quartic. both are polynomials with positive integer coefficients. so just by looking at the degrees, there exists an $N$ such that for all $n \geq N$, we have $n^4 + 1 > 8n^3 + 16n^2 + 16n$. you can check that for $n = 9$, the inequality does not hold true but for $n=10$, it does. and by the degree, you can conclude that this is true for all $n\geq 10$, so $k\leq 9$
grand pondBOT
fastrack_and_backtrack
novel sedge
the general idea is that a polynomial of higher degree with positive integer leading coefficient will outgrow a polynomial of lower degree as we approach +infinity 😄
solemn silo
Guys i did it
I rewrited it so lhs would have only k^4, and then I multiplied all by 1/k3 , and it gives smth true only if k>=9
.close
gusty falcon
hi, probably stupid question
midnight plankBOT
hi, probably stupid question
gusty falcon
but what does it mean for a system to be "solvable"
is it the same distinction as consistent and inconsistent?
nova yoke
like a linear system of equations?
yea it means the same as consistent, i.e. a solution exists
gusty falcon
like a linear system of equations?
yeah
Well i've seen this
It's kinda informal but 😭
that makes no sense, right?
simple field
i think better language would be "a unique solution exists"
simple field
i wouldnt deep it too much tbh
if theyre saying "its all very sad" in their answers lmao
gusty falcon
lolll
😭
yeah the class isn't really a formal math class
but things become very hazy
when the definitions aren't precise
so now i'm left wondering what "solvable" entails
i'm guessing i just go with conventions here?
like uh solvable -> unique or infinite solutions
nova yoke
that makes no sense, right?
who wrote this? it's totally wrong
gusty falcon
and unsolvable -> no solutions
simple field
yeah that makes the most sense
gusty falcon
who wrote this? it's totally wrong
yeah 😭 but also the other students haven't learnt lin alg
so ig maybe it's a pedagogy thing
idk yeah
nova yoke
that system is solvable, it's just that the second equation doesn't introduce any new solutions nor does it remove any
gusty falcon
indeed
i mean ig it's fine to define solvable your own way
but unless i'm missing something
the two images contradict themselves, right?
simple field
yeah
gusty falcon
yeah so ig
simple field
but again, not too big of a deal
gusty falcon
that's the crux of the issue perhaps
nova yoke
yes they do, they're basically the same scenario with slightly different numbers
gusty falcon
but again, not too big of a deal
right yeah
simple field
like i doubt youre ever going to be asked to define solvability
gusty falcon
yes they do, they're basically the same scenario with slightly different numbers
correct
simple field
it should always be clear from context what theyre asking you for
so just go with the flow
midnight plankBOT
midnight plankBOT
Available help channel!
nova furnace
guys so d is suppose to be tan theta but im having trouble understanding how to get there
midnight plankBOT
guys so d is suppose to be tan theta but im having trouble understanding how to ...
sharp coral
,rccw
grand pondBOT
nova furnace
oh yea i forgot to label that one of the triangles the one with the angle theta has a hypotenuse of 1
sharp coral
in terms of soh cah toa, think about what side "d" represents in the triangle
nova furnace
hypotenuse
sharp coral
is it?
nova furnace
im assuming
the triangles are similar and d is tangent to the unit circle and perpendicular to the other hypotenuse which is 1
nova furnace
yes
sharp coral
because it is a triangle that has both side d and angle theta
nova furnace
im assuming yes
sharp coral
which side is the hypotenuse? remember that the hypotenuse is always opposite the right angle
nova furnace
the long ahh line at the bottom
sharp coral
yes, so in that case what does side d represent relative to angle theta
nova furnace
opp
nova furnace
1 im assuming
sharp coral
yes, and what side is relative to theta is that (for this triangle specifically)
nova furnace
uh i dont know maybe adjacent or like cos
but tge thing is that those 2 are similar triangles
sharp coral
but i would just use the big blue triangle here
we don't need to worry about any triangle similarity here. just pure soh-cah-toa on the blue triangle indicated here
i mean you can also do it by similarity but i think that would be more work
nova furnace
i guess but i am watching along for a youtube video and it says the triangles are similar so their ratios gotta be the same and the guy wrote d/1 = sin theta/cos theta
im just having trouble understanding the similarity part and the equation
so then d = tan
sharp coral
let's start by doing it the way i am suggesting and then we can get to the youtube method
nova furnace
like i kind of get it but it feels like i have a concussion
aight bet so whats your suggestion im listening
sharp coral
but i would just use the big blue triangle here
we have a blue triangle marked here, and we know two sides: d and 1
we know that d is the opposite side, and the bottom is the hypotenuse
so side is 1 for this triangle?
nova furnace
wait what rephrase “so side is 1 for this triangle”
sharp coral
like out of opposite, adjacent, or hypotenuse, which one is the side with length 1
nova furnace
adjacent
sharp coral
ok. so we have two known side lengths, d which is the opposite side and 1 which is the adjacent side
nova furnace
tangent is o/a which is d/1
sharp coral
yes
sharp coral
sure
nova furnace
so that long ahh side is suppose to be 1/cos theta which is sec theta im assuming if i got my trig right but how do i get there
sharp coral
well you know that that side is the hypotenuse, and the adjacent side is 1
nova furnace
i dont know the hypotenuse
right now i got (cos theta)+x
cuz from the 2 triangles we dont know the under side from the second one only cos
sharp coral
ok let's say that the bottom side is "x"
so the bottom side x we said is the hypotenuse, and the adjacent side we said is 1
so we can use a similar strategy as before to solve for x
nova furnace
but how is it just x if we also got a cos
imma send another pic to show u what im seeing
why is my shit sideways
,rccw
grand pondBOT
sharp coral
ok you have a different x than me
nova furnace
but remember in the first one since they are 2 triangles we combined into on the unit circle before we got the d and the other traingle we just had the plain triangle w the adjacent side was cos
sharp coral
let's instead call the entire long bottom side "y"
sharp coral
the entire long bottom side "y" is the hypotenuse of the big triangle
nova furnace
solve for y?
sharp coral
from trig using only the big triangle, yes
nova furnace
how do i use trig if its a side do i do inverse
sharp coral
you have a big triangle:
- we have said previously that 1 is the adjacent side
- and d is the opposite side
- and y is the hypotenuse
that is all the information we need for this problem
sharp coral
you have a known side: 1 (adjacent) and an unknown side: y (hypotenuse)
what trig function relates the adjacent side and hypotenuse?
sharp coral
yes
nova furnace
fuh do i do with that info
oh
1/cos theta = y
which is sec
damn thats cool af
imma try to find the other function using this knoweledge
ion know how to do ts
i think itll be easier when i actually reach trig
but thanks cuz u made me get some things
thanks a lot
like fr not sarcastix
how do i close ts
sharp coral
use .close
midnight plankBOT
@nova furnace Has your question been resolved?