#help-17
vast shale
Is this okay?
Yes, looks good
Not sure if this is factorable, likely not
So it should be fine as is
Thank you I really appreciate you for your help!! I don’t have to stress about my homework anymore 😭😭 .close
All good, have a good day now :D
I too! 🙂
U too* ! 😭
copper yarrow
guys i am strugling with how to calulate "1x1"
vocal sleetBOT
guys i am strugling with how to calulate "1x1"
lost salmon
guys i am strugling with how to calulate "1x1"
huh?!
lost salmon
did you no here me?
wdym
drifting jackal
guys i am strugling with how to calulate "1x1"
Don't troll
copper yarrow
Don't troll
I am not trolling, I just am not smart.
onyx juniper
me when i lie for no reason
vocal sleetBOT
@copper yarrow Has your question been resolved?
vernal violet
Incredibly important and productive conversation right here
storm sigil
hello i need help
vocal sleetBOT
hello i need help
storm sigil
i have this probelm
so far
i have figured out d is 5
a is 4
and i so far i got
4cos( ) +5
but i dont know how to get the period
ornate terrace
normally, from min to max is pi
look at the difference here and find bpi=difference
storm sigil
?
ornate terrace
if we look at a normal cos graph
storm sigil
uyea
ornate terrace
from min to max, its pi right?
storm sigil
from min to max, its pi right?
mhm
storm sigil
3pi/4 and half of the period later its pi?
so half of period is 1pi?
im so confused 😭
ornate terrace
a normal cos graph
just think about from max to min rn
its pi
so if the max to min on your graph is pi/4, by what is your graph dilated by
storm sigil
2pi/4?????
ornate terrace
:’)
storm sigil
IM SORRYY HAHHAH im o confused
ornate terrace
IM SORRYY HAHHAH im o confused
its alr
proper dock
And put it together
storm sigil
Find the amplitude
i already did? the amp is 4
storm sigil
were thinking about b rn
yup
ornate terrace
cause its thinner than a normal cos graph, obv b has a value right?
we can find that value through comparing points
vernal violet
Akira are you having trouble again
storm sigil
we can find that value through comparing points
mhm
ornate terrace
normally we have a comparison of pi between the max and minimum
but now its pi/4
so b has a value that makes b(pi/4) equal pi
ornate terrace
yep
storm sigil
Akira are you having trouble again
again 🫠
proper dock
Agreed
storm sigil
the worst thing about cos graph is d has multiple values
the worst thing about cos graph is the graph and the cos and the math
ornate terrace
lol
fair enough
ahh wait
they said exact expression
you might need to make d an expression
storm sigil
you might need to make d an expression
nope
i already put it in it was correct
thx
ornate terrace
oh nice
storm sigil
now i have word problem im gonna cry 😭
ornate terrace
glgl
storm sigil
thxx
vernal violet
Lowkey the strat for word problems is to look for keywords
Like if it says how far something varies from another thing
Its probably amplitude
storm sigil
To read the damn english
the real strat is not doing it hehehe why think harder or smarter when u can just not think
proper dock
the real strat is not doing it hehehe why think harder or smarter when u can jus...
Bro will never get any work done with that mindset
vocal sleetBOT
Available help channel!
storm sigil
Bro will never get any work done with that mindset
i have other things to do mannnn
vocal sleetBOT
i have other things to do mannnn
marble pagoda
hello in an equation where there's a negative in front of the brackets do you put a one there?
for example y=-(x-8)^2
vocal sleetBOT
hello in an equation where there's a negative in front of the brackets do you pu...
marble pagoda
It becomes -1(x-8)(x-8)
(-1x+8)(x-8)
-1x^2+8x+8x-64
-1x^2+16x-64
but you can't factorise a negative number so
1x^2-16x+64
yes?
yeeeeeeeeeeeeeeeeees?
the full equation was y=-(x-8)^2 + 7 and y=3x+4
ending up with 1x^2-13x+53
and 53 is a prime number
so it isn't easily factorisable
gentle thicket
so it isn't easily factorisable
What's your query then?
marble pagoda
wait sorry I realise the 4 on the end was actually p
marble pagoda
hello in an equation where there's a negative in front of the brackets do you pu...
just this
if the negative becomes -1
y=-(x-8)^2
vocal sleetBOT
vocal sleetBOT
Available help channel!
storm sigil
god i need more help 😭
vocal sleetBOT
god i need more help 😭
storm sigil
how do i graph this qn
i already did all the wrok
the period aka b is 3pi/2
amp aka a = 2
the phase shift aka c =pi
midline aka d is 4
but i dont know how to graph it
😭
outer panther
if u can just move the point on x=pi and it'll graph the rest i guess you could just plug in pi into g and move the point to (pi,g(pi))
so 4 i think
storm sigil
like s0?
