yep
#help-49
twilit jetty
lusty python
which solves for $m = -1$ and $n = 1$
grand pondBOT
1 divided by 0 equals Infinity
twilit jetty
yep, thats the only solution
lusty python
ty
midnight plankBOT
midnight plankBOT
Available help channel!
gusty sky
How is infinity one of the answers ?
midnight plankBOT
How is infinity one of the answers ?
wooden badger
wht is the ques?
compact copper
Looks odd
gusty sky
compact copper
ooooo, cengage?
gusty sky
ooooo, cengage?
Yeah
Took 2 days to reach here 😭
So does anyone have any idea why infinity is one of the solutions
gusty sky
that implies the slope is 90 degrees
I know that m=infinity implies 90 degrees but how does solving that equation give m=infinity?
gusty sky
They aren't technically WRONG
By their logic every linear equation has infinity as one of its solutions though!
compact copper
like put m2 = infinity in the first equation after (2) and it makes somewhat sense
wooden badger
it reduces the degree of the equation
gusty sky
it reduces the degree of the equation
Wdym by that ?
then, If a quadratic equation gets reduced to a linear one then infinity is one of the solutions?
wooden badger
then, If a quadratic equation gets reduced to a linear one then infinity is one ...
well if the quad is in m the equation of slope
gusty sky
well if the quad is in m the equation of slope
Umm, so if we were supposed to get 2 slopes but only got 1 at the end then infinity is one of the slopes ?
hard umbra
it's abuse of notation
if instead you consider the equation ax + by - 5/2 b = 0 for the line
you end up with the following equation for a and b
,w (a + 3b - 5/2 b)^2/(a^2 + b^2) = 1
grand pondBOT
hard umbra
in such a case, you arrive at the lines x = 0 and 3x + 4y - 10 = 0
wooden badger
Umm, so if we were supposed to get 2 slopes but only got 1 at the end then infin...
yeah, bcz slope of vertical line is undefined and we can't get to the answer by algebra (someone else might explain better, my internet is horrid tdy)
hard umbra
i mean it is just abuse of notation and when interpreted literally is utter nonsense
one shouldn't set up the equation like that because a vertical line has no representation as y - 5/2 = mx
to set up the line equation that is agnostic to whether it's vertical, you should just use a form like ax + by + c = 0 where there is no problem
midnight plankBOT
@gusty sky Has your question been resolved?
somber saffron
just gave my maths exam
midnight plankBOT
just gave my maths exam
somber saffron
.close
midnight plankBOT
pearl hull
just gave my maths exam
Congrats
lyric charm
just gave my maths exam
midnight plankBOT
Available help channel!
inland patio
In my book there's the following problem:\
Suppose that we have $n$ boxes numbered $1,\ldots, n$, and that the $i$th box contains $r_i$ red balls and $n_i$ black boxes, where $r_i,n_i\geq 1$. Imagine that one chooses a box uniformly at random, and then picks a ball (again at random) in the chosen box. Compute the probability that the $i$th box was chosen knowing that a red box was picked.\
Is there a typo in the last sentence? First the author speaks about red balls, then about red boxes. How do I approach this?
midnight plankBOT
In my book there's the following problem:\\
Suppose that we have $n$ boxes numb...
grand pondBOT
psie
runic hamlet
yes likely a typo
inland patio
yes likely a typo
So should it be red balls in the end instead of red box? Or perhaps black box instead of red box?
runic hamlet
red balls
inland patio
.close
floral ruin
midnight plankBOT
floral ruin
How should I start in this question?
lyric charm
simplify the Re(trig bs)
maybe writing c and s for cos(θ) and sin(θ) could save you some effort
slow thorn
yes
floral ruin
alright
$Re(\frac{2\cos^2{\theta} + 7\iota \sin{\theta}\cos{\theta} -3\sin^2{\theta}}{9\sin^2{\theta} + \cos^2{\theta}})$
grand pondBOT
Prathmesh
floral ruin
ohh
yea
thanks
in this question should I first use that property that z + z' = 2Re(z) and then put z = x+iy or is there any better and short way of solving this ?
slow thorn
in this question should I first use that property that z + z' = 2Re(z) and then ...
for the first section of the problem yeah
if would simplify to $Re\left(\frac{z-1}{2z+i}\right) = 1$
grand pondBOT
slow thorn
and then i guess nothing to do other than z = x+iy
floral ruin
yea
midnight plankBOT
Available help channel!
pallid granite
Can i get a hint about the following integral
midnight plankBOT
Can i get a hint about the following integral
pallid granite
I did complete the square but what i do with the x in the numerator
umbral timber
Can i get a hint about the following integral
P(D') + Q = x, where D' is the derivative of the denominator
P and Q are real numbers
grim vector
x/((x+1)² + 1) = (x+1 - 1)/((x+1)² +1) no ? And then you separate and proceed with u = x+1 for the first integral
umbral timber
then you can separate into two integrals. one of them can be solved via a u-sub and the other by trigonometric sub
cerulean temple
how to find residue of $\frac{1}{(x^2+1)^3} $ at $e^{i\frac{\pi}{2}}$
midnight plankBOT
how to find residue of $\frac{1}{(x^2+1)^3} $ at $e^{i\frac{\pi}{2}}$
grand pondBOT
Slowaq
cerulean temple
solving this limit seems to he hard
$$\lim _{z\to e^{i\frac{\pi}{2}}}\left((z-e^{i\frac{\pi}{2}})^3\frac{1}{(z^2+1)^3}\right)^{''}$$
grand pondBOT
Slowaq
cerulean temple
nvm i can factor out denominator
lethal path
**Slowaq**
oh be careful, you're missing a factor of 1/2! in front
cerulean temple
ah of course thanks for pointing it out!
lethal path
no worries!
