I don’t understand it
#help-23
sand palm
safe radishBOT
inland ivy
Np
safe radishBOT
Available help channel!
queen parcel
Euler's formula is $e^{ix} = \cos{x} + i\sin{x}$.
For what values of x is this valid? All real numbers? All complex numbers?
flat frigateBOT
Shen
pulsar condor
iirc all x in C
queen parcel
Okay - I'm going to think about this for a bit
queen parcel
If z = 1 + 2pi i, then the LHS is e - but is the RHS equal to e as well?
i looked on wikipedia and i think I'm good for now
.close
cyan niche
Hi.
I am wondering how the definition of integration as the 'limit of a sum' relates to the actual process of integration, and how integration is the antiderivative. At the moment, this other definition involving the limit of a sum feels separate to me. I know if you use the limit of a sum formula, you can calculate an integral - but how does it relate to differentiation and stuff?
stoic dune
You're right, how the area under a curve relates to differentiation is not obvious, and this equation doesn't explain it
That is, this equation is not the same as the second fundamental theorem of calculus
cyan niche
hmm ok
so if I were to understand the origin of integration, and why it gives the area under a curve when you apply an upper and lower bound, I should look to the second fundamental theorem of calculus?
stoic dune
The above formula:
How area under a curve relates to a sum of infinitely many tiny parts
The second fundamental theorem of calculus:
How the area under a curve relates to the derivative of a different curve
Both are important viewpoints for the same thing
cyan niche
Right, I see. But when thinking of the second fundamental theorem, don't we still do approximations, where we take a small change, dx to the function, and find the resulting change in area? So surely there is a link to the 'limit of a sum' definition somewhere?
safe radishBOT
@cyan niche Has your question been resolved?
cyan niche
but Kaynex has been very good so far Bot
safe radishBOT
@cyan niche Has your question been resolved?
pale fern
Uh
I don't know if its considered a troll or not but
If you got the answer what's the point of the method
torpid rampart
I have a test and I need to study
torpid rampart
I don't know how to get that answer
pale fern
I have a test and I need to study
Does it say " show your work" ?
pale fern
I don't know how to get that answer
Oh
pale fern
You said " the answer is -3,4 "
torpid rampart
Yes
torpid rampart
Idk my teacher just gave me an answer sheet from the text book
torpid rampart
.close
zealous ginkgo
did vertical method on the top and the denominator and know I'm trying to figure out what's the undefined is.
safe radishBOT
did vertical method on the top and the denominator and know I'm trying to figure...
sour abyss
when the denominator is 0 it is undefined
so values of x which make the denominator 0
zealous ginkgo
But
I’m confused on what to do next
i know i need to denominator to be 0 but i dont know what steps to getting a 0 as the denominator
sour abyss
find what makes x^2 - 5x - 14 = 0
because that’s the denominator
for the values where it’s 0 the function is undefined
sour abyss
well the numerator can equal anything and the function will always work
even if it = 0
because 0/x = 0
but x/0 is undefined
so you only have to worry about the bottom of it
zealous ginkgo
would the answer be, -6 - 2 when undefined, x=7 & x=2
zealous ginkgo
ah right forgot about the negative
sour abyss
yeah so it’s undefined for x = 7 and x = -2
zealous ginkgo
alright thanks
alpine meadow
can someone explain the steps to get the answer to this question?
royal kiln
can someone explain the steps to get the answer to this question?
- where is the vertex of this parabola?
- does this parabola open up or down?
(waiting)
If you just want to turn your brain off and get the answer, solve this
$0 \ge -3(x+1)^2 + 12$
flat frigateBOT
Disorganized
flat frigateBOT
Disorganized
safe radishBOT
@alpine meadow Has your question been resolved?
royal kiln
...and I will warn you to remember what happens to the constant side of the equation (and the inequality) when you take an even root of both sides.
lusty shard
https://gyazo.com/62c8b734f8ce29daad4b5b7a558577bc hey, I thought i could do this...but i don't think i get it. I tried to solve for the "same" euler and indicial equations to see if i get the same characteristic equation but i dont think that's gonna happen (this is an extra credit)
idk what my prof means by make x= e^t... i mean obviously plug that into the equation shown but idk
safe radishBOT
https://gyazo.com/62c8b734f8ce29daad4b5b7a558577bc hey, I thought i could do thi...