storm sigil
outer panther
that makes more sensep
outer panther
sorry back
outer panther
been a while since i did this, but you def need a somewhere to input each point
the y intercept isnt an integer
the y coordinate of it^
can you only drag them
storm sigil
can you only drag them
i can drag both them but thats it why
outer panther
the x coordinates of the crest isn't rational
by not rational i mean it has too mant decimal places to be dragged accurately (not necessarily irrational)
storm sigil
oh yea i dont know if i mentiond this or not, but we can also drag them into the in between
storm sigil
gosh im looking at the wrong thing
oh lol hahhaha
storm sigil
yea
i know that, but how do we place it, i figured out most of it, exept where to actually place it on the graph, i thought maybe 5p/2p is at the x loction 2p but i dont know
outer panther
and since ghe period is 6pi you'd get the other point at x=-pi/2
cause it'd be half a period less than 5pi/2
storm sigil
oh
vocal sleetBOT
@storm sigil Has your question been resolved?
storm sigil
.close
median nacelle
am I doing the calculations right for division 5? I tried to use the addition rule but idk if ive applied it correctly here
vocal sleetBOT
am I doing the calculations right for division 5? I tried to use the addition ru...
@median nacelle Has your question been resolved?
median nacelle
<@&286206848099549185>
vocal sleetBOT
@median nacelle Has your question been resolved?
median nacelle
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
elder trail
Can someone help me with this question? I am trying to review for my class and I got this question.
The ages of all females from the freshmen students in a university is a/an ____________ .
frequency
population
statistic
sample
vocal sleetBOT
Can someone help me with this question? I am trying to review for my class and I...
hybrid cove
it's a population
plain minnow
Can someone help me with this question? I am trying to review for my class and I...
What is the definition of each? Does anyone of them fit the specific scenario?
elder trail
I don't know if it's population or a sample
plain minnow
Do you know the difference between the two?
elder trail
Do you know the difference between the two?
I know the difference between the two but im confused how i am gonna categorize the given in the question.
Because the question gives a specific set of ages (female - freshman) so i wondering if it is a sample of all the ages in the university
elder trail
this is not the general chat
peak matrix
75 pings
(you should remove Helpers role in id:customize if you dont want pings)
plain minnow
Because the question gives a specific set of ages (female - freshman) so i wonde...
If you systematically exclude a certain group (people who are either non-freshman or non-female), then they are not to be considered as part of the underlying population. If you're doing statistical analysis on a group, A, with N people and you take data from all people in that group, it doesn't really matter that there exists another group B. You could always argue that there is a bigger group (female freshman ⊂ university students ⊂ people in the university's city ⊂ people in the country ⊂ people on Earth ⊂ beings in our galaxy ⊂ ...). So when they say that we are exclusively taking samplings from female freshmen at a certain university, then that is our population. And when the question says that we have taken samples from all female freshman at that certain university, then we have a...?
elder trail
so it's population?
plain minnow
vocal sleetBOT
vocal sleetBOT
Available help channel!
past stag
I have no clue how to even start
vocal sleetBOT
I have no clue how to even start
civic otter
Start with writing an expression for the 3rd, 4th and 7th term of that arithm. sequence
past stag
an = a1 + 2d + a1+3d + a1 + 6d
civic otter
You mean this?
$\a_3 = a_1 + 2d \
a_4 = a_1 + 3d \
a_7 = a_1 + 6d$
past stag
yeh
twin meteorBOT
Alberto Z.
civic otter
yeh
Right
Now you have to translate mathematically what "a3, a4, a7 are in geometric progression" means
past stag
u1, u2, u3
u1(1),u2(r),u3(r^2)
civic otter
This is correct, but there's another property (actually, the definition) of geometric progression
past stag
un = u1(r^n-1))?
past stag
a progression with a constant ratio between each number
civic otter
a progression with a constant ratio between each number
Perfect, that's it
So now you can apply this to our exercise
waxen hawk
Guys so I have 6 pants that fit me the same but are in different sizes, my bf said to add them all up and divide by the number to find the average would this work?
civic otter
a progression with a constant ratio between each number
(I believe you wanted to say the ratio between a number and the previous one, but yeah)
past stag
so how can I keep solving
civic otter
!occupied
vocal sleetBOT
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).
civic otter
so how can I keep solving
Write an equation that expresses this: "the ratio between each number and the previous is constant"
past stag
Write an equation that expresses this: "the ratio between each number and the pr...
r = u(n+1)/u(n)
civic otter
And in your case what are u(n+1) and u(n) ?
twin meteorBOT
Alberto Z.
past stag
but yeh I see where u goin
civic otter
I dont think a7/a4 makes r tho
Why not?
Read the question carefully
They say that they are the first three terms
past stag
My math teacher said this is gonna be the hardest quiz of my life so far
civic otter
so I just say a7 = u3
Yep
past stag
Okay, so how can we proceed
civic otter
Solving that equation
past stag
a4^2 = a7*a3?
civic otter
a4^2 = a7*a3?