cerulean temple
.close
midnight plankBOT
midnight plankBOT
Available help channel!
dawn dagger
,, (P) \begin{cases} \sum_{i=1}^4 z_i \ra \max & \ 3z_1+z_2+4z_3 &\le 1 \ z_1+2z_2+5z_4 &\le 1 \ z \ge 0 \end{cases}
grand pondBOT
midnight plankBOT
,, (P) \begin{cases} \sum_{i=1}^4 z_i \ra \max & \\ 3z_1+z_2+4z_3 &\le 1 \\ z_1...
dawn dagger
can somebody spot my mistake
midnight plankBOT
Available help channel!
mint ravine
Help please
midnight plankBOT
Help please
pure wraith
Have you set an equation up?
mint ravine
like with modular arithmetic?
pure wraith
No just the equations you need to do this
I don't think you'll need any modular arithmetic
lilac finch
mint ravine
ok
mint ravine
Have you set an equation up?
so x+y+z=20 and 108=2x+5y+7z where x, y, and z are positive integers greater than or equal to 1
mint ravine
i heard of it but i forgot what it was
let me search it up rq
ok yeah i remember it now
pure wraith
Do you know how to use it in this case
mint ravine
oh ok
pure wraith
so x+y+z=20 and 108=2x+5y+7z where x, y, and z are positive integers greater tha...
From these two, can you get a simpler equation in 2 variables that we can work with
mint ravine
yeah we can take mod x and mod y for both equations
oh wait or you could just get mod x i think that'll work
mint ravine
yeah i am taking the mod with the smaller coefficient to make it easier
vivid yoke
you can get it down to 1 variable lol, you have 2 equation for 3 variables
Plotting it into 3d graph gives a line
pure wraith
like with modular arithmetic?
I assumed this was what his unit's about
Besides it reduces the tedious casework
vivid yoke
Ig so
midnight plankBOT
@mint ravine Has your question been resolved?
midnight plankBOT
fallow scarab
Jesus christ
sudden yacht
<@&268886789983436800>
fallow scarab
<@&268886789983436800> idk
brittle edge
Don't ping everyone
It’s not that deep bro calm down
sudden yacht
<@&268886789983436800> idk
Same thought 🙈
summer kestrel
Ok
lofty beacon
<@&268886789983436800>
brittle edge
I need help
sudden yacht
I need help
midnight plankBOT
tribal temple
<@&268886789983436800>
(we already have the other two pings, thank you 
)
midnight plankBOT
Available help channel!
chrome swift
hello
dreamy lichen
this set of vectors in r3 are linearly dependent right?
indeed
the third vector is 3*first + 2*second
small jasper
chrome swift
then this is false
midnight plankBOT
sharp coral
then this is false
that is indeed false. it is true only for sets of two vectors
chrome swift
.reopen
midnight plankBOT
✅ Original question: #help-49 message
pure wraith
that is indeed false. it is true only for sets of two vectors
It says "others"?
chrome swift
It says "others"?
naw cloud is right
sharp coral
it's not a scalar multiple of either
it's a linear combination of the others, which would be a more generally true condition
sharp coral
does my answer here work
that works
chrome swift
thanks
a set of more than 2 vectors in R^n is lin dep if at least one of the vectors can be written as a linear combination of the other vectors right?
midnight plankBOT
@chrome swift Has your question been resolved?
pure wraith
a set of more than 2 vectors in R^n is lin dep if at least one of the vectors ca...
Yeah
chrome swift
hi
frank wolf
Yes
chrome swift
if we deal with n by n matrcies it's just always n then right
wraith dirge
if we deal with n by n matrcies it's just always n then right
yeah cause m = n
wraith dirge
the co-domain is smaller than the range right
no, the co-domain is always greater than or equal to the range
chrome swift
so the range is a subspace of R^m
chrome swift
ok
chrome swift
it can be all of R^m when?
chrome swift
identifiy matrix right
hybrid crow
identity matrix from R^m to R^m
but it doesn't have to be identity
you will learn later
it just has to be "surjective"
hybrid crow
idk where you are at in linear algebra
but there is a ton of theory to develop
regarding surjectivity
injectivity
and such
chrome swift
the last thing we learned is single value decomp
hybrid crow
as I've now realized, the order of math curriculum is far from homogenous
chrome swift
we hit IMT, eigenvectors and markov chains
hybrid crow
I haven't hit any of that
chrome swift
yes imt mb
hybrid crow
but I am quite knowledgeable in linear maps
wraith dirge
it can be all of R^m when?
for n x n matrices, the codomain will be the same as the image unless the determinant is zero
and other equivalent conditions
chrome swift
but I am quite knowledgeable in linear maps
ok good
help me out u two ima cover all of linear algebra in 2 hours
let's see what yall know
hybrid crow
is your class proof-based
chrome swift
nope
hybrid crow
oh great
chrome swift
a lotta shit though
hybrid crow
than u probably can cover it all
hybrid crow
well idk if u need to know injectivity and surjectivity then for your test
you tell me
and I can explain them
chrome swift
i don't think we're that theory based
chrome swift
wait
hybrid crow
I guess my version of it is far different from a non-proof based course
wraith dirge
so you've just covered matrix operations, determinants, vector spaces, inner product spaces, eigenvalues, and linear transformations?
chrome swift
does injectivity have different phrases
hybrid crow
one-to-one
chrome swift
so you've just covered matrix operations, determinants, vector spaces, inner pro...
sounds right
i mean not yet
we covered all that in class, but not in my self review session
also we covered a lot of stuff on orthogonality
then some symettrical matrices stuff
and markov chains (im bad at)
wraith dirge
yeah thats typical
chrome swift
ok good
chrome swift
im reviewing everything while doing problems on them
then i'll do some practice tests
still gonna bomb this final but i'll take a B
so image is the specific output vector pretty much
wraith dirge
you should be good, the only "proofs" you will do are just showing something is a vector space, or something is an inner product
everything else is just memorizing how to solve a specific kind of problem basically
chrome swift
you should be good, the only "proofs" you will do are just showing something is ...
