So you're trying to find 35% of 300 right
#help-28
wraith river
wraith river
Try 105
wraith river
What the
wraith river
also i rly appricite u helping
I probably shouldve let an actual professional help LOL
Do u lose marks if u get it wrong what are u doing
atomic grove
no
long story short i got confrinciss and i have to do all 20 assiments for homework
sorry for my grammar and spelling
atomic grove
im almost done
atomic grove
its right
wraith river
Oh how come it wasnt right before
atomic grove
it only syas its right when u get both correct
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wraith river
Okk
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torn jolt
hello i’ve been stuck on this
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hello i’ve been stuck on this
lilac valve
From the second to the third line, I don't think you grouped the dy/dx on the left side with the other instances of dy/dx
by the time you reach the third line it should be dy/dx(-5-x)=-3+y
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torn jolt
what is cmd
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what is cmd
onyx glen
a particular string of three letters
if it is an abbreviation then there are many things it could stand for
haughty temple
hello, may I ask why when we define principle argument, we bound the argument to the interval -180 < arg(z) < =180? I understand why we have to do this so as to have a unique value instead of having multiple arguments, but what I don't understand is why it cannot be equal to -180 but equal to 180
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hello, may I ask why when we define principle argument, we bound the argument to...
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potent creek
hi
torn jolt
passive in what sense?
potent creek
cant i just say hi
gritty rose
.close
errant crow
Let ABCD = rectangular and AB = 40 , BC = 30 , CQ = 16 , PG = 14 find the length of AP
round gust
maybe label the drawing
errant crow
Wdym
Let ABCD = rectangular and AB = 40 , BC = 30 , CQ = 16 , PQ = 14 find the length of AP
Sorry I type wrong
round gust
yes
onyx glen
draw a line from C parallel to PQ and extend AP past P. let R be the point where these two lines meet
PQCR is a rectangle
ACR is a right triangle whose diagonal AC is known
round gust
we can't just give you the answer
you have to figure it out
ann told you what the steps are
errant crow
Cause I use another method
I draw A to C and I do similar triangle
But Ann method seemed better
onyx glen
what triangles are you saying are similar?
errant crow
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cyan flicker
how do i know the apothem of this triangle?
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how do i know the apothem of this triangle?
cyan flicker
each edge is equal to a
and they are asking me to calculate the area of the side triangle
which is 4a * apothem
rough plaza
first off, is this a 3D-shape, which in this case a square pyramid? (since you said each edge is equal to a)
cyan flicker
yeah, i didnt know if it was square or quadrangular pyramid
cause the exercise is in my native language
rough plaza
i see
rough plaza
and they are asking me to calculate the area of the side triangle
if each side is a, what is the length of apothem?
try using bird view to easier spot the apothem
rugged harbor
do i need to calculate it using Pythagorean theory
You can use that
rugged harbor
Yup
cyan flicker
alright, let me give that a shot, 1 minute
rugged harbor
If you learned it, you can use trigonometry too, but this approach is fine
rough plaza
nice!
cyan flicker
how do i use that?
i have learned the basics i think
just gotta remind myself
on what that is
rugged harbor
There's a very nice formula to find the area of a general triangle using two sides and the angle they share, but if you are not familiar with it, what you are doing is more than enough
cyan flicker
what grade do you learn that in?
deft zodiac
think its cos rule
rugged harbor
I don't know, it changes from country to country, I learned it when I was 16, but mine was just a suggestion, follow the path you are already taking and you'll get the correct solution
rugged harbor
How did you get it?
cyan flicker
ap^2 = (a/2)^2 + a^2
ap^2 = a^2/4 + a^2
ap^2 = a^2/4 + 4a^2/4
ap^2 =5a^4
ap = square root of 5^4/2
rugged harbor
Yes looks good
cyan flicker
this is what a classmate of mine did
and this one is correct
ill send the whole exercise and the translation after sending the pic, give me 1 minute
rugged harbor
Post it in the original language too, somebody may know it
cyan flicker
Consider a regular square pyramid whose edges measure a with a belonging to the positive irrational numbers
the lateral area of the pyramid can be given as a function of a by:
and then I have 4 options
rugged harbor
Oof we used pythagoras wrong
The apothem isn't the hypothenuse of that right triangle
cyan flicker
ohhhhh yeah
rugged harbor
mb, should have checked better
cyan flicker
mb, should have checked better
thanks for the help, should've payed more attention
rugged harbor
me too 
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cyan flicker
.reopen
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cyan flicker
i'll be back in a few hours 
deft zodiac
5 hours TM
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cyan flicker
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sharp shadow
convert everything to fractional powers eg fourth root a^5=a^5/4, (sqrt b)^-2/3=(b^1/2)^-2/3
deft zodiac
,rccw
glossy valveBOT
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@cyan flicker Has your question been resolved?