idle dove
dy/dx = y' = y'(x(t)) * x'(t), and you know what x'(t) (and in fact also every derivative of x(t)) is
You can show something similarly for the second derivative. Because the derivatives of y don't vanish with this substitution, it's still linear and constant-coefficient.
lusty shard
alright! let's see if i can apply this
could i set alpha and beta to 1 or would i need to keep those in to aid with my answer?
safe radishBOT
@lusty shard Has your question been resolved?
lusty shard
like i plugged in using the conditions above but im not sure if thats right so i just plugged it in first and then did the derivatives next for the 2nd one
safe radishBOT
@lusty shard Has your question been resolved?
lusty shard
<@&286206848099549185>
idle dove
rotate the image and make it legible
lean otter
can someone help me?
safe radishBOT
can someone help me?
outer ravine
So are you familiar with differentiation @lean otter ?
In that equation you differentiate all the x variables
And when you differentiate the y variables you add a dy/dx next to it
That's a vid to show you how to solve that question
safe radishBOT
@lean otter Has your question been resolved?
normal geyser
can someone help me?
safe radishBOT
can someone help me?
normal geyser
At the gym Luna swims every 6 days, runs every 4 days and cycles every 16 days. If she did all three activities today,
in how many days will she do all three activities again on the same day?
safe radishBOT
@normal geyser Has your question been resolved?
normal geyser
<@&286206848099549185> 15 minutes
velvet nebula
perhaps try solving a simpler problem, if you just consider swimming and running, in how many days will she do both?
safe radishBOT
@normal geyser Has your question been resolved?
safe radishBOT
bright crow
You have not made a mistake.
Remember that there is a constant term at the end +C, the extra +1 is also a constant
Yes
twilit bolt
safe radishBOT
twilit bolt
How would I use u substitution here to get the answer
normal cosmos
divide the numerator and denominator by 4
twilit bolt
Apparently it’s supposed to be (sqrt3x)/2 for u
normal cosmos
and use this result
twilit bolt
divide the numerator and denominator by 4
Why do I divide by 4
normal cosmos
Why do I divide by 4
by 3 sorry
or if you want to approach it step by step, make the substitution
twilit bolt
or if you want to approach it step by step, make the substitution
I have to approach it showing steps for points
which brings me to the question of how because my substitution doesn't get me the answer for some reason
normal cosmos
t=\frac{2}{\sqrt{3}}\tan \theta
$t=\frac{2}{\sqrt{3}}\tan \theta $
$ t=\frac{2}{\sqrt{3}}\tan \theta $
pastel verge
by 3 sorry
Or 4?
spark sleet
Help
pastel verge
$t=\frac{2}{\sqrt{3}}\tan \theta $
No space before $
pastel verge
No space
normal cosmos
$t=\frac{2}{\sqrt{3}}\tan \theta$
flat frigateBOT
the.dorkest.knight
twilit bolt
this is how its supposed to be done but i don't get the u sub
normal cosmos
this is how its supposed to be done but i don't get the u sub
just substitute u=tan(theta)
and solve the resulting integral
safe radishBOT
@twilit bolt Has your question been resolved?
low jungle
Trying to find P in terms of ONLY x and ONLY y and I thought these were right but apparently they are both wrong. Could I get any help to where I went wrong?
safe radishBOT
Trying to find P in terms of ONLY x and ONLY y and I thought these were right bu...
@low jungle Has your question been resolved?
upbeat dragon
Looks to be right, except for the part where you put in your value for y to obtain P(x)
You're missing a few brackets and the division by 2
Not entirely sure if this helps tho lol
low jungle
from the choices I have from P(x) none of them have x^3((8-x)/2) lol thats why I didnt include the division by 2
upbeat dragon
Okay, there is a right answer among them
upbeat dragon
from the choices I have from P(x) none of them have x^3((8-x)/2) lol thats why I...
You need to multiply your x^3 into (8-x)/2
low jungle
isnt it just (x^3(8-x))/2 ??
pastel verge
It's xy = x³((8-x)/2)
safe radishBOT
@low jungle Has your question been resolved?
safe radishBOT
flat frigateBOT
toutoi sonzai.
safe radishBOT
grizzled fossil
What is 3^4 mod 10?
grim sandal
yea makes sense
grizzled fossil
Now can you deduce what 3^2020 is knowing that 3^4 = 1 mod 10?
grim sandal
wait so
is it wrong to assume that the reason why we get $3^2$ is because $3^{2020} \equiv 1 \mod 10 \therefore 3^{2022} \equiv 1 \times 3^2 \mod 10$
flat frigateBOT
toutoi sonzai.
grizzled fossil
No but weren’t you asking why 3^2020 = 1 mod 10?
safe radishBOT
Available help channel!
grim sandal
No but weren’t you asking why 3^2020 = 1 mod 10?
yea my bad
safe radishBOT
yea my bad
jolly delta
hello. a simple question: why are we multiplying 1 when simplifiyng exponents?
safe radishBOT
hello. a simple question: why are we multiplying 1 when simplifiyng exponents?