yeah exactly this
chrome swift
everything else is just memorizing how to solve a specific kind of problem basic...
we have hard true or false questions
wraith dirge
we have hard true or false questions
like conceptual questions (e.g. definitions, properties)?
chrome swift
yeah pretty much
wraith dirge
could you give an example
wraith dirge
this was my paper
yeah for stuff like this you will also need to remember theorems from your textbook and/or lectures
chrome swift
yeah they're tricky and niche
some of them are easy theorems, others are kinda a stretch on theorems like the 4 sub-spaces one was kinda hard (question 1)
wraith dirge
yeah they're tricky and niche
yeah these problems are pretty hard, like for the first one you kind of need to memorize that its true
wraith dirge
you got 3, 4, and 6 wrong
wraith dirge
oh i misread it nvm
chrome swift
how tf did u know
wraith dirge
some of them you can intuitively think through
chrome swift
i see why 4 is false tho DMANNIT
4 is false because if the vectors are linearly dependent, u can't do gram schmidt
wraith dirge
yeah, its tricky
chrome swift
oh P^2
wraith dirge
it requires you to remeber the definition
but also, this will probably just be one part of your final. since your time is limited, you should focus more on problems that you're struggling with
chrome swift
for sure ima lock in
linear transformations is something I need to master bbecause i don't have a good grasp on it
wraith dirge
what textbook are you using?
chrome swift
not as intuitive as eigenvectors
im using my professors slides who uses the book linear algebra and it's applications 6th edition by Lay
hybrid crow
oh lord
chrome swift
yeah these problems are pretty hard, like for the first one you kind of need to ...
i thought that since W is a m-subspace of R^n, it must be less or equal to n. I know the dim(W) + dim(Wperp) = n, so i just did n-m. does that make sense or is my approach ass
we were given that W = m btw
wraith dirge
i thought that since W is a m-subspace of R^n, it must be less or equal to n. I ...
yeah
chrome swift
wow i explained that bad i edited it
is consciousness something from our brains or a metaphysical thing
@wraith dirge @hybrid crow
professor said this is very powerful
but what does this mean in words?
hybrid crow
is consciousness something from our brains or a metaphysical thing
consciousness is physically located in our brains
and is just like electrical impulses and neurons fired collectively
hybrid crow
<@1239339626410545273> <@531301351272087552>
oh yeah this is op
chrome swift
consciousness is physically located in our brains
i disagree but im not gonna argue that
because i sent this in the wrong chat mb
hybrid crow
this basically means every linear map is uniquely determined by its action on basis vectors
because T is a linear map and has the properties of both additivity and homogeneity
knowing where the basis vectors goes
allows us to derive
where any other vector goes
since every vector is just a linear combination of those basis vectors
chrome swift
naw you lost me hold on
hybrid crow
i'ma give u an example rn
wraith dirge
the map of any vector is a combination of the maps of the basis vectors
geometrically, if you know where the basis vectors are mapped by T, you can figure out where any vector will be mapped
hybrid crow
Suppose $v_1,...,v_n$ is a basis of $V$. Define $T \in \mathcal{L}(V,W)$ by $T(v_k) = w_k$ for each $k = 1,...,n$. Now, let $v$ be an arbitrary vector such that $v \in V$. Since $v_1,...,v_n$ is a basis, we know that $v = a_1v_1+\dots + a_nv_n$ for $a_j \in \textbf{F}$. From this, we know that $T(v) = T(a_1v_1+\dots + a_nv_n) = a_1Tv_1+\dots + a_nTv_n = a_1w_1+\dots + a_nw_n$.
grand pondBOT
holo_morph
hybrid crow
Essentially, this shows that you can define a linear map T just based on its action on basis vectors
then you can "linearly extend" it across the domain
chrome swift
naw you lost me
hybrid crow
to see what it does to any other vector
chrome swift
is V a subspace?
hybrid crow
V is an arbitrary vector space
chrome swift
like is it R^n
wraith dirge
it's like you get the simpliest vectors e1 e2....ep right
bc you can represent any vector as a linear combination of the basis vectors right
hybrid crow
as is W
chrome swift
bc you can represent any vector as a linear combination of the basis vectors rig...
yeah this
let's say u have a mystery transformation matrix right
wraith dirge
since the transformation map is linear, you can also represent the map of any vector as a linear combination of the maps of the basis vectors
chrome swift
you know what the outputs are for e1 and e2
and u wanna find the output for an arbuitrary vector in R^2
you just do the first entry times e1 and second times e2 right
chrome swift
ok
hybrid crow
v = a_1e_1 + a_2e_2
where the a's are scalars
then apply the linear map/matrix to both sides of the equation
Tv = T(a_1e_1+a_2e_2)
now the right side can be simplified
chrome swift
that's essentially the matrix A
hybrid crow
using the properties of a linear map
Tv = a_1Te_1 + a_2Te_2
so we have expressed Tv
as a linear combination
chrome swift
damn ur proofs class got u messed up if u know this shit like the back of ur hand
hybrid crow
of the images of e_1 and e_2 under the map T
hybrid crow
damn ur proofs class got u messed up if u know this shit like the back of ur han...