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@cyan flicker Has your question been resolved?
brittle robin
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brittle robin
why does $\frac{1}{\sqrt{x^2-0.1}} \ge \frac{1}{x}$
glossy valveBOT
Conquest
brittle robin
make sense?
vast fossil
sqrt(x^2 - 0.1)/x = sqrt(1 - 0.1/x^2)
brittle robin
confused
vast fossil
sqrt(x^2 - 0.1)/sqrt(x^2) = sqrt((x^2 - 0.1)/x^2) = sqrt(x^2/x^2 - 0.1/x^2) = sqrt(1 - 0.1/x^2)
And sqrt(1 - 0.1x^2) <= sqrt(1) = 1
So sqrt(x^2 - 0.1)/x <= 1
Hence 1/x <= 1/sqrt(x^2 - 0.1)
frozen spade
$$x>1 \implies \frac{1}{\sqrt{x^2-0.1}}>\frac{1}{\sqrt{x^2}}=\frac{1}{x}$$
glossy valveBOT
秋水
vast fossil
Ah yeah that's easier actually
brittle robin
yeah im still slow to get this one, what values is this even true for? everything I plug in the domain doesn't satisfy
oH
ty nvm
.close
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barren wagon
Helloo
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worthy ginkgo
Did you draw anything?
barren wagon
<@&286206848099549185>
gritty rose
I dont kno how to do this
Try drawing a picture and share it here
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kind kraken
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fast peak
well how would the epsilon-delta definition look like for this
kind kraken
mod (x-a)
torn jolt
have you worked with anything epsilon related before?
kind kraken
for epsilon >0, if mod (n-inf) < delta then mod (an-l) < epsilon
torn jolt
what is mod in this case?
kind kraken
have you worked with anything epsilon related before?
Not really , I have not gone through that chapter yet in spivak
distance/.
distance?
torn jolt
but how can n have a distance to infinity
kind kraken
Yea It is wrongg :///
torn jolt
try not using delta
kind kraken
you mean how close n is to the infinity ?
glossy valveBOT
rbit ✨
kind kraken
since an is defined like a: N----> R
torn jolt
yes thats why instead of using a real number delta to restrict the argument, we use a natural number
now whats another way for getting something like "distance to infinity" which makes more sense
just remember that measuring the distance to infinity doesnt make sense and would always give infinity
kind kraken
just remember that measuring the distance to infinity doesnt make sense and woul...
I understood this one
so delta should be some natural number
torn jolt
maybe think of this graphically, if the sequence converges, then the further you go to the right, the points of the sequence will get closer and closer to some specific horizontal line (arbitrarily close), so for any distance you give (epsilon), you just need to go to the right for long enough until all upcoming points are inside an epsilon region of the limit right?
torn jolt
so delta should be some natural number
yeah, but instead of making delta very very small, we make it very large, so instead of calling it delta we just call it something like N0
kind kraken
I mean I should understand these concepts before attending this question because I am not able to understand what you are saying
torn jolt
ok let me find some pics
kind kraken
yeah, but instead of making delta very very small, we make it very large, so ins...
yea because if I put some natural number over delta, It still does not hold right? because n-inf is inf and inf can not be less than whatever the natural number which was put over delta
so making delta bigger , so that is how I should use N0
kind kraken
ok let me find some pics
yes pleasw
torn jolt
this is how a convergent sequence may look like
the red line is the limit (3)
and the dotted lines represent the epsilon region (like, how close you are to the limit, the smaller epsilon is the smaller the region)
kind kraken
Yea now I can understand what you previously said
torn jolt
now you can see how at the beginning some points may fall out of this region, but if you just go to the right for long enough (increase the n), eventually all values will lie inside that region. Can you see how this holds for any epsilon region, no matter how small we choose epsilon?