@jolly delta Has your question been resolved?
pale fern
hello. a simple question: why are we multiplying 1 when simplifiyng exponents?
Huh
Because your simplifying exponents
Oh yeah
Your right
fringe thicket
hello. a simple question: why are we multiplying 1 when simplifiyng exponents?
idk why are u lol
fringe thicket
hello. a simple question: why are we multiplying 1 when simplifiyng exponents?
the only reason i could think is maybe theyre trying to show why 2^0=1
not a very good explanation tho
jolly delta
yes, it's a video about zeroth power
fringe thicket
a number to the power of 0 is 1 coz u want it to satisfy the multiplication index law
(n^0)*(n^1)=n^1
divide both sides by n^1 and u get n^0=1
jolly delta
what's the "multiplication index law"
old oriole
safe radishBOT
old oriole
i dont know how to count the area and i dont get question ii
flint ledge
question 2 just asks for the area of all the shapes
@old oriole do you know how to do question 2?
mild tangle
probably the best way to go around this is by dividing the given shape into familiar shapes, and then solving for their areas and adding them together
r u able to do the rest @old oriole
they r pretty simple and the formulas can be found on google if you haven't been familiarised with them
quartz delta
Let $v_1, v_2, v_3, v_4$ be vectors of $\mathbb{R}^n$ that are orthogonal 2 by 2. Prove that $(v_1, v_2, v_3, v_4)$ are linearly independent.
flat frigateBOT
TC159
quartz delta
I was thinking of centering $v_1$ and doing the dot product of $v_1$ with all of them to get a homogeneous system
flat frigateBOT
TC159
pastel verge
What does orthogonal 2 by 2 mean?
safe radishBOT
@quartz delta Has your question been resolved?
quartz delta
What does orthogonal 2 by 2 mean?
every pair of vectors is orthogonal
$v_1 \cdot v_2 = 0$
pastel verge
I wrote linear combination adding to 0
quartz delta
what?
quartz delta
Oh, I mean, I don't see how you proved anything. How does that say that those (c_1, c_2, c_3, c_4) are the only solutions?
pastel verge
No that wasn't proof I was just stating it
quartz delta
stating that they're linearly independent...?
pastel verge
You need to start with finding projections to show they are linearly independent
pastel verge
There's one more way by dot product
inland ivy
Dot product is best
safe radishBOT
@quartz delta Has your question been resolved?
covert fable
safe radishBOT
main kelp
Look at the bases of A,B,C, what do the angles there have in common?
Something that D does not satisfy
covert fable
they all have a rectangular base
main kelp
So the angles are...
covert fable
90 degrees?
main kelp
Yes
So, an example would be that the prisms are sorted in descending order of the number of right angles in the base.
covert fable
but none of the rules can fit A and D but this one that I just gave doesn't fit D but it does fit A
main kelp
I don't understand what you just said
covert fable
okay nevermind but I still don't understand what the 2 rules could be
solemn willow
hi
solemn willow
just a simple question
if i were being asked to write the element of a domain and co-domain
should i write
1,2,3,4,5 only or i need to write {1,2,3,4,5} in bracket too
like 1,2,3,4,5 ∈ A right?
or no need for ∈ A
median flint
$1,2,3,4,5 \in A$ is a shortcut for $"1 \in A$ and $2 \in A$ and $3 \in A$ and $4 \in A$ and $5 \in A"$
flat frigateBOT
polikuj2
median flint
${1,2,3,4,5}$ is the set containing exactly 1,2,3,4,5
flat frigateBOT
polikuj2
median flint
So you could write ${ 1,2,3,4,5} \subset A$, but ${ 1,2,3,4,5} \in A$ makes no sense
flat frigateBOT
polikuj2
solemn willow
$1,2,3,4,5 \in A$ is a shortcut for $"1 \in A$ and $2 \in A$ and $3 \in A$ and $...