I know it better than that
chrome swift
are u a TA or something
chrome swift
ok need to grind thanks tho
chrome swift
not even for school ngl
can't lie i heard gpa doesn't matter that much
so idc if I get a b here
hybrid crow
I've never heard that
chrome swift
in college not high school
chrome swift
homogeneity?
hybrid crow
btw Mittens
wraith dirge
in college not high school
it depends what you do after college
hybrid crow
that second property
you've probably seen it used in integrals
integral of 3x = 3 * integral of x
typa deal
or when differentiating
chrome swift
integral of 3x is 3x^2/2 right
wraith dirge
that second property
and the first property
wraith dirge
integral of (f + g) = integral of f + integral of g
hybrid crow
anyways what I am trying to highlight here
is that differentiation and integration are both linear maps
because they follow the properties
and you can even
represent them
as matrices
chrome swift
is this a multivariable thing
cerulean oyster
Property of a linear transformation can be summed up as follows:
$T(a\mathbf v + b\mathbf u) = aT(\mathbf v) + bT(\mathbf u)$
grand pondBOT
cerulean oyster
Just to leave visual reference
chrome swift
thanks
cerulean oyster
You can see that line combines both core concepts.
And V and U are vectors
Always remember that vectors are in reality mostly anything
chrome swift
looked up green's thorem it looks like multivariable is gonna be hard
cerulean oyster
Functions can be vectors, etc...
chrome swift
You can see that line combines both core concepts.
And V and U are vectors
yeah i see it
wraith dirge
looked up green's thorem it looks like multivariable is gonna be hard
you dont need to prove greens theorem or anything in multi
applying it isnt that complicated
cerulean oyster
for calc3, greens and stokes are just a tool
wraith dirge
the problems will be simple unless you have a bs professor
cerulean oyster
if youre doing real analysis tho 
chrome swift
do electrical engineers need to do real analysis
cerulean oyster
Depends on your college and country, but most engineering carreers settle with Calculus instead of Analysis
Luckily.
cerulean oyster
Thats calculus too.
wraith dirge
the theoretical side of electrical engineering definitely needs real analysis, but if you are working as a regular electrical engineer you dont. i would assume you will fall into the latter category
chrome swift
the theoretical side of electrical engineering definitely needs real analysis, b...
couldn't tell you
cerulean oyster
Depends on your college and country, but most engineering carreers settle with C...
They have a lot in common even if they have different scopes, and Analysis is certainly a lot harder since its centered around proving what Calculus has been doing since Newton.
Most techniques sat "unproven" for a while > more like not a formal proof of them actually working
wraith dirge
linear algebra is harder than multi i heard is that true
it depends on the professor you have and how easily you understand the concepts. multi will probably be around the same difficulty though
wraith dirge
multi and differential equations involve a lot of calculus, so it will depend a lot on how well you did in calc 1-2
linear algebra is (mostly) unrelated
chrome swift
ask me a question in calc 1 and 2 real quick
wraith dirge
ask me a question in calc 1 and 2 real quick
uhh integrate 1/(x^3sqrt(x^2 - 1)) from sqrt(2) to 2
chrome swift
wdym by sqrt 2 to 2
wraith dirge
the limits of the integral
wraith dirge
it is
chrome swift
i failed bc but took a placement exam in my uni and passed
but in bc we didn't learn trig sub
wraith dirge
you didnt?
chrome swift
yeah i don't recall it
wraith dirge
im pretty sure thats on the exam
chrome swift
uhh integrate 1/(x^3sqrt(x^2 - 1)) from sqrt(2) to 2
is it secant or sin
wraith dirge
is it secant or sin
secant
chrome swift
naw bro this is kinda hard i'd have to re-learn it
but re-learning is very easy if u learned it once already
wraith dirge
that was just an example problem, i was just saying if you did okay in calc 1-2 you will probably do okay in multi and diffeq
chrome swift
that's good
wraith dirge
differential equations is lowkey easier than multi, a lot of it is just memorizing specific techniques
if you're an ee major, you probably are only required to take those 5 classes (calc 1-3, linalg, diffeq)
depends on the program though
chrome swift
yeah we do need to do all that
chrome swift
midnight plankBOT
chrome swift
this matrix is size 3 by 3
hybrid crow
it is
junior flower
chrome swift
<:AA_Sus:702402047852413009>
does this mean its a dumb question
junior flower
missing some details here
chrome swift
junior flower
also are you sure co-domain is the word you're looking for?
chrome swift
should i say range
junior flower
that seems better yes
chrome swift
i think co-domain is better actually
junior flower
and by 2 and 3 you mean dimension 2 and dimension 3 right
sharp coral
i think co-domain is better actually
i think it would be better to say the codomain of a 3x3 matrix is R^3 regardless of its range
sharp coral
R^3 and R^2
2-dimensional is not the same as R^2
chrome swift
oh so it's range is R^2
junior flower
no
junior flower
its range is like, a subspace of R^3 that looks 2 dimensional (but still made of elements of R^3)
sharp coral
my professor said this
see how in this case, it actually outputs vectors which have 2 components, they are literally in R^2
unkempt sluice
Ax is linear combination of columns of A with components of x being coefficients. So the range is simply the column space
chrome swift
Ax is linear combination of columns of A with components of x being coefficients...
wait this kinda makes sense add more stuff so it clicks
unkempt sluice
Its dimension thus is the dimension of the column space, namely the rank
chrome swift
i gotchu thanks bro
unkempt sluice
Okay, Ax is this
unkempt sluice
Okay
chrome swift
so it's dim(col A) which is basically rank
unkempt sluice
Yes
unkempt sluice
range
sharp coral
range = column space, they are exactly the same
chrome swift
range = column space, they are exactly the same
what? really
junior flower
the co-domain isn't something you compute
sharp coral
well for matrices yes
unkempt sluice
codomain is just what the target of the map is, can be as large as you want.
sharp coral
the codomain is just set of all vectors of the type the matrix spits out. so because the matrix outputs vectors in R^3, its codomain is R^3, even though it will not output every possible vector in R^3
unkempt sluice
When you give a map you have to give a domain and a codomain. The codomain can be anything, image is another thing. Like f: R->C; f(x)=x. Codomain is C but image(f)=R
chrome swift
wait we talking about complex stuff now?