Exactly this one
past stag
how can I proceed from here
civic otter
That was the hardest question of my life
Really? What kind of math have you done so far? 😅
past stag
Really? What kind of math have you done so far? 😅
We just started sequences and have just been blasting through it
but we havent done anything like this
civic otter
I can assure this is somewhat the easy part of math, at least for what I have experienced
But so far it's probably not, all good
Do you need help also with the second part of the exercise?
past stag
Im do the rest later
vocal sleetBOT
Available help channel!
rough hinge
can someone give me the beginning steps to start answering this question
rough hinge
i think it will go by partial fraction but how do i split it into 3 parts
copper yarrow
have you leaned bedmas yet?if you have ont then learn it.
dense cedar
Just factor in two you will get three parts
You know how to decompose ?
When there is quadratic in deno?
rough hinge
A/x2+1 + B/x+4
rough hinge
(X+1) (X-1) (X+4)
copper yarrow
Bedmas is the order of operations in math, Brackets,expoenents,Division,multiplication,Addition,subtraction.
dense cedar
(X+1) (X-1) (X+4)
x²+1 != (x+1)(x-1)
dense cedar
(ax+b)/x²+1 + c/(x+4)
This is the correct way of decomposition
rough hinge
i see
i will proceed with the process and tell you where i get to
1 sec
i got A for -1 /3 and B for 1/5
now i need to get C
vast shale
Can someone check if this is right?
vocal sleetBOT
Can someone check if this is right?
tranquil trellis
yes
vocal sleetBOT
mellow void
Can someone check if this is right?
writing D = {Reals} is inaccurate; it's just Reals, not {Reals}
vast shale
writing D = {Reals} is inaccurate; it's just Reals, not {Reals}
my teacher wants it formatted like that
😭
mellow void
my teacher wants it formatted like that
seriously? I want to confirm this... if anything, it should be $D=\mathbb{R}$ and not $D={\mathbb{R}}$
twin meteorBOT
Flip
vast shale
welp
mellow void
because ${\mathbb{R}}$ is the set containing exactly one element, being the set of all real numbers
twin meteorBOT
Flip
vast shale
should i tell her this
edit: kk
rare swift
For x it's (x)ⁿ and for a either of them works
hearty finch
For x it's (x)ⁿ and for a either of them works
if a was -1, then it would be (-1)^-n-k right ?
rare swift
if a was -1, then it would be (-1)^-n-k right ?
Yes
rare swift
Thank you so much 💟 🫶
Welcome
final herald
Problem: Paths in this problem have no repeated nodes. Paths have no orientation, if you reverse the nodes' order it is the same path.
a) How many paths of length 5 is there in K_{3,7}?
b)How many paths of length 4 is there in the same graph?
So what I'm understanding is that I should look at the two cases, we either start in the collection with 3 nodes, or the one with 7, we then add these 2 cases together. Lets say we start in the one with 3 we get 3*7*2*6*1*5. What confuses me is how do I account for the fact that paths are not oriented? Do I simply divide by 2?
vocal sleetBOT
Problem: Paths in this problem have no repeated nodes. Paths have no orientation...
@final herald Has your question been resolved?
ruby onyx
vocal sleetBOT
ruby onyx
im looking to draw the CDF from this PDF
so when pdf is decreasing the cdf will stay horizontal?
round canyon
im stuck on the sinx = +-1/2 part
vocal sleetBOT
im stuck on the sinx = +-1/2 part
round canyon
i got pi/6 and 5pi/6 and i think u elimate the negative values since the angle in original problem is postive
unless im on the completely wrong track
mellow void
i got pi/6 and 5pi/6 and i think u elimate the negative values since the angle i...
if m is some random function living in a vacuum, defined for every single real number, then we may want every single solution
that's also so wacky to use the half-angle formula on this, well done lol
*double
round canyon
if m is some random function living in a vacuum, defined for every single real n...
wha
mellow void
too many cooks man
round canyon
so would it be pi/6, 5pi/6, 7pi/6, 11pi/6?
mellow void
yeah, but also its friends
elder fable
pi/6 + 2kpi
round canyon
the teacher showed example in class and there was only 2 answers but i forgot what it was
mellow void
13pi/6, etc
elder fable
where k belongs to Z
for the other solution
sin x = -1/2
its -pi/6
- 2kpi
k belongs to Z
mellow void
it's all context-sensitive
elder fable
I know it works , but for me its just not intuitive at all
elder fable
wait a sec i will draw the unit circle
round canyon
is it not all t hose values
round canyon
-pi/6 is 5pi/6 - pi
how come u subtract pi
elder fable
mellow void
you're on the line sin(x) = 1/2 or -1/2
split it into two cases
suppose sin(x) = 1/2
we have one solution via the well-known unit circle: x = pi/6
but we also know that sin(x + 2kpi) = sin(x) for all integers k
hence x = pi/6, x = pi/6 + 2pi, x = pi/6 + 4pi, etc., as well as x = pi/6 - 2pi, etc. are all solutions
we also know that x = 5pi/6 is a solution, so that gives another infinite set of solutions in the exact same way
that exhausts all such solutions for sin(x) = 1/2
does that make sense so far?