like, is epsilon = 0.0001, it might take long but eventually, with n≥1000 all values will lie inside that region
kind kraken
yeah yeah I understand
torn jolt
if epsilon 0.0000001 then it might be n≥100000
torn jolt
now what you said in the beginning with the epsilon was correct, if
|an - L| < epsilon
then we are inside the epsilon region
na forget about epsilon-delta, its more like epsilon-N
kind kraken
if epsilon 0.0000001 then it might be n≥100000
Hey one quick question
Have you read Spivak calc book?
torn jolt
nope
kind kraken
aight Is this something related to upper bound theorem ?
N is not bounded from above blah blah
I thought about reading that first and then going through what you said
nvm
torn jolt
in some sense, i guess, but its not important to know about that
kind kraken
Its just that I am understanding some parts but can not connect the dots to arrive at the answer
Can I try for a few mins?
kind kraken
You said that we want delta as big as possible right I meant we want it big
torn jolt
yeah but use N instead of delta (usually you use the symbols epsilon and delta for things that can become really small)
kind kraken
yea so the thing inside the mod that is n- inf will always be inf
should be less than a biggggg N , Am i missing something? Other than some few brain cells here?
torn jolt
well, instead of measuring the distance to infinity, how about we just measure how far we are (thats why instead of becoming smaller and smaller, the N becomes larger and larger)
torn jolt
no no how far we are from 0 actually
kind kraken
no no how far we are from 0 actually
I thought we were talking about mod (n-inf) < delta
torn jolt
yeah, but instead of doing that, why not do mod(n-0) ≥ N
kind kraken
I wrote it from the definition mod (x-a) < delta and in this case it is (n-inf)<delta since n is approaching inf
how can it be zero?
torn jolt
from the definition of limits for functions?
kind kraken
yeah
torn jolt
well its just that mod(n-inf) doesnt make sense, so we have to change the definition
instead of having a delta close to 0, we have an N that can become larger and larger, and instead of looking at mod(n-inf) < delta, we instead check if n≥N
kind kraken
But why should it only be zero in place of inf while changing the definition
torn jolt
because "coming closer and closer to infinity" is about the same is "going further and further away from 0", just that the first one cant be well expressed mathematically
kind kraken
yess got that got that
So its like we are moving farther away from 0 such that n.>or equal to N
torn jolt
yeah
btw, was the function definition
for any epsilon > 0:
there exists delta > 0:
so that for any x with 0 < mod(x, a) < delta:
mod(f(x), L) < epsilon?
kind kraken
THis is the definition that I know
torn jolt
so can you modify it accordingly?
kind kraken
It is not defined in the question,
This is what the question is
So it is like for any epsilon>0, there exists N , mod(n-0)> or equal to N then mod (fX - L)< epsilon
torn jolt
yeah, but instead of mod(n-0) you can just say n, so n≥N, that simplifies nicely
$N \in \mathbb{N}$
glossy valveBOT
rbit ✨
torn jolt
oh and i guess just a small overlook but instead of fX its an
torn jolt
i mean you are done at this point, that is the definition
kind kraken
Could you please put this in words for me from the beginning?
I got what we are doing but I can not connect the dots ..It is like I have to write it on my own words so I will have to understand the problem really well
If you can not I will type it out and could you point out my mistake?
torn jolt
just try, taking this as a reference, imagining how for smaller and smaller epsilon, the N will increase more and more so that all points still fit in the region
kind kraken
oh god I guess I got it now
like as we approach infinity eventually all points will fall under the interval epsilon no matter how small it is
torn jolt
yeah
kind kraken
Thank youu sm
torn jolt
np
torn jolt
yes convergent sequences are always bounded
kind kraken
Could you tell if this problem was like hard??