cuz the question asked me to state the elements only
solemn willow
like 1,2,3,4,5 ∈ A right?
so i can do like this right?
pulsar condor
if i were being asked to write the element of a domain and co-domain
Well what's the domain and codomain?
solemn willow
Well what's the domain and codomain?
pulsar condor
They ask for Q
So answer with Q
If you write Q in roster, you're writing out all elements of Q
cerulean sparrow
safe radishBOT
https://i.imgur.com/T23vt6E.png
cerulean sparrow
Whats the diff between into and onto in discrete math?
pulsar condor
Onto means surjective
Into means the codomain is B
Ie it's how Im(f) relates to B in terms of proper subsets
Mapping into just means you output stuff from B
Mapping onto means you map to everything in B
cerulean sparrow
Thanks!
cerulean sparrow
Mapping into just means you output stuff from B
Mapping onto means you map to ev...
Wait so, into just means There exists a A->B?
and onto is For all B, there exist an A
pulsar condor
Yes
safe radishBOT
@cerulean sparrow Has your question been resolved?
solemn willow
kekw
steel stag
ya
quasi path
solemn willow
90 minus ?
pulsar condor
why would you claim a channel... then not post in it?
solemn willow
😅
flat spoke
If you're good mifuyu, close this one. They have their own channel.
lost field
solemn willow
.close
safe radishBOT
quasi path
The problem is cos theta+ sin theta= √2 sin (90-theta) then cos theta- sin theta=?
Please help
pulsar condor
The problem is cos theta+ sin theta= √2 sin (90-theta) then cos theta- sin theta...
You have a channel.
safe radishBOT
Available help channel!
ivory pawn
safe radishBOT
ivory pawn
hello can you help me answer this
lost field
For 1-6, you see that it's a product of (x-...), as when it's just x, think of it like (x-0)
When are these products 0? This is the same as asking what 0 * anything is
for 7, you can pull out an x by factoring
Then you are left with something like x^2 + bx + c
@ivory pawn
ivory pawn
i dont get it 😦
lost field
That's okay, I'll work through a similar one for you
Let $P(x) = (x-2)(x+3)(x-5)(x+7)$. You want to find the zeros of the polynomial $P(x)$. What this means is the values of $x$ such that $P(x) = 0$. Recall that 0 times any number is 0, so you just need to find where (x-2) = 0, (x+3) = 0, (x-5) = 0, and (x+7) = 0
flat frigateBOT
kirby
lost field
Well, if x - 2 = 0, then x = 2
Then if x + 3 = 0, then x = -3
The rest follow similarly
lost field
Here's a graph of the P(x) I showed you
so you can see the visual sort of thing you're doing
lost field
yup
What do you mean the P?
You already have it, they give you P(x) = ...
You just need to find P(x) = 0
P(x) is a function
lost field
2,-4,-5,7
You put in these values, then you see that P(2)=0, P(-4) = 0, P(-5)=0, and P(7) = 0
queen parcel
The problem is just asking you for information about P
lost field
Yup
queen parcel
x = 2, x = -4, …
lost field
Yea
queen parcel
Because those are the x-values the problem is asking for
Namely, the x-values that make P(x) equal to 0 for this example
lost field
nono, you just need to write x = 2, x = -4, x = -5, x = 7
queen parcel
You’re not necessarily trying to show that they make zero.
You’re trying to say that they do
lost field
yes
worthy hemlock
queen parcel
x counts as a factor as well, just like the rest of the stuff inside parentheses
ivory pawn
aw what
so what do i do
i use the x on the p
like put 0
so i put
1 and -1
orrr
uhhh idk
queen parcel
You are just adding x = 0 to you solution set
Because that value of x (namely 0) will make that P(x) zero
queen parcel
Perfect!
tawny thorn
Can someone pls help with 3) a and b. For a) I’m literally getting 60987.06 but the answer at the top ain’t even close
safe radishBOT
Can someone pls help with 3) a and b. For a) I’m literally getting 60987.06 but ...
tawny thorn
<@&286206848099549185>
safe radishBOT
@tawny thorn Has your question been resolved?
tawny thorn
Anyone?
ember bough
it's compounded monthly?
ember bough
and also she deposits $1500 every year
tawny thorn
Yes
ember bough
so each month she earns 8.2/12 % of the amount of money in her account
tawny thorn
Yes
ember bough
so each year she'll earn (roughly) 8.515% of her money in the back plus $1500
ember bough
wait what?
also printed out the list of money she has each year
1623
3379.09
5279.17
7335.06
9559.54
11966.4
14570.7
17388.5
20437.3
23736.2
27305.5
31167.6
35346.3
39867.7
44759.9
50053.2
55780.6
61977.6
68682.7
75937.7
83787.6
92281.2
101471
111415
122174
133815
146411
160040
174786
190741
208005
226685
246896
268764
292426
318028
idk
tawny thorn
Can someone pls help with 3) a and b. For a) I’m literally getting 60987.06 but ...