mellow void
draw a circle
not a hypothetical
have a circle that you can touch ready
pick a point on that circle and mark it
draw the line between the center of the circle and that point
(draw the axes so that the origin is centered at that center)
indicate the angle between the positive x-axis and the line from the center to your point, and call the angle measurement A
good so far?
round canyon
indicate the angle between the positive x-axis and the line from the center to y...
wha
mellow void
yeah
now add 2pi to that angle
well, easier
add pi to the angle
and trace your finger from the marked point to the point that corresponds to the new angle A+pi
where are we?
round canyon
and trace your finger from the marked point to the point that corresponds to the...
wdym
like up the lien
line
mellow void
around the circle
so imagine that as you're incrementing the angle A, the line from the origin to your point is "spinning"
it's sweeping around the origin, and the point that lies on the end of this line is tracing around the circle
your finger will indicate this point's position, so as you change the angle, you move your finger around the circle
round canyon
so when u move the the line around on the circle, the dot on the end goes around too
mellow void
yeah
round canyon
so ur tracing the shape of the circle
mellow void
correct
so when you add pi to the angle A, the point is moved via this sweeping action
where is this new point, in relation to its previous position?
mellow void
do the instructions and premise make sense?
round canyon
i think
so when u add to the angle value the line spins around the origin of the circle
mellow void
right
round canyon
so if u added pi it would be in the bottom no?
round canyon
since its 180 degrees
mellow void
exactly
in particular it's like, at the point exactly opposite of the original point
now add another pi to the angle
where are we now?
round canyon
back at where we started
round canyon
around again
mellow void
and if we subtract 2pi?
mellow void
exactly
and so, no matter how many times we add or subtract 2pi from our angle, we're always at the same point we began at
the very important thing here is, your point actually has coordinates
its x-coordinate is cos(A), and its y-coordinate is sin(A), by definition
when we add or subtract k copies of 2pi to the angle A, by definition, the corresponding point has x-coordinate cos(A+2kpi) and y-coordinate sin(A+2kpi)
but we've just said that these two points are exactly the same
round canyon
so if u add 2pi its just going around the circle so value is still the same?
mellow void
thus we have (cos(A), sin(A)) = (cos(A+2kpi), sin(A+2kpi))
yeah, exactly
and since these two points are the same, it follows that cos(A) = cos(A+2kpi) and sin(A) = sin(A+2kpi)
for any choice of angle measure A, and for all integers k, this applies
so therefore, whenever we have sin(x) = c, it immediately follows that every integer k yields another solution, sin(x+2kpi) = sin(x) = c
round canyon
so you add 2pi to account for all the other values outside of the period of 2pi?
mellow void
yes
so long as the domain of our function is every real number
or, like, a lot of them lol
if this were a more geometric question where our angle was declared to be in the half-open interval [0, 2pi), or worse, then we wouldn't care for all these extra solutions
round canyon
if the interval was given do u still have to add the 2pi thing on the end?
mellow void
only for as long as the extra solutions promised by the symmetry were truly in the domain of our original function
or for as long as we cared for these extra solutions otherwise
round canyon
so how do u apply this to the original problem
mellow void
hence x = pi/6, x = pi/6 + 2pi, x = pi/6 + 4pi, etc., as well as x = pi/6 - 2pi,...
we proceed off of this talking point, newly reassured that we can always produce a whole class of solutions whenever we get one
we're imagining sin(x) = 1/2 and we obtain two "easy" solutions, x = pi/6 and x = 5pi/6
now we have a whole class of solutions: x = pi/6 + 2kpi and x = 5pi/6 + 2kpi, for each integer k
then we do the same for the case where x satisfies sin(x) = -1/2
find me two solutions for this that work, and describe the entourage of solutions that follow them
round canyon
find me two solutions for this that work, and describe the entourage of solution...
entourage of solutions?
what ist hat
mellow void
an entourage is a group of people attending an important person
I'm being funny by calling the family of numbers {x + 2kpi : k is an integer} an entourage for the very important number x
laugh
then live, then love. in that order
round canyon
mellow void
yeah
now you can slap them all together
there's a tiny optimization actually, because 7pi/6 = pi/6 + pi and 11pi/6 = 5pi/6 + pi
so in fact, these solutions aren't very different from the prior ones. they're a factor of pi away
round canyon
there's a tiny optimization actually, because 7pi/6 = pi/6 + pi and 11pi/6 = 5pi...
so theres 4 solutions?
round canyon
mb
mellow void
what I'm saying is that we can tidy up the solutions
making not four classes of solutions, but only two
mellow void
so theres 2 input values
input values isn't the right term, it's more like there's two "generators" that give you all the solutions
vocal sleetBOT
@round canyon Has your question been resolved?
round canyon
input values isn't the right term, it's more like there's two "generators" that ...
the two values -pi/6 and pi/6?
mellow void
sure
round canyon
🗿
vocal sleetBOT
https://gyazo.com/acb1baa6a30c679d28a0df11b4d5f658
edgy sigil
For question b, does this mean find the values for B which mean the determinant of the matrix is 0?