For me it is I mean most of my friends are struggling with it
torn jolt
oh i think coming up with the definition is harder, once you have a definition to work with that makes it easier to prove, but still not that easy id say
kind kraken
thank u sm there are like 4 sub questions based on this first question .. Now ill watch and learn what convergent series are then get into the next question but by that time this thread would close
is it okay if I pm you?
torn jolt
sure
kind kraken
ayyy ayy
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hallow garden
What is a arithmetic:sequence:
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What is a arithmetic\:sequence\:
hallow garden
Pls help me solve it people learned this in 7th grade and i forgot how to do it
sharp vine
,w arithmetic sequence
glossy valveBOT
hallow garden
also whats the ratio for 3,6,9
sharp vine
also whats the ratio for 3,6,9
rather common difference, which is 3 in this case
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hallow garden
.close
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torn moon
hey can some1 explain the last two steps pls
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hey can some1 explain the last two steps pls
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worthy ginkgo
The result is correct but the redaction is awful
Never do such a thing
Let me reformulate this for you
$a_{n} = \left(1+ \frac{7}{n}\right)^{n} = \exp\left[ n\ln \left(1+\frac{7}{n}\right)\right]$
glossy valveBOT
black_couscous
worthy ginkgo
But, $n\ln\left(1+\frac{7}{n}\right) = 7\frac{\ln\left(1+\frac{7}{n}\right) }{\frac{7}{n}}$. And you know that $f:x\longmapsto \ln(1+x)$ admits a derivative in $0$ that is $f'(0)= 1$ (moreover $f(0)=0$). So $\frac{f(x)}{x} \underset{x\to 0}{\longrightarrow} 1$ and thus $7\frac{\ln\left(1+\frac{7}{n}\right) }{\frac{7}{n}}\underset{n\to \infty}{\longrightarrow} 7\cdot 1$
glossy valveBOT
black_couscous
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stoic parcel
water is flowing at 4km/h through a circular pipe of diameter 16 cm into a rectangular tank 22×20×16m how long will it take to fill the tank
worthy ginkgo
The section is $S= \pi \cdot 0,16^{2}$
glossy valveBOT
black_couscous
worthy ginkgo
In an hour the volume going through the pipe is $S\cdot 4\cdot 10^{3}$
glossy valveBOT
black_couscous
worthy ginkgo
Does that help?
worthy ginkgo
No the area of the tube
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crimson sleet
how do I solve part b?
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how do I solve part b?
crimson sleet
x is sqrt2 btw
because 1^2 + 1^2 = x^2
so that's 2 = x^2
and then x = sqrt2
but for y, I'm not sure how to approach the question
I've tried drawing this imaginary line (the black line)
but I don't know it's length yet, so that doesn't help
and yea, I just dk
<@&286206848099549185>
quartz sleet
the answer's gonna be sqrt(3)
crimson sleet
it's halfway between 1 and y
crimson sleet
oh god, my paint skills are terrible lol
lol all g
crimson sleet
sqrt2
quartz sleet
and what's x in the geometric sense?
quartz sleet
well in the picture, they marked out an arc using x, right? so what would x be?
crimson sleet
1?
quartz sleet
hmm, i thought it was sqrt(2) lol, i'm not talking about any numerical value
quartz sleet
x would be the radius of the circle if you were to extend the arc
quartz sleet
do you think you can do something with that info?
crimson sleet
thinking
quartz sleet
gotcha
crimson sleet
can I have a hint
quartz sleet
what would this be?
crimson sleet
will that length also be sqrt2
oh!!
I see
ok
so that's sqrt2
and then the other side is 1
okay gotcha
quartz sleet
crimson sleet
.close
rapid laurel
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rapid laurel
i dont get the step of r is the largest integer
so that
step
@topaz valley @fast peak 😭
this question is similiar
by defining r
but i just dont quite get how its working
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rapid laurel
<@&286206848099549185>
@native flicker @frozen spade lol
BRO @崩溃, u cant just leave me bro
@native flicker
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gritty rose
i dont get the step of r is the largest integer
this one?
rapid laurel
yes yes yes
rapid laurel
like if i were to do this question, i wouldnt define n?
gritty rose
r is an integer. log_2(n) is a slow growing growing
r is defined for every n so it doesn't matter
n is arbitrary, and r is defined in terms of that arbitrary n
if it helps, think of r = f(n)
gritty rose
because it gives a nice answer?
rapid laurel
ahahah
this feels so odd
to me
i dont have ur intuition
ok assuming
u get r= log 2 n ceiling thing
like why is it useful?
also why are u sure that the floor of log 2n is r
for sure
gritty rose
all your "why" questions have the same answer
it gives the right answer
there are many ways to do the proof. this is just one of the easiest ones
try finding a simpler one, you'll have a hard time
rapid laurel
can u explain how each steps get me the right answer? It's quite obvious that they do
the motive behind doing each step
gritty rose
i don't really know what you're stuck on
rapid laurel
im stuck on why would u set 2^r<n<2^r+1
if i were to do this
quesiton
i would have no clue to think of that
gritty rose
can you ask a math question instead of a philosophical one
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past sinew
Hi, I need some help with understanding how to solve this!