sharp coral
domain and codomain are things that apply to all functions
unkempt sluice
I just gave an example saying that codomain doesn’t mean anything
chrome swift
so i can ignore co-domain
unkempt sluice
Yes
chrome swift
all i need to know is every output vector b is in R^m
unkempt sluice
It’s just the target of a mapping. You have to give it for the sake of definition
Doesn’t imply anything
sharp coral
all i need to know is every output vector b is in R^m
if every output vector b is in R^m, then the most consistent thing to do is to say the codomain is R^m
sharp coral
it's standard for the codomain to be all of R^m because then you don't have to do any calculations, it is evident purely from the matrix's size
unkempt sluice
Put it this way, change your linear map to this: R^3->R^100
x|->Bx, B=(your A, 3 times 97 size 0 matrix). Range is still of dimension 2, codomain is R^100, It doesn’t matter.
Domain and codomain is for the sake of definition. We compute range/image, codomain we leave it behind knowing there is such a thing we don’t need to think about it.
chrome swift
ok i guess i get it
unkempt sluice
(Mapping (x_1, x_2, x_3) to (x_1, x_2, 98 many 0 in the example above)
chrome swift
range and image are the only important things because it's the actual answers we get or outputs we get
unkempt sluice
Yeah
chrome swift
range is the output space i mean
where all the images live
or the span of all images?
unkempt sluice
Same thing, image of a linear map is still a linear space
unkempt sluice
It is correct
chrome swift
but range is bigger than image right
sharp coral
no?
unkempt sluice
Synonym
chrome swift
i thought image is the specific output whereas range is where all the outputs live
unkempt sluice
I treat range as image, it’s like in some fields, people give special names. But image is for all cases
chrome swift
guess im confusing myself. main take away is forget about codomain for this final
sharp coral
well you can say "the image of vector x under T is T(x)" but you also say "the image of T" which is the set of all images of elements of the domain, i.e. the range
sharp coral
guess im confusing myself. main take away is forget about codomain for this fina...
well you should still be able to tell what the codomain of a matrix is given the matrix
just don't overthink its importance
unkempt sluice
Exactly. It’s just something you have to give when you define a map.
chrome swift
ight
this is also important to know right
v= c1e1 + c2e2
Tv = T(c1e1 + c2e2)
Tv = c1T(e1) + c2T(e2)
hybrid crow
i'm proud of you mittens
chrome swift
naw thank you @hybrid crow LOL
sharp coral
as long as the domain of T is 2-dimensional, sure
hybrid crow
i thought image is the specific output whereas range is where all the outputs li...
range of T = image of T, both are subsets of codomain
chrome swift
range of T = image of T, both are subsets of codomain
the same subspace?
hybrid crow
yes
sharp coral
although the math doesn't change much for higher dimensions other than making the linear combination longer
hybrid crow
image can also be used for an individual vector
namely, the image of v under T is Tv
chrome swift
as long as the domain of T is 2-dimensional, sure
ye it is
unkempt sluice
Exactly the same thing. Synonym, literally no difference in my mind, range and image.
chrome swift
although the math doesn't change much for higher dimensions other than making th...
makes sense ye
unkempt sluice
Your case domain is R^3, codomain is R^3, image is isomorphic to R^2, kernel is isomorphic to R
My case domain is R^3, codomain is R^100, image is still isomorphic to R^2, kernel is still isomorphic to R
chrome swift
ok
sharp coral
i would describe the image as being a 2-dimensional subspace of R^3
same thing but i'm not sure isomorphism has been introduced yet
hybrid crow
yeah I don't think mittens knows that
chrome swift
never heard of that
hybrid crow
image is a 2d subspace of R^3
unkempt sluice
Anyway, we think now you get the idea that codomain is not important.
hybrid crow
never heard of that
basic idea of an isomorphism is a link between two things with the same underlying structure
chrome swift
image is a 2d subspace of R^3
that's only for rank 2 right
so image's subspace is the R^Rank A
sharp coral
rank = dimension of image
the image is always a (rank)-dimensional subspace of (codomain)
chrome swift
so image is related to column space?
sharp coral
image = range = column space
unkempt sluice
You construct any linear map from R^3 to R^a million, rank, which is the dimension of image, will still be <=3. Codomain can be as large as you like.
Same thing
sharp coral
all three are exactly the same subspace
chrome swift
what about null space
sharp coral
null space = kernel
sharp coral
don't worry about it then
but kernel is always a (nullity) - dimensional subspace of (domain)
chrome swift
never heard of kernel in my life. when do i need to
hybrid crow
that's for popcorn gang
yeah frick kernel we only fw null space around here
chrome swift
like what math is that
sharp coral
for m x n matrix domain is R^n and codomain is always R^m
sharp coral
never heard of kernel in my life. when do i need to
sometimes kernel is used instead of null space but they mean the same thing
chrome swift
ok good that's why we don;t know it
unkempt sluice
Linear algebra itself is just a special case of algebra, like linear spaces, just modules over PID. I think reading algebra first can completely cover linear algebra. The latter seems abundant.