Meaning the matrix is singular
limber peak
For question b, does this mean find the values for B which mean the determinant ...
Yes
edgy sigil
Any ideas on how I'd go about solving it?
Just assign an integer value to them and work from there?\
or make it a system of equations?
limber peak
Whattt do you meannnn?
Cant you just solve for the determinant and set it equal to 0?
edgy sigil
So leave the values as B and the solve for the determinant?
edgy sigil
Can you clarify in non mathematical terms what you mean by get the determinant in terms of beta
limber peak
Uhhhhh that is kindof difficult without mathematical terms but i’ll try
You want an equation which has beta as the variable
If you find the determinant of M2 youll automatically have an equation in which beta is the variable
You get what i mean?
So det(M2) = ..beta^2 + ….beta + …. For examle
Set that equak to 0
And then you can just solve for beta using algebra
Do you know how to find the determinant of a matrix?
edgy sigil
Do you know how to find the determinant of a matrix?
Yes
Now you've explained it
It makes sense
Setting it equal to zero enables me to just solve it using algebra
Thank you
Any chance you could shed some light on question D?
I'm struggling to understand what the question means in terms of what I am actually supposed to do
mellow void
I'm struggling to understand what the question means in terms of what I am actua...
just as you had found det(M2) in terms of B(eta), you'd like to determine M_1^n in terms of n
matrix powers are defined just like integer powers: you compute the matrix product M_1 * M_1 * ... * M_1, n times
edgy sigil
And how would I compute it n times?
mellow void
do you know about diagonalizable matrices?
edgy sigil
yes
mellow void
in part (c) you were asked to determine the eigenvalues of M_1, with eigenvectors corresponding to the eigenvalues
do you have these?
edgy sigil
I don't
mellow void
you need them for part (d)
edgy sigil
I just wanted to clarify some of the questions before I started working on it
edgy sigil
yeah carry on, I can use your explanation to aid in my working out
mellow void
you're hoping that $M_1$ is diagonalizable, so that there is an invertible matrix $P$ and a diagonal matrix $D$ so that $M_1=P^{-1}DP$
twin meteorBOT
Flip
mellow void
the diagonal matrix D will consist entirely of the eigenvalues of M_1
and the invertible matrix P will have the corresponding eigenvectors as its columns
e.g. if $v_1=(a_1,b_1)$ and $v_2=(a_2,b_2)$ are eigenvectors for $\lambda_1$ and $\lambda_2$ respectively, then $D=\begin{pmatrix}\lambda_1&0\0&\lambda_2\end{pmatrix}$ and $P=\begin{pmatrix}v_1&v_2\end{pmatrix}=\begin{pmatrix}a_1&a_2\b_1&b_2\end{pmatrix}$
twin meteorBOT
Flip
mellow void
that's what's going on in the background, so now let's discuss why diagonalization helps us with computing matrix powers
we have that $M_1=P^{-1}DP$. what is $M_1^2$?
twin meteorBOT
Flip
mellow void
write it in terms of P's and D's
edgy sigil
Honestly mate, I have no clue
I've heard about diagonalization of matricies before but never actually worked through the topic
edgy sigil
What does the 2 represent?
edgy sigil
okay sorry that was obvious I'm just being stupid
mellow void
no problem
edgy sigil
I understand what you're asking but I just don't know how to work it out
M1^2 is just M1 x M1 right?
mellow void
one trick is to just use what you're given
mellow void
M1^2 is just M1 x M1 right?
correct
edgy sigil
I'm not sure how to write it in terms of P's and D's
edgy sigil
honestly I don't even know where to begin
mellow void
who is M_1 in terms of P and D?
edgy sigil
I don't know
edgy sigil
I'm sorry if this is ridiculous, I've just been in the library all day and my brain is running on empty right now
edgy sigil
if D is the eigenvalues and P is the eigenvectors
mellow void
like you don't see the path immediately so you're waiting around
edgy sigil
Does P^-1 mean the inverse of the eigenvectors?
mellow void
P^-1 is the inverse of the matrix P. in fact, I no longer care about the specifics of P and D at the moment
they're just matrices
and one of them happens to be invertible
square matrices I should say
mellow void
no problem at all
edgy sigil
So they're both square matricies
P is invertible
I think the wording "in terms of" is whats confusing me
are you able to explain a different way?