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Hi, I need some help with understanding how to solve this!
oak basin
what have you tried?
past sinew
I'm not really sure where to start at all. I don't understand what the question's asking or how to approach it.
oak basin
right then, can you recall the limit definition for the derivative of f(x)?
you can write it out and take a picture if typing here is difficult for you to express fully of what the definition is
oak basin
no, i was asking about the first principle
anyway, seems like you know what it is, i'll skip ahead
so basically, all we need to do here is to prove that $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h} = f(x)$ for $x\in \mathbb{R}$
glossy valveBOT
waler
oak basin
are you clear of the goal of the question now?
past sinew
So do I solve for this?
oak basin
right, lets not skip ahead here
i need your confirmation if you have understood what we need to do first
oak basin
**waler**
do you understand why we have to do this?
lets just forget about the properties given, and focus on the goal first
past sinew
I'm not entirely sure, no...
oak basin
do you agree that $f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$?
glossy valveBOT
waler
past sinew
I... don't know...
oak basin
this is just the limit definition for the derivative of f(x)
oak basin
or as you can call it, the first principle definition
right then
now that we can agree on that, the question is asking us to prove that f'(x)=f(x) for x in R
oak basin
great, now it's time for us to apply the properties given for f(x)
now firstly, can you rewrite f(x+h) in terms of f(x) and f(h)?
oak basin
yes
past sinew
So, like this? And then replace it with the actual functions?
oak basin
great, that's what i was expecting
so basically $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}=\lim_{h\to 0} \frac{f(x)(f(h)-1)}{h}$
glossy valveBOT
waler
oak basin
correct?
past sinew
Yeah.
oak basin
right, and since f(x) is just a constant wrt to the limit, we can take it out
so we should have $$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\=\lim_{h\to 0} \frac{f(x)(f(h)-1)}{h}\=f(x)\lim_{h\to 0} \frac{f(h)-1}{h}$$
glossy valveBOT
waler
oak basin
right, now can you apply the second property of f(x), i.e (ii) for f(h)?
oak basin
no, im just asking you to rewrite f(h) by applying the second property
past sinew
Oh, so just the second part?
oak basin
im not sure what you meant by that
can you express f(h) in terms of h and g(h) for me?
past sinew
?
oak basin
yes
glossy valveBOT
waler
oak basin
are you clear up until now?
past sinew
Yeah
oak basin
alright, now can you apply the last property to evaluate $\lim_{h\to 0}\frac{1+hg(h)-1}{h}$?
glossy valveBOT
waler
past sinew
Like this?
oak basin
yes
glossy valveBOT
waler
compact forge
Anybody know how to solve question number 5?
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Anybody know how to solve question number 5?
compact forge
I did everything correct but got different answer to the answer
hot herald
show work, your answer and the expected answer
compact forge
Ok
So
What i did was
^3 fit in n^2 0 times so I left it
And the cube root of 448 was 8 and 56
So my answer was
The real answer was
Wuestion 5
hot herald
continue with the same idea you began with
compact forge
Do I keep going or do I look back with the number I cubed with
hot herald
you look at the number under the radical, currently 56
and see whether that has any factors that are perfect cubes
(just like how you supposedly started with the 448)
e.g. if the question was to simplify
$$\cbrt{56n^2}$$
how would you have approached that?