hybrid crow
indeed
chrome swift
explain single value decomposition in terms of linear transformations
didn't learn SVD yet but i need to later
unkempt sluice
That involves a decomposition, A=UDU^H
For any complex square matrix A, there exists a hermitian matrix U and a upper triangular matrix D such that A=UDU^H
chrome swift
why is x in R^m not in R^n
unkempt sluice
You can prove this by induction. This will be a lemma in SVD. You need to know this decomposition first
sharp coral
why is x in R^m not in R^n
typo, it should be R^n
chrome swift
That involves a decomposition, A=UDU^H
isn't it just a rotation, an axis dilation, and a reflection
chrome swift
typo, it should be R^n
ok thanks
unkempt sluice
isn't it just a rotation, an axis dilation, and a reflection
You will need to use this decomposition to prove existence of SVD, I believe it’s called Schur decomposition or something: A^H A=VDV^H, for hermitian matrix V and diagonal matrix with non-positive real entries D. Thus D=Λ^2 , taking square root of entries. Let U=AVΛ^-1, clearly U is hermitian, thus A=UΛV^H, the SVD you wanted
unkempt sluice
is it a coincidence that the columns are derivatives
I view it as coincidence
chrome swift
You will need to use this decomposition to prove existence of SVD, I believe it’...
no idea about hermitian dude
unkempt sluice
You wanted to know SVD then you need to know those definitions and the Schur decomposition first
It involves those
When A is real, U and V can be orthogonal matrix instead of hermitian matrix. Parallel proof in real case. That’s why you said rotation and stretch. U and V will represent rotation, Λ will be the stretch
chrome swift
we only know about the orthogonal matrix part
unkempt sluice
U and V will represent rotation or rotation and reflection I mean, depending on the determinant of U and V being 1 or -1
chrome swift
so we find eigen vectors of A^TA
chrome swift
U and V will represent rotation or rotation and reflection I mean, depending on ...
ooh intresting
unkempt sluice
Yes, which let you obtain D
chrome swift
ngl i need to lock tf in rn this is a lot of stuff
tryna do a whole semester in a day is a bad idea
unkempt sluice
A^T A=VDV^T, you need to keep this V btw, since you will define U=AVΛ^-1. So don’t throw away V when you got D
It’s a good idea I think. Your thoughts will be continuous right, no interruption by taking breaks
chrome swift
shoot taking one right now is that bad
unkempt sluice
I think you already got what you needed. A symmetric real (Hermitian complex) matrix B equals UDU^T (UDU^H) for orthogonal real (hermitian complex) U and diagonal non-positive real matrix D is a special case of this Schur decomposition. So you already know it. It just when B is regular (BB^H=B^H B, when B is real it becomes BB^T=B^T B), D, the upper triangular matrix, by calculation has to be diagonal.
You don’t need the general Schur decomposition if you want to skip it for now. It seems you already know the special cases so good to go. I didn’t use the general case in my proof.
chrome swift
naw we didn't dive into the depths
u sure did
we learned one way to calculate it
it's with orthogonal matrices
unkempt sluice
I prefer to read the general cases first, then take a glimpse at the special cases and say, oh I knew it already, skip.
By a row of skipping, linear algebra is done. The reading part appears in algebra, my case.
unkempt sluice
Many cases like this, like Jordan form is a special case of Jacobson form. Depending on what you need. If your interest is not math but something else, chemistry, engineering… it’s perfectly fine to know the usual linear algebra version. General forms won’t be used at all.
chrome swift
yeah im an engineer
unkempt sluice
Then your way is more efficient. You are doing well.
midnight plankBOT
Available help channel!
twilit field
midnight plankBOT
twilit field
Just to clarify
twilit field
What is i’, U’, V’? And what is U^0, what is i’ tilde?
U',V' aare the dual spaces
unkempt sluice
tilde thing and U^0 definitions please
unkempt sluice
What do you mean annihilator of U?
unkempt sluice
c) comes from a) and b), as i’ tilde: V’/ker(i’) \cong im(i’). By a) ker(i’)=U^0, by b) im(i’)=U’. Thus we only need to show a) and b):
a) directly comes from definition.
b) you simply need to show that any linear f: U->F can be extended to V->F, i.e. there exists g:V->F such that g|U=f. And you can use Zorn’s lemma to show this
twilit field
Why do I need zorn's lemma for a fin dim case
unkempt sluice
Consider the set of (W, g), where U<=W<=V, and g:W->F such that g|U=f. This set is not empty since (U,f) belongs to it. Any chain has upper bound: a chain of some (W_j, g_j), you let W=union of all W_j, and g(x)=g_j(x) if x is in W_j. Done
twilit field
I mean it's not necessary is it
unkempt sluice
Okay I guess we can. No zorn, induction on dim(V)-dim(U)
twilit field
Lemme send my proofs?
unkempt sluice
Sure
twilit field
I'll start with (a)
unkempt sluice
f: U->F choose a v in V not in U, and expand f to U’=U+Fv ->F mapping v to zero, dim(V)-dim(U’)=dim(V)-dim(U)-1, thus b) holds by induction on dim(V)-dim(U). Done, no Zorn
twilit field
for (c).Can't I just use the 1st iso theorm
unkempt sluice
(Actually on second thought, you can just take W to be the direct summand complement of U in V, i.e. V=U \oplus W, then f can extend to V simply by mapping W to 0)
unkempt sluice
for (c).Can't I just use the 1st iso theorm
This is what I said
unkempt sluice
Anyway, direct sum or induction on dim(V)-dim(U), both work for b). Though latter requires this finite dimension
twilit field
(b) can probably be done by double inclusion too
unkempt sluice
Whatever works
unkempt sluice
Sure
twilit field
.close
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humble bronze
G is a finite abelian group and a positive integer n is prime to o(G) . Prove that every element $ x \in G $ has an nth root in G.
midnight plankBOT
G is a finite abelian group and a positive integer n is prime to o(G) . Prove th...
humble bronze
How do I solve this
unkempt sluice
Good
humble bronze
.close
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midnight plankBOT
Available help channel!
waxen shoal
how would we graph this on paper if we wanted to
midnight plankBOT
how would we graph this on paper if we wanted to
waxen shoal
$$ f(x,y) = (x^2 - 1)(y^2-1) $$
grand pondBOT
Ved
tacit vortex
I want to ask for the green part, which one is correct.. or both are accepted
tacit vortex
Karnaugh, logic gates
grim vector
Both are correct, also should be accepted i think
Wait in the draw you need to add a not
And change D by B
shut cargo
3rd times the charm
midnight plankBOT
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Available help channel!
wispy mortar
yo
twilit field
<@796939110815629322> what happened, u were typing <:angerysad:60232954013155329...