mellow void
I should first tell you that the task of telling someone to writing something "in terms of" something else is very common, so it's worth learning what it means
mellow void
it's exaclty like how you found the determinant of M_2 "in terms of" beta earlier
you computed the determinant of M_2, and you see betas floating around in there
so if you compute M_1^2 in terms of P and D, you should see P's and D's floating around in your expression when you're done
edgy sigil
ahhhhhhhhhh
mellow void
in the case where P and D are also expressible "in terms of" other things, I ask that you don't substitute P and D with these other things
edgy sigil
Right I'm following you now
mellow void
because like I had alluded to before, at this moment, it's actually easier to follow what's really going on if you ignore all the pretense that D is a diagonal matrix with eigenvalues and P is a collection of columnized eigenvectors
edgy sigil
yeah you're making sense now
mellow void
all that's important for this calculation is: P and D are square matrices and P is invertible
coolio
edgy sigil
yeah now we're cooking on gas
mellow void
nice lol
edgy sigil
You explained that so well
Thanks mate
at the n'th hour of my revision you've done gods work
sorry carry on
limber peak
at the n'th hour of my revision you've done gods work
This line goes incredibly hard in this context
edgy sigil
alright, so now we can continue on with computing M_1^2
please do
mellow void
please do
ah I was still asking you to do this actually lol
knowing only that M_1 = P^-1 * D * P, compute M_1^2
edgy sigil
ah I was still asking you to do this actually lol
yeah give me two seconds, im just working it out now
I have m1^2
mellow void
what'd you get?
mellow void
abstracting the situation
without numbers
just P's and D's
maybe numbers in powers, that's alright
mellow void
you've got a bunch of symbols and rules for how to slap them together
$M_1^2=(M_1)(M_1)=(P^{-1}DP)(P^{-1}DP)$
twin meteorBOT
Flip
mellow void
using exclusively what we're given and nothing more, this is the situation
mellow void
can we do better?
edgy sigil
can you expand the brackets out and simplify?
mellow void
what do you mean by expand?
mellow void
matrix multiplication doesn't distribute over itself, if that's what you're suggesting
like I can't say $a(xyz)=(ax)(ay)(az)$
twin meteorBOT
Flip
edgy sigil
Yeah okay I was suggesting the wrong thing
Give me 2 minutes I just need to run to the toilet
mellow void
best of luck
the vector norm of your message makes it a spoiler
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we'll get there when we get there
double vertical bars surrounding your message
based
edgy sigil
I am back
mellow void
welcome
edgy sigil
Thank you
mellow void
any additional thoughts on how to proceed?
edgy sigil
is this still reliant on me working out the eigenvalues and eigenvectors
mellow void
nah, we're not there yet
mellow void
**Flip**
we're still pushing symbols on this step with the hope in our hearts to simplify this expression for M_1^2
and I promise it's for a good reason
because it'll motivate the key observation that makes part (d) feasible
edgy sigil
in that case I would be extremely in your debt if you could tell me how to simplify it
as I don't want the hope in your heart to run out
mellow void
so first you need to understand the rules of matrix multiplication, so that you can push symbols with the greatest of ease
edgy sigil
okay
mellow void
let's say I want to compute A(BC) for appropriately-chosen matrices A,B,C
BC is annoying to compute and I refuse to do so
are there any alternatives?
that is, A(BC) is the product of A and [the product of B with C]
edgy sigil
that is, A(BC) is the product of A and [the product of B with C]
I'm sure there is, I just cannot provide you with an answer right now
mellow void
ah
edgy sigil
If I am wasting your time
mellow void
unfortunately it's required
edgy sigil
Please say so
mellow void
you're not wasting my time, I'm on board lol
edgy sigil
I know how to multiply two matricies together I just don't understand the intricacies of the rules enough to apply them to what you're asking
If you get me
mellow void
I think so
maybe you know this property of matrix multiplication by name
starts with an 'a'
has more than two syllables
edgy sigil
associative
edgy sigil
a(bc) = b(ac)
mellow void
that's actually not it
mellow void
that's it
mellow void
the implication of this is, if I have a bunch of characters wrapped in parantheses floating around, like (AB)((C(DE)F))
I can do away with all of them and, without any real consequences, write their product as ABCDEF
so long as we understand that if we want to do some multiplication-based shenanigans with C, we can only do wo with B or D
because in general, matrix multiplication isn't 'commutative', i.e. AB != BA
mellow void
so therefore, $M_1^2=(P^{-1}DP)(P^{-1}DP)=P^{-1}DPP^{-1}DP$
next thing
P is invertible and has inverse P^-1
why do we care? what does that mean?
twin meteorBOT
Flip
edgy sigil
Right
I'd have never gotten there myself but I understand now
From what you've told me
mellow void
so far the only things we've used (in this scenario) have been a substitution (M_1 = P^-1DP) and associativity of multiplication to de-couple everything
substitution mostly by just, using what you're given; you have exactly one assumption so you might as well use it somehow
mellow void
so why do we care about P being invertible? what does it mean for a matrix to be invertible?
mellow void
there's no such thing as an inverse product but yeah, there is an inverse for P
edgy sigil
yeah thats what I meant sorry
mellow void
what does the inverse of P do?
edgy sigil
give us the inverse of the eigenvectors?
mellow void
forget about eigenwhatsits
we're only talking about matrices and matrix multiplication
we've completely forgotten where P came from, and we only care that P is invertible at this time
edgy sigil
okay
edgy sigil
what does the inverse of P do?