compact forge
Oh
glossy valveBOT
ℝamonov
compact forge
Oh I think I’m getting it
Thanks
But for 8 under radicals it can’t be 4
OHH WAIT
Thanks I get it now everything makes sense
compact forge
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compact forge
working: ^3 fits in p^5 1 time
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working: ^3 fits in p^5 1 time
compact forge
384 cube root is 24 and 16
16 cube root is 2 so 8 x 2 is 16
24 cube root is 8 and 3
8 cube root is 2
so 16x2 =32
32p^2=radicals 3p^2
answer
my answer
32p cube radicals 3p^2
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tardy quiver
Yes 
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last locust
Does anybody understand exercise 1 b) and c)? I have found in a that p=3, also I know how to calculate the basis of a nullspace of a matrix and I also know how to calculate the dimension of the nullspace. What confuses me is that I have to do it with respect to the value of p, and I dont know how to incorporate it into the equations
last locust
any help apreciated!
fast peak
well just do the calculations as normal but leave p as a variable
just make sure that whenever you have to divide by something, you don't divide by 0
from there you then have to consider different cases
so for example if at one point you want to divide by p-3, you have the two cases p=3 and p!= 3, which you have to calculate separately from there
last locust
will you be online in a few minutes when I am that far? because i dont know how to calculate them seperatly with p!=3
fast peak
maybe. or someone else might be. we'll see
fast peak
well that's the point
either p-3=0
in which case no
or p-3 != 0
in which case yes
last locust
okay so I do it twice, once with the pivot being a 3 and once 3!? Or do I still calculate it with the variable?
fast peak
I use != here as meaning "not equal" in case that confuses you
last locust
so what are my options to continue with the equation system (x1+x2+x3)?
lets take case p=3 first
do I just input 3 in row 3 column 3?
or do I do the steps with the variable p
fast peak
yes just input p=3 into the last row
last locust
just 3 then?
last locust
oh yeah of course because it cant be 3=0
and what about p -3 not equal to 0?
which numbers does that yield?
fast peak
well just p not equal to 3
could be anything
but the important part is that now p-3 is not zero
and so you can divide by it
and then treat it like any other nonzero number and do all the usual calculations with it
if you divide the last row by p-3 then you now get (0,0,1,0)
instead of the zero row which you had previously
last locust
clear, so the one input in i=3,j=3 is 0 because 3-3 is 0, and the other one is 1 because p-3/p-3=1?
fast peak
in the second case we can make it 1 by dividing, yes
and now you have two different cases from which you can do the usual stuff
last locust
perfecto
so the BColAp of option 1 where 3-3=0 is the first 2 columns of the original matrix, because the third column is linearly dependent, and the second option where we have a third pivot, is all of the columns of the original matrix?
Im just testing my vocabulary
fast peak
no clue what BColAp is supposed to be
last locust
well technically its a greek "beta"
fast peak
if p!= 3, then a basis are the first three columns, yes
(I'm assuming that you didn't make a mistake during row reducing btw, I don't wanna check)
fast peak
ehh mistakes happen
in an exam hopefully they wouldn't treat that too harshly
assuming this was an exam situation and you did indeed make a mistake, at least you did everything correctly from there, made all the correct conclusions etc
that counts
imo much more than being able to do the calculations in the first place
last locust
if I do a mistake in the determinant of a 5x5 matrix. I understand, but this is very simple row reduction and theres no way they give me points if I make a mistake
But here I’m unable to factor out or subtitute for any numbers, because x3=0 only appears one time
fast peak
yes so x_3=0 for all x in the null space
last locust
but what about x1 x2 and x4
dont I need to collect values into those vectors as well
fast peak
well you have a relationship between them
if for example x_1 =t is some parameter, then x_4=t and x_2 = -t
so every vector has the form (t, -t, 0, t)
fast peak
well first that is not a basis
second none of the vectors (-1, 1, 0, 0), (1, 1, 0, 0) or (1, 0, 0, 0) are actually in Nul(A)
last locust
no, but its the vectors that span NulA no?
fast peak
if they are not even in the space, how could they span it
last locust
how are they not in the space if I got these values from the echelon form
fast peak
I am not sure how you got these values
fast peak
p not equal to 3 I hope
last locust
yes sorry
we just get 1 as we said, so thats why theres a 1 in row 3 col 3
so considering that, isnt the basis, the span of all the linearly independent vectors of a matrix/ all the vectors with a pivot position in the REF?
last locust
no but the basis for the column space has values of the original matrix
not the reduced one
so you would put variables into the vectors of the Basis of NulA?