Could some pls help
ember bough
<@&286206848099549185>
tawny thorn
Is anyone there
tawny thorn
<@&286206848099549185>
azure storm
hola
what is the question
ah i see
im not an expert but
i think this best sums up how to do it @tawny thorn tho econ isnt my strong suit
lean otter
Help
lean otter
with workings pls
sorry you wont get this here
lean otter
we'll help you
ty
normal cosmos
what 's 2h when h=7, 3k when k=5
lean otter
what 's 2*h when h=7, 3*k when k=5
IDK
lean otter
bruh
queen parcel
Substitute the values you’ve been given into the expression (what others have said).
lean otter
I don't want
normal cosmos
I don't want
then why did you ask on this channel?
queen parcel
^^
@sleek forge You have to learn how to do it yourself
If you read #❓how-to-get-help, you’d know this
safe radishBOT
@sleek forge Has your question been resolved?
broken void
safe radishBOT
https://media.discordapp.net/attachments/812076295680163841/912924298770018434/u...
safe radishBOT
@broken void Has your question been resolved?
broken void
<@&286206848099549185>
broken void
so it's preferences?
wait so I add 12+5+8+1+10+3+2
and then I get 41
I'm gonna close it then idk...
.close
winter pivot
@lean otter
safe radishBOT
<@456226577798135808>
lean otter
A wheel of cheese in the shape of a right circular cylinder is 18cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15 degrees is cut from the wheel, find the volume of the cheese
Afaik i use the s=r theta theorem
winter pivot
find the volume of what?
winter pivot
the wedge that you cut? or the entire cylinder?
lean otter
the wedge i think
winter pivot
this is simple.
winter pivot
15/360 * volume of cylinder
lean otter
the other dude was telling me i use s = r theta
as a hint it says "use a new triangle area formula the multiply by the thickness"
winter pivot
you don't need to do that really.
it works but it's much simpler to just think of the wedge of cheese as a certain percentage of the entire cylinder
lol!
lean otter
is that where the s = r theta comes in
wait
look in #help-0
is what the other dude is saying right
winter pivot
it's right, but as i said it's not the best way to go about it. you really do not need theta = s/r
lean otter
ok
lean otter
btw what is my unit
winter pivot
for what
lean otter
volume of the wedge
winter pivot
cm^3
winter pivot
.close
fallen sail
hello i have a question please why do we have in this proof |α|<\frac{ε}{M} where does the 1/M come is it because of this theorem 2 (second image)
fallen sail
isn't it normally |α|<ε ?
safe radishBOT
@fallen sail Has your question been resolved?
fallen sail
<@&286206848099549185> pls 
safe radishBOT
@fallen sail Has your question been resolved?
fallen sail
<@&286206848099549185> 
pulsar lynx
ERM how do I LT this? my formula book says the top part must be equal to the sqrt25 number 😦
safe radishBOT
ERM how do I LT this? my formula book says the top part must be equal to the sqr...
pulsar lynx
and our lecturers solution just writes it out as if it acted like the 1 on top was a 5
like dis
me confusey
flat frigateBOT
safe radishBOT
@pulsar lynx Has your question been resolved?
proven lava
Hey! Can someone explain what f^-1 means?
safe radishBOT
Hey! Can someone explain what f^-1 means?
quasi bison
inverse function of f
f^-1(a) is the solution of the equation f(x)=a
some functions do not have inverses
proven lava
uh i didnt understand😬 can u give an example
pastel verge
f(1)=2 then f^(-1)(2)=1 if inverse exists
As an example
The given function in your question is for non negative reals
pastel verge
Should have mentioned the domain I think
proven lava
ohh so ive to take 17 as the solution of x^2+1?
proven lava
ah gotcha, thanks
pastel verge
And x>0 looking at options
proven lava
yeah but it doesnt work for the 2nd one, so ig the ans is D)
proven lava
yessir got it, thanks
pastel verge
I shouldn't have assumed the domain to be non negative actually the inverse doesn't exists for R
proven lava
for the 2nd one?