I'm not sure I get teh question
Like if G is of order 2
all H will be a subgroup
If |H|>2, consider {e,x} where x is an element of some order less than the order of the group abnd x isn't its own inverse
junior flower
but what if G isn’t order 2
twilit field
If |H|>2, consider {e,x} where x is an element of some order less than the order...
then consider this
wispy mortar
If |H|>2, consider {e,x} where x is an element of some order less than the order...
Z mod something?
twilit field
Z mod something?
well, We don't know if G is finite
wispy mortar
1 or even
junior flower
his ritual is 
wispy mortar
What is |x|
do we just have to get a subgroup H that is not abelian?
bold peak
You want elements of even order that multiply to an element of odd order hmm
junior flower
all subgroups will be abelian
hard umbra
i think those aren't abelian
junior flower
hard umbra
i think you should Z/6
bold peak
Huh idt that works
junior flower
i think those aren't abelian
what about S_1
hard umbra
what about S_1
shhhh
bold peak
Huh idt that works
Nvm it does
junior flower
snow is so smart
bold peak
Who is snow
flat spire
snow is @silent dock
bold peak
I mean, looking at their roles, they're definitely a prodigy 
unkempt sluice
I don’t get the question. H seems to always be a subgroup
bold peak
Z_6 is a counter example I believe
hard umbra
bluds never heard of snow 
unkempt sluice
But H={0,3}, isn’t it a subgroup?
hard umbra
mfw 6 is even but has an odd factor
hard umbra
abelian't
midnight plankBOT
junior flower
thanks for helping snow
it was all pure
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Available help channel!
rapid yew
Can someone pls help with these I'm so stuck
midnight plankBOT
Can someone pls help with these I'm so stuck
umbral timber
Can someone pls help with these I'm so stuck
have you started with anything?
and are you allowed to use graphing tools or do you have to sketch this by hand
rapid yew
I have these facts also
rapid yew
and are you allowed to use graphing tools or do you have to sketch this by hand
U have to actually draw it
umbral timber
I have these facts also
perfect
umbral timber
U have to actually draw it
then what would be helpful is that you actually set a value for 'a' and A and work with those values
umbral timber
is there like a specific relationship between them or can it be anything
'a' here is just a distance so doesn't matter what we chose, the graphs will take a similar form . use something simple like a=5 and A=1
rapid yew
I don't know how precise it wants it but that looks like what I have on my graphing calc 😞
maybe the ends matter
the sketching thing is really bad so I can't really get a levelling line
umbral timber
I don't know how precise it wants it but that looks like what I have on my graph...
hm this does look okay.
midnight plankBOT
@rapid yew Has your question been resolved?
rapid yew
no
lusty python
Can anyone give me a brief explanation of the chain rule? And I have an example question right here: find the derivative of $2^x$
midnight plankBOT
Can anyone give me a brief explanation of the chain rule? And I have an example ...
grand pondBOT
1 divided by 0 equals Infinity
dreamy lichen
do you wanna know why chain rule works or how to apply it?
lusty python
so far im just trying to learning derivatives by myself
and i proved some stuffs through the definition
late rover
Aren't you 9th grade?
lusty python
Aren't you 9th grade?
it's called i want to learn ahead
late rover
Have you learnt limit?
lusty python
Have you learnt limit?
a little bit
dreamy lichen
Have u checked khanacademy & 3b1b? khanacademy shows how to apply it and the proof through definition, 3b1b offers great intuition
gaunt imp
so far im just trying to learning derivatives by myself
If you're learning by yourself instead of as a requirement for a course, you should take the time to learn the concepts right and how and why they work
late rover
do you wanna know why chain rule works or how to apply it?
You haven't answer his question tho
lusty python
do you wanna know why chain rule works or how to apply it?
i want to know why chain rule works and how to apply it
gaunt imp
If I were you, well, I'd first make sure I know everything before it, that is, the basics, like the palm of my hand
dreamy lichen
Have u checked khanacademy & 3b1b? khanacademy shows how to apply it and the pro...
then try this, calculus with khanacademy + 3b1b is a pretty good combo
late rover
Okay lol, just watchi 3b1b videos
gaunt imp
And then look for a somewhat gentle introductory text with proofs
lusty python
Okay lol, just watchi 3b1b videos
lel
$f'(x) = \frac{dy}{dx} f(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
grand pondBOT
1 divided by 0 equals Infinity
lusty python
not really scary for me but at least it can scare some secondaries
dreamy lichen
there are simpler ways to scare ur classmates than learning calculus lol
late rover
Pull out some IMO geometry problem, it can even scare a 12th grader
gaunt imp
Pull out some IMO geometry problem, it can even scare a 12th grader
Hell, that'd scare me as well
lusty python
Pull out some IMO geometry problem, it can even scare a 12th grader
hell, IMO already scares me
these kiddos in my class only finished through quad formula and intersections between $y = mx + b$ and $y = ax^2$
grand pondBOT
1 divided by 0 equals Infinity
dreamy lichen
sounds normal
gaunt imp
these kiddos in my class only finished through quad formula and intersections be...