fast peak
we can do it without the parameter if that confuses you
if we set x_1 = 1 arbitrarily, then it follows that x_4=1 and x_2 =-1
so (1, -1, 0, 1) is in the nullspace
and {(1, -1, 0, 1)} is a basis
fast peak
yes
if the column space has dimension 3, then the null space has dimension 1
remember that the dimensions of those two add up to 4
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last locust
perfect I understood it now
But for option 2, how do I solve these 2 equations and put them into a Basis vector?
fast peak
well in the second option p=3 the column space now has dimension 2
so now you need to find 2 basis vectors
so you need to set 2 of the variables arbitrarily in two different ways and see what you get
last locust
But didnt the column space before have 3 dimensions, but there was only 1 basis vecotr?
fast peak
column space has dimension 3 => null space has dimension 1, so the basis for the null space has 1 vector
now column space has dimension 2 => null space has dimension 2 and a basis has 2 vectors
last locust
clear, so let my try to set the variables
so we need to factor the 2 equations out in a way that we get:
x_1= [...]; x_2=[...]?
fast peak
what do you get if you for example set x_1=1 and x_2=0
last locust
So this is right?
fast peak
that also works
fast peak
well there is no x3 in the formulas for x1 and x4
acoustic pebble
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acoustic pebble
So I have to show vec(OC) in terms of vec(a) and vec(b)
Is it correct??
I am also given that 4vec(AC)=vec(AB)
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acoustic pebble
<@&286206848099549185>
polar wharf
OC = a + AC
OC = b + BC
2OC = a + AC + b + BC
acoustic pebble
Yes
acoustic pebble
polar wharf
OC = 1/2 ( a+b)
polar wharf
why not
acoustic pebble
I'm given that
acoustic pebble
$4\vv{AC} = \vv{AB}$
glossy valveBOT
Basudev
glossy valveBOT
acoustic pebble
It's figure 20 and last question
acoustic pebble
How
polar wharf
C is 1/4th the way from A to B
acoustic pebble
Ye
polar wharf
right so BC is 3/4 BA
acoustic pebble
YE
polar wharf
so back to 2OC = a+b+AC + BC
2OC = a + b + 1/4 AB - 3/4 AB
2OC = a+b - 1/2 AB
AB = a + b
should be correct
polar wharf
2OC = a + b - 1/2 (a+b)
2OC = 1/2(a+b)
OC = 1/4(a+b)
no ?
wait no
because AB = b-a
it's just some directions you get the gist of it
express OC in terms of a, b AND AB, and then express AB in terms of a and b
then you simplify all in terms of a and b
polar wharf
ok correctly then
OC = a + AC = a + 1/4AB
OC = b + BC = b + 3/4 BA = b - 3/4AB
AB = b-a
2OC = a + b - 1/2AB
2OC = a+b - 1/2(b-a)
2OC = a+b + 1/2b + 1/2a
2OC = 3/2a + 1/2b
OC = 3/4a + 1/4b
acoustic pebble
Are you telling (3a-b)/4 is the answer??
polar wharf
polar wharf
looks good
acoustic pebble
Isn't my approach valid as well?
acoustic pebble
But why we get diff answers then
polar wharf
isn't it the same ?
acoustic pebble
Ok
polar wharf
AC + BC = -1/2 AB
line 3
you wrote AB/4 - b
i'll let you figure out out, good luck
acoustic pebble
Ok
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acoustic pebble
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fringe anvil
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scenic wren
we have one slight problem
consider f being f(v)=0 for all v
iirc f is still linear
oh wait
nevermind
my bad
{f(v1),...}={0,0,....,0}={0}
what i got so far is this:
assume the set of v_i are linearly independent and f is linear
now lets look at the set of all those f(v_i)
assume this the set is linearly dependent
meaning:
af(v1)+bf(v2)+...=0 with a,b,... not all being 0
since f is linear we have given:
af(v)=f(av)
f(v)+f(w)=f(v+w)
now use this:
f(av1)+f(bv2)+...=0
f(av1+bv2+...)=0
fringe anvil
but then how can I prove that av1+bv2+ .. = 0 has a non trivial solution of a=b... = 0?
scenic wren
that is the thing
fringe anvil
does not have*
scenic wren
we want to show that
f(av1+bv2+...)=0
is only true if
av1+bv2+...=0
and we know that
av1+bv2+...=0
is never true (unless a,b,...=0)
proof by contradiction
fringe anvil
is there no chance of writing one of the vectors as a linear combination of other vectors in the V space?