I have no idea
mellow void
so there is a so-called inverse for P, let's call it Q because I'm tired of writing P^-1 lol
what can you tell me about their product, PQ?
mellow void
singular?
mellow void
what's a singular matrix?
edgy sigil
own
mellow void
oh
edgy sigil
sorry used the wrong word there
mellow void
lol I was gonna say singular means noninvertible
edgy sigil
then we've opened a whole other kettle of fish haha
mellow void
but wait a minute, that's not exactly a spectacular property for the product PQ to have
the product of any two matrices AB (with appropriate dimensions) is another matrix
so what gives?
edgy sigil
I am not sure in this context
mellow void
what does the word "inverse" really mean?
edgy sigil
the opposite
mellow void
yeah
and "invert" is like
to undo, probably
reverse something
so Q being an inverse for P is like saying that Q acts as an undoing action of whatever P does
edgy sigil
I see
mellow void
and in this context, talking only about matrix multiplication, the only thing that P could ever act on is, other matrices, via matrix multiplication
and we're saying that Q reverses this action of matrix multiplication by P
so (sorry I'm writing them in a different order for a moment) the product QP is a matrix that represents doing something, then undoing something
for a net result of doing nothing
edgy sigil
so they cancel out
mellow void
in the context of matrix multiplication... what matrix 'does nothing'?
edgy sigil
a matrix with a determinant of zero?
mellow void
not quite
edgy sigil
a singular matrix?
mellow void
same thing, so no
mellow void
yeah, like, it doesn't change anything it acts on
i.e. it doesn't change anything it gets multiplied to
edgy sigil
I don't know
mellow void
it has a name
edgy sigil
An identity matrix?
mellow void
that's right
mellow void
for any square matrix X, there is this (necessarily unique) matrix I, called the identity matrix, for which IX = XI = X
edgy sigil
I get you
mellow void
and for a matrix P to have an inverse Q, that means that QP = PQ = I
edgy sigil
yes
mellow void
they are opposites, they 'cancel out' and become to do-nothing action I
(and they cancel out in both directions, notice)
edgy sigil
yeah I get you
mellow void
I don't know where you're going with your math education, but associativity, identity, and inverses are hot properties for a binary operation to have
edgy sigil
Yeah I've been introduced to those in other subjects
mellow void
and crucially, the idea of being able to undo stuff is very very nice
as we should now see
edgy sigil
I see
mellow void
$M_1^2=P^{-1}DPP^{-1}DP$
twin meteorBOT
Flip
mellow void
what comes next?
edgy sigil
The P and the P-1 cancel out?
mellow void
exactly
their product is the identity, and the identity times D is just D
hence $M_1^2=P^{-1}DPP^{-1}DP=P^{-1}DDP$
twin meteorBOT
Flip
mellow void
now that juxtaposed D-D pair is silly, and we can just write that as D^2
so finally $M_1^2=P^{-1}D^2P$
twin meteorBOT
Flip
edgy sigil
which makes a bit more sense
mellow void
that's good because now you're going to do this again
I'll now ask for M_1^3 in terms of P and D
edgy sigil
Please don't 🤣
I don't think my understanding has quite developed to the point where I can work things like that out
mellow void
you've got it backwards
you don't wait for the development to come so you can work things out
you work things out to encourage the development
edgy sigil
I think I'm going to take that question as a bit of homework
And take a break
If you don't mind
mellow void
that sounds excellent
mellow void
of course, I appreciate your patience in turn while I try to get you to do things lmao
edgy sigil
But I've got some great learning points here
edgy sigil
of course, I appreciate your patience in turn while I try to get you to do thing...
Thank you
That means a lot
Although I don't really understand I am still willing to try and work through things the best I can
mellow void
based
edgy sigil
I really appreciate your help
Thank you so much
I've copied what you've written into a seperate note so I can come back to it
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atomic wagon
Idk how to rewrite this sentence as a biconditional statement “the midpoint of a segment is the point that divides the segment into two congruent segments”
vocal sleetBOT
Idk how to rewrite this sentence as a biconditional statement “the midpoint of a...
round ridge
Idk how to rewrite this sentence as a biconditional statement “the midpoint of a...
a point is the midpoint of a segment if and only if...
and then you add the rest
atomic wagon
ty
round ridge
np
atomic wagon
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short kraken
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vast shale
and why didnt they sub in f(0)
tan 2*0 / 0
short kraken
why
vast shale
0*2=0 too
tidal dock
if you get 0/0 as your limit, you gotta do more work
it's inconclusive as is
vast shale
so 0/0=2 too
short kraken
if you get 0/0 as your limit, you gotta do more work
i still dont understand how they broke it down
vast shale
0/0 equals all numbers
tidal dock
tan = sin/cos
vast shale
0/0 is called the indeterminant form
short kraken
sin/cos * 2x
tidal dock
tan = sin/cos
do you know that?
short kraken
do you know that?
yes lol
vast shale
0/0 is called the indeterminant form
so you have to do more work to know what it is
like maybe have more conditions
short kraken
so you have to do more work to know what it is
but what are they subbing 0 into at the end when the result is just 2
short kraken
i dont see them make that sub
so sin2x/cos2x
xcos2x mb
but howd they get sin2x/2x * 1/cos2x * (2)
tidal dock
but what are they subbing 0 into at the end when the result is just 2
lim x-> 0 sin(x)/x = 1 is a well known limit, you should memorize it
vast shale
but what are they subbing 0 into at the end when the result is just 2
you are aware that it is a limit right?