pastel verge
For both there will be no answer cuz f inverse doesn't exists
pastel verge
Not now uranium
pale fern
Wait r u the helper @pastel verge or him
pastel verge
I'm helping him
proven lava
For both there will be no answer cuz f inverse doesn't exists
what why
pastel verge
what why
For inverse to exists the function needs to be one-one and onto
Or bijective if you use that term
proven lava
ive never heard of either of those terms lol
proven lava
hmm yeah
pastel verge
But then if this is an inverse function 17 can't map on two elements
inland ivy
Wouldnt the inverse be pm
proven lava
ohhh u mean that thing
domain cant have 2 functions but a function can map to 2 domains?
pastel verge
It won't stay function is what I'm saying
inland ivy
Oh reasonable
quiet linden
hey im trying to solve this, i did a factorization of the left hand side but now im not sure how to continue
safe radishBOT
hey im trying to solve this, i did a factorization of the left hand side but now...
analog jackal
safe radishBOT
analog jackal
I'm trying to solve these three equation. So far I have try to simplify the first equation as X1, the second as Fs and the third as X3. But I'm stuck on how to get the final equation
Can give me any idea on how to start with these three
<@&286206848099549185>
safe radishBOT
@analog jackal Has your question been resolved?
safe radishBOT
@analog jackal Has your question been resolved?
safe radishBOT
@analog jackal Has your question been resolved?
red glen
Does x1,x2,x3 are only variable here?
lean otter
Just want to know if my approach to this problem is correct
safe radishBOT
Just want to know if my approach to this problem is correct
@lean otter Has your question been resolved?
safe radishBOT
lean otter
.close
safe radishBOT
Available help channel!
lean otter
can someone help me solve x^2 = 9
safe radishBOT
can someone help me solve x^2 = 9
flint ledge
square root both sides
add a +- onto the left side
because x can be positive and negative because it’s squared
lean otter
so, 3?
flint ledge
and -3
lean otter
Why
flint ledge
(-3)^2 = 9
lean otter
ohhh true ok thx
delicate bobcat
x = 3 OR -3
Not and for these (minor quibble)
safe radishBOT
Available help channel!
median lynx
HELPP
median lynx
it is known that the value for the Basic Statistics course is normally distributed with an average of 76 and variance 8.
if the highest 10% score gets an A, what is the limit for the lowest score to get A?
pls tell me the method
safe radishBOT
@median lynx Has your question been resolved?
safe radishBOT
normal cosmos
lg = log base ten?
yeah
so represent the radical as an exponent
and use this
$\log x^{m} =m\cdotp \log x$
flat frigateBOT
the.darkest.knight
normal cosmos
$\sqrt[a]{b} =b^{\frac{1}{a}}$
flat frigateBOT
the.darkest.knight
safe radishBOT
safe radishBOT
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ornate vigil
safe radishBOT
ornate vigil
I dont get the question for this one
<@&286206848099549185>
and this one too:
i thought i got this one but no i will show what i did
<@&286206848099549185>
toxic tide
A^t is?
ornate vigil
it is the transpose
wait
let be try to write the tranpose
and see if it is correct
toxic tide
And do at.a
toxic tide
X = [ x, y]
short rover
A^T is a (2x1) matrix
A is a (1x2) matrix
Multiplying A^T * A you'll get a (2x2) matrix
pastel verge
as the order is not the same
,w inverse {[2,-2],[0,-2]}
flat frigateBOT
pastel verge
Find A^(-1)B
ornate vigil
how does 1/2 get in the pic
ornate vigil
A^T is a (2x1) matrix
A is a (1x2) matrix
Multiplying A^T * A you'll get a (2x2)...
it worksssss yes thanks
short rover
a few things:
matrix division isn't a thing, from AX = B you cannot divide by A ; writing X = B/A doesn't make sense
however you can multiply by A^-1 on the left
AX = B, multiply by A^-1 on the left
A^-1 (AX) = A^-1 (B)
X = A^-1 (B)
this is most likely different from BA^-1
ornate vigil
but there is not B
pastel verge
Wdym
ornate vigil
wait there is
short rover
can you show your work
safe radishBOT
@ornate vigil Has your question been resolved?
ornate vigil
<@&286206848099549185>
frosty grove
what is the question?
ornate vigil
this one
ornate vigil
i dont know what that is
frosty grove
ok
1 min
so we get this equation right?