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scenic wren
given:
f:V->W is linear
v1,...,vn in V
if {f(v1),...,f(vn)} is linearly independent, prove that {v1,...,vn} is linearly independent
suppose {f(v1),...,f(vn)} is linearly independent
af(v1)+...+nf(vn)=/=0
f(av1)+...+f(nvn)=/=0
f(av1+...+nvn)=/=0
f cannot be the zero-function for which f(v)=0 for all v
since f is linear we have no translation, only uniform scaling and rotation
lets look at when f(av1+...+nvn)=/=0:
-f is not the zero function (which is cannot be because we know that f(av1+...+nvn)=/=0)
-av1+...+nvn is not in the kernel of f, whic hwe know because f(av1+...+nvn)=/=0
-av1+...+nvn is not 0, otherwise we would have f(av1+...+nvn)=f(0)=0
lets put this in other words:
we know:
f(av1+...+nvn)=/=0
now assume:
av1+...+nvn=0
then we would get
f(av1+...+nvn)=f(0)=0 which is a contradiction
therefore av1+...+nvn cannot be 0
only exception is if a,...,n=0
since av1+...+nvn=/=0, {v1,...,vn} are linearly independent
@fringe anvil
fringe anvil
=/= is not equal in your notation?
scenic wren
yes
fringe anvil
why did you write af(v1)+...+nf(vn)=/=0 instead of af(v1)+...+nf(vn)=0?
scenic wren
we know that {f(v1),...,f(vn)} is linear independent, meaning it does not equal to 0
fringe anvil
unless a,b,...n are 0
I got what you did now
but you should have also included "for a,b...n not equal to the trivial solution 0"
thanks
fringe anvil
yeah mate, but I prepare for an exam xd
fringe anvil
no
scenic wren
my semester just started
scenic wren
ah ok
fringe anvil
now we take the exams for period 1
fringe anvil
I do AI master at University Paris-Saclay
scenic wren
oh nice
i do math-physics-education
to become a teacher
i had linear algebra for 2 semesters
fringe anvil
the fields in maths and the nobel in physics came to our professors this year btw
xd
fringe anvil
this one?
fringe anvil
that is the soleil laboratory
it's for physics
my uni si a confederation of universities and research labs
scenic wren
ah ok
my uni is like spread all over the city, which is quite nice
means you are always close to street food haha
scenic wren
haha nice
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tardy lotus
Are you a collage student?
Ohh sorry nvm I didn't read these chats.I actually wanted to know that if this topic (the linear transformation) was a uni topic.
hollow wigeon
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surreal zodiac
How do you construct a triangle with the given information: a = 7cm, altitude of a = 3cm and α = 60°
brittle parcel
How do you construct a triangle with the given information: a = 7cm, altitude of...
I had to construct this monstrosity with touchpad, so sorry for that
I'm not sure, but here's my guess
shrewd hamlet
How do we know if “a” is the base or the hypotenuse
atomic blade
Altitude
shrewd hamlet
O wait we don’t even know if it’s a right
atomic blade
That word
hot herald
triangle most likely won't even be a right triangle
surreal zodiac
I drew a sketch, because english isnt my first language and im not sure about the terminology
atomic blade
So idk why they used the word altitude
Well I guess the altitude itself is not a side
shrewd hamlet
I mean, it could be a right
atomic blade
It would be very scuffed
surreal zodiac
a more accurate version
hot herald
that setup won't work
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sharp vine
I confused on parts a-c. I tried working on part a) and got 5m, 5.1,5.01m. Then for B). I simply used the area formula and plugged in the previous radii from part A). But im unsure if what i've done is correct so far.
sharp vine
For c) Im assuming I have to use the derivative of the area formula?
safe trench
Show work
sharp vine
This is what I have so far. A). Was basically based off the fact that it’s moving at 1 m/s. Which is how I got those answers for A) then for B) I used the Area formula
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languid whale
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languid whale
Idk where to start. Ik the small angle rules but they dont make sense here
smoky wing
Don’t you just like use the formula
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smoky wing
Can someone help me?
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little heron
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gaunt lynx
little heron
nope