i am trying to give you the intuition of how to solve indeterminant limits
you are getting close to the equation 0/0 but not reaching it, so you don't have and actual indeterminant equation
tidal dock
but howd they get sin2x/2x * 1/cos2x * (2)
$\frac{a}{bc} = \frac{a}{b}\cdot \frac{1}{c}$
twin meteorBOT
artemetra
vast shale
you are aware that it is a limit right?
i am trying to give you the intuition o...
so you can just do algebraic manipulations
look
in general
there are 3 cases for your limit
short kraken
$\frac{a}{bc} = \frac{a}{b}\cdot \frac{1}{c}$
where'd the extra two come from then
tidal dock
but howd they get sin2x/2x * 1/cos2x * (2)
the (2) in the end is there to cancel out with the 2 we added to have 2x instead of x in the denominator
^^^
short kraken
we added a 2?
vast shale
there are 3 cases for your limit
1- substitute, if you got a actual number, then that's the limit
2- if you got 0/0 do algebraic manipulations to make it in an another equal form
3- x/0 is DNE
tidal dock
tidal dock
yep
short kraken
nvm im confused we didnt pull 2 out at the end
if we did
we would have 2 * tanx/x
vast shale
about why lim x-->0 sinx/x = 1
here is the graph of it
short kraken
we rewrote 1/x as 2/2x
yea im still not following
tidal dock
yea im still not following
you are aware that $\frac{2}{2x} = \frac{1}{x}$, right?
twin meteorBOT
artemetra
tidal dock
great
short kraken
oh thats what u mean added a two
tidal dock
yea
short kraken
why did we do that tho
tidal dock
multiplied by a two
short kraken
if it just equals 1
tidal dock
tidal dock
we want sin2x/2x
yes
tidal dock
in order for the blue thing to be 1, we need the green things to be the same
that's the whole point of all of this really lol
tidal dock
yep
tidal dock
btw 1/cos2x is also 1?
yes
short kraken
1/(1*2)?
tidal dock
no
tidal dock
yep
short kraken
and the 2 is the initial 2 we had
tidal dock
yep, from tan(2x)
vast shale
so initially we have tan2x/x we rewrote tan to say sin/cos
$tan 2x / x
= sin 2x/cos2x * 1/x
= sin 2x/cos2x * 1/x * 2/2 (multiply by 1)
= sin 2x/cos2x * 1/2x *2
= sin 2x * 1/cos2x * 1/2x *2
= sin 2x * 1/2x * 1/cos2x *2
= sin 2x/2x * 1/cos2x *2$
that's now an equivalent form of tan 2x / x, NOW you take the limit with the usual substitution
try substitue 0 in the equivalent form now
short kraken
but didnt we use that 2 to get sin2x/xcos2x? 😭
twin meteorBOT
Abstract Algebra Simp
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
tidal dock
but didnt we use that 2 to get sin2x/xcos2x? 😭
wut?
short kraken
just...why are we multiplying 2 at the end if we didnt pull it out at the beginning
tidal dock
pull it out of what
short kraken
tan2x/x
tidal dock
how?
vast shale
just...why are we multiplying 2 at the end if we didnt pull it out at the beginn...
we didn't pull out
we multiplied the whole equation by 1 which is 2/2
short kraken
why isnt it this
tidal dock
you mean why didn't we do $\frac{\tan(2x)}{2x}\cdot 2$ in the cery beginning?
short kraken
why is there a * 2 at the end im trying to piece that together
twin meteorBOT
artemetra
short kraken
**artemetra**
no
vast shale
why isnt it this
becaust this means that you multiplied the whole equation by 1/2
not by 1
tidal dock
why is there a * 2 at the end im trying to piece that together
i mean it doesn't matter where it is
vast shale
so it is not equivalent
tidal dock
i mean it doesn't matter where it is
ab = ba
short kraken
i mean it doesn't matter where it is
ugh no why is there a 2 at all in the last step
vast shale
**Abstract Algebra Simp**
Compile Error! Click the <:errors:504266556054962196> ...
@short kraken do you get this one or not?
tidal dock
because in the step above we just have x
short kraken
lemme look at it
tidal dock
one singular x
in the denominator
but in the next step we have 2 x
to preserve equality we multiply by 2
short kraken
so sin2x/cos2x * 1/2x * 2
but how did the 1/x * 2/2 turn into 1/2x*2
it would be 2/2x no?
vast shale
it would be 2/2x no?
yes
which you can simplify to be 1/x
remember we don't want to change the value of 1/x
short kraken
okay but where did 1/2x specifically come from
im sorry for being frustrating its just not clicking
vast shale
okay but where did 1/2x specifically come from
you get that 1/2x*2 = 2/2x right?
like there is no difference