when X is the < a b c d > matrix
@ornate vigil are you here?
so, after this, you can multiply the matrices at the left size and get this equation:
then, you can solve the equation (you get these for equations)
and that's it.
ornate vigil
so, after this, you can multiply the matrices at the left size and get this equa...
how come there are -2c
i think now my problem is to understand the multiplication
frosty grove
you multiply the second row of the first matrix with the first column of the second matrix
ornate vigil
the 2nd now for the first is 0 -2 right and the first collum is a and c
frosty grove
yea
frosty grove
in the first line?
ornate vigil
thats the first row
frosty grove
then, you can solve the equation (you get these for equations)
of this?
ornate vigil
so, after this, you can multiply the matrices at the left size and get this equa...
here
in the first parentese
frosty grove
you multiply the first row of the first matrix and the first column of the second matrix
(2, -2) * (a, c) = 2a -2c
If you just learned matrix multiplication I suggest you work with the definition near you
you can also use a matrix multiplication calculator to check yourself. I love this one https://matrixcalc.org/en/
ornate vigil
you multiply the first row of the first matrix and the first column of the secon...
this explination explained what i was confused about
thanks a lot
i can finaly move onnn
frosty grove
Great, Good Luck!
ornate vigil
thank uuu
safe radishBOT
delicate bobcat
To find inverse do this
Think of f as
y = (1/5)x -7
Solve for x in that equation
What you get for x =
From doing the above is the inverse of f
35 not 30
Yea
So g(x) = 5x + 35
Works for all functions that have an inverse
Yw
ornate vigil
then, you can solve the equation (you get these for equations)
so I am geetting the same numbers again T_T
<@&286206848099549185>
I think the b is the problem
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Available help channel!
cosmic grove
and what's the problem ? @tidal bronze did you try it ?
to prove that $U^{\perp}$ is a subspace of $\mathbb{R}^n$, there is a classic method for this kind of question, and you can find it in your lessons
flat frigateBOT
Herels
cosmic grove
First you show that
$$U^{\perp} \ne \emptyset$$
flat frigateBOT
Herels
cosmic grove
the classic way to do it is to show that $0_{\mathbb{R}^n} \in U^{\perp}$
flat frigateBOT
Herels
cosmic grove
after this, you show the stability of U^T by linear combination of two vector of this space
pulsar condor
show that <u,0>=0
safe radishBOT
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Available help channel!
hazy nebula
safe radishBOT
hazy nebula
i not sure how to even solve the first qns
although i took the u1 and subbed the 1 in, i still did not get u=0.25
marsh walrus
hmm
marsh walrus
although i took the u1 and subbed the 1 in, i still did not get u=0.25
wait, how do you mean?
marsh walrus
you need induction
hazy nebula
for part (a)?
flat frigateBOT
jan Niku
marsh walrus
or i guess you could do it by hand
marsh walrus
okay
safe radishBOT
@hazy nebula Has your question been resolved?
hazy nebula
i still cannot solve it
i'm not sure where to start
My workings are just a bunch of sribbles
marsh walrus
noble attempt uhh
have you figured out what u_2 and u_3 are?
i cant tell
@hazy nebula
hazy nebula
uhm
u2 is 4/9 and u3 is 7/16
don't i just use the equation in part(b) and sub in the values?
marsh walrus
i dont think youre supposed to
but you can
do you know how induction works like in general?
marsh walrus
i mean i just wanna be sure if we start going through it
if youve done other induction problems successfully then im sure its fine
hazy nebula
marsh walrus
so weve done the basis step right?
hazy nebula
yes
marsh walrus
cool
then we need the inductive step
lets assume that $u_n = \frac{3n-2}{(n+1)^2}$
flat frigateBOT
jan Niku
marsh walrus
and show that $u_{n+1} = \frac{3(n+1)-2}{((n+1)+1)^2}$
flat frigateBOT
jan Niku
flat frigateBOT
jan Niku
marsh walrus
now for the mystery part
my guess is
youll have to start with u_n+1 as given by part b
and end up with the relation proposed in part a
lmc
,w factor 3x^2-x-9
flat frigateBOT
marsh walrus
flat frigateBOT
marsh walrus
like
heres my thought
if you wanna try as well
we want some nice form of u_n+1
so we use $u_{n+1} = \frac{3n+1}{(n+2)^2}$
flat frigateBOT
jan Niku
marsh walrus
then we do $\frac{3n+1}{(n+2)^2} = u_{n+1} - u_n + u_n$
flat frigateBOT
jan Niku
marsh walrus
then we know the process that connects the two forms
youll still have to rewrite without assuming the conclusion though
this looks like it will work
,w simplify (n+1)^2 (3n+1) - (n+2)^2 (3n-2)
flat frigateBOT
hazy nebula
huh
marsh walrus
so thats my argument for the inductive step
start by like
we have assumed that u_n = ....