As should be
lusty python
As should be
mhm
gaunt imp
One thing you should keep in mind, not by learning faster are you learning better
lusty python
One thing you should keep in mind, not by learning faster are you learning bette...
i try to make sure i can prove smth first
like, learning by rote is just a worse way to kill yourself
gaunt imp
If you want my opinion, you'd get more benefit out of making sure you've got what you should know covered, and maybe proving some results you've seen in class or playing a bit around with them
lusty python
there are simpler ways to scare ur classmates than learning calculus lol
am i allowed to choose courses in khan academy?
forest iron
Yea
dreamy lichen
No, they force u to take one
lusty python
im cooked
gaunt imp
Jumping straight to learning calculus will, most likely, force you to learn by rote as you're saying, do keep that in mind
lusty python
If you want my opinion, you'd get more benefit out of making sure you've got wha...
geometry
late rover
You're Vietnamese, right?
lusty python
get a triangle ABC, and get 3 median lines of each side in ABC, get the interesection point of it. and the distance between A, B, C with the intersection is always = 2/3 vertex line
lusty python
You're Vietnamese, right?
how do you know
late rover
I do know what vietnamse ( the language )looks like
lusty python
I do know what vietnamse ( the language )looks like
hm
the only thing is that i don't think if you see i use the language before
lyric charm
$f'(x) = \frac{dy}{dx} f(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
d/dx, not dy/dx
late rover
I have a friend, he always complaint that 9th grade geometry is crazy diffucult there
So you probably should get better at those first
lusty python
my bad
dreamy lichen
Idk at what level you're rn, but maybe to test ur understanding of algebra and trig, try taking the unit test here. If you pass it without any issue, you should be more or less ready to learn calculus. That said, there is probably still a ton of simpler topics to be learnt in more depth.
https://www.khanacademy.org/math/precalculus
lusty python
I have a friend, he always complaint that 9th grade geometry is crazy diffucult ...
i agree with that
forest iron
Calc isnt hard
lusty python
I have a friend, he always complaint that 9th grade geometry is crazy diffucult ...
the diagram gets messy in my eyes
forest iron
If a 14 year old can learn it by himself then why cant you
lusty python
If a 14 year old can learn it by himself then why cant you
aight
visual elk
1/0 is NOT infinity
gaunt imp
Calc isnt hard
Oh but it's definitely annoying
A type of annoyance I believe people should deal with in due time
late rover
Tbh, you should only learn calcs only if you confident with your work at school
And you have time
lyric charm
If a 14 year old can learn it by himself then why cant you
what 14yo
late rover
You're studying for exam mostly
dreamy lichen
what 14yo
Someone green does real anal at 14, so i guess there'll be quite a few 14yo's doing calc in here
forest iron
Oh but it's *definitely* annoying
Nah thats the best part of math
gaunt imp
Nah thats the best part of math
You and I have different tastes in mathematics then, I guess
lyric charm
oh yeah ok because expecting yourself to be the model for all ppl your age worldwide is such a reasonable thing to think
gaunt imp
I'm starting to get the feeling the guy came here with some strange intention to flex
late rover
Hey, if you love hard problems, try probability and combinatorics instead
dreamy lichen
@lusty python any questions left?
midnight plankBOT
lusty python
1/0 is NOT infinity
,calc 1/0
grand pondBOT
Result:
Infinity
i let you say it again
visual elk
1/0 is NOT infinity
eternal pawn
Checkmate
midnight plankBOT
Available help channel!
visual elk
,w 1/0
grand pondBOT
visual elk
Look
dreamy lichen
i forgor the factord
eternal pawn
da2a i think
dreamy lichen
!da2a
midnight plankBOT
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question!
visual elk
.close
midnight plankBOT
visual elk
I accidentally opened bro
round skiff
serene coral
then do .close
visual elk
I already did
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Available help channel!
gilded oasis
HI
lyric charm
hello and welcome to the server
you have just claimed a help channel
do you have a math question to ask?
gilded oasis
NO
lyric charm
ok, then you can .close this channel and head over to #discussion or #chill
gilded oasis
ok
midnight plankBOT
ok
If you are done with this channel, please mark your problem as solved by typing .close
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Available help channel!
ocean hare
lim x to a of ((ln(x)-ln(a))/(x-a))
The answer is 1/a but I don't understand my profs justification. This is the justification:
ln(x)=y
y'=1/x
y'(a)=1/a
I'm confused what this means since we haven't solved problems like this before
midnight plankBOT
lim x to a of ((ln(x)-ln(a))/(x-a))
The answer is 1/a but I don't understand my...
sudden yacht
They've just recognized the limit definition of the derivative
And consequently used the fact that the derivative of lnx is 1/x
fallow scarab
A bit circular I suppose
midnight plankBOT
@ocean hare Has your question been resolved?
calm hazel
if you had f(x)=1 if x has more 1's than 0's in its digits and f(x)=0 if x has more 0's than 1's in its digits, or 0.5 if the amount is equal, would f(x) be defined for irrationals?
midnight plankBOT
if you had f(x)=1 if x has more 1's than 0's in its digits and f(x)=0 if x has m...
hard shard
,calc 1/99
grand pondBOT
Result:
0.01010101010101
true
mystic condor
oh my god stop changing the message lol
calm hazel
sorry
fathom onyx
You'd need to define what f's domain is
grand pondBOT
mystic condor
thats a rational number tho
calm hazel
yeah i forgot rational numbers can also have infinite digits
fathom onyx
Sure, but you can't explore irrationals here without having this checked first
fathom onyx
You'd need to define what f's domain is
Because, like I said, ^
calm hazel
both are infinite
are there meaningful ways to compare the size of infinite sets other than can/cant have an injective function?
calm hazel
You'd need to define what f's domain is
im curious if you could have numbers with infinite digits be in f's domain
fathom onyx
See 10-adic numbers
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chrome swift
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sharp coral
to find U in an LU decomposition, you perform gaussian elimination but only the adding rows to other rows part, not any other row operations
chrome swift
i gotchu
i learned the theory i think it makes sense
you put A in echelon form without row swapping, and you construct u as the inverse of every row operation
chrome swift
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