then after simplification
we add u_n to both sides and isolate u_n+1
to get $\frac{3n+1}{(n+2)^2} - u_n + u_n = u_{n+1}$
flat frigateBOT
jan Niku
marsh walrus
simplifying the first two terms on the LHS we obtain the relation given in part a
(algebra left as an exercise :p)
but it just kinda happens
turn into a single fraction, then simplify the denominator
youll get exactly what you want
that one was fun 😄
marsh walrus
well its like
we did the basis step right
now we need to show that if we assume that it holds for some u_n, we get something that still holds true for u_n+1
aka we showed we can push the first domino over, we need to show that every domino after that will knock over the next domino
so we get to assume for free that $u_n = \frac{3n-2}{(n+1)^2}$
flat frigateBOT
jan Niku
marsh walrus
what we'd like to show, using everything were given, is that $u_{n+1} = \frac{3n+1}{(n+2)^2}$
flat frigateBOT
jan Niku
marsh walrus
good up to there? @hazy nebula
hazy nebula
yes
marsh walrus
so, part of the given information is that relation
$u_{n+1} = u_n + \frac{9+n-3n^2}{[(n+1)(n+2)]^2}$
flat frigateBOT
jan Niku
hazy nebula
yes
marsh walrus
this is just true
theres no uhh
we just get this for free
anything we do to this starting from here is just true, as long as the algebra works out
were not assuming anything by starting from this equality
since it defines the relation
recall that we also get to assume that $u_n = \frac{3n-1}{(n+1)^2}$
flat frigateBOT
jan Niku
marsh walrus
by inductive hypothesis
hazy nebula
ya
marsh walrus
so lets substitute
$u_{n+1} = \frac{3n-1}{(n+1)^2} + \frac{9+n-3n^2}{[(n+1)(n+2)]^2}$
flat frigateBOT
jan Niku
marsh walrus
still havent assumed anything, just algebra
well i guess we assumed the inductive hypothesis
but thats okay
okay, heres the part where its up to your teacher like
the algebra to discover this is the algebra you need to do would probably be considered scratch work
aka we are doing a proof, not a derivation
so I would just write this, if it were me
$u_{n+1} = \frac{3n-1}{(n+1)^2} + \frac{(n+1)^2(3n+1)-(n+2)^2(3n-2)}{[(n+1)(n+2)]^2}$
flat frigateBOT
jan Niku
marsh walrus
im not sure about finding the algebra that gives you this form backwards
youd need to do some scratch work forwards to know this is what you need to do
hazy nebula
how did the 9+n-3n^2 become that
marsh walrus
i wanna say
thats a good question, we should pause it since its gonna get confusing
since were so close
finish this part, then we can look
so just assume its reasonable, and i can show after
we continue from here, by some argument this is equal to the thing above it, well see the argument in a sec
we split along the middle minus in the numerator of the second fraction
sorry i gotta see it rendered
something went awry 
flat frigateBOT
jan Niku
\begin{align*}
u_{n+1}
&= \frac{3n-2}{(n+1)^2} + \frac{(n+1)^2(3n+1)-(n+2)^2(3n-2)}{[(n+1)(n+2)]^2} \\
&= \frac{3n-2}{(n+1)^2} + \frac{ (n+1)^2(3n+1) }{ [(n+1)(n+2)]^2 } - \frac{(n+2)^2(3n-2)}{[(n+1)(n+2)]^2} \\
&= \frac{3n-2}{(n+1)^2} + \frac{3n+1}{(n+2)^2} - \frac{3n-2}{(n+1)^2} \\
&= \frac{3n-2}{(n+1)^2} + \frac{3(n+1) - 2}{((n+1)+1)^2} - \frac{3n-2}{(n+1)^2} \\
&= u_n + \frac{3(n+1) - 2}{ ((n+1)+1)^2} - u_n \\
&= \frac{3(n+1) - 2}{((n+1)+1)^2}
\end{align*}
