im just trying to help rick?
#help-19
frail comet
heady smelt
idk how to do it with fractions tho
subtle depot
I dont have a problem rick does
heady smelt
SOME1 JUST HELP B4 I EAT MYSELF
frail comet
hahahahahhahaha
heady smelt
BRO SOMEONE HELP
heady smelt
i did
mystic saffron
ok shut
mystic saffron
js work on the problem
heady smelt
pls someone just help
heady smelt
i’m about to die
subtle depot
Ok so when you raise to a fraction you use the numerator as what you raise it by
frail comet
i’m about to die
DOXXED
mystic saffron
if u continue to have multiple questions, js leave ur thing open till ur done
heady smelt
if u continue to have multiple questions, js leave ur thing open till ur done
Sorry sir
subtle depot
And the denominator as ur root number
mystic saffron
or if sm helper has already helped u in the past 20 minutes and u repeatedly dont understand, dm them
heady smelt
*Ma'am lol
Oh sorry
mystic saffron
...
frail comet
THANK U BB
bro this man is a troll aahhahahhahahahah
mystic saffron
💀
heady smelt
SOMEONE JELP
frail comet
W RICK
frail comet
L RATIO
heady smelt
@subtle depot let’s get back to business bbg
frail comet
bbg?????????
heady smelt
bro send the question again
no alana
mystic saffron
omg
frail comet
mans is wildinnn
frail comet
NAh
mystic saffron
robin stfu
frail comet
I MOG EVERYONE
mystic saffron
help him or hush
heady smelt
have adhd or sm
i have ADD
frail comet
BRO???????
heady smelt
attention deficiency disorder
frail comet
I LOOKMAXX EZ
mystic saffron
robin f u wanna chat, go to his dms
frail comet
robin f u wanna chat, go to his dms
nah im trying go into your dms?
frail comet
AHHAHAAHHAHAHAHAHAHAHAHAHA
heady smelt
this is basically the cube root of 1/1000
Kk
mystic saffron
so u can get the cube root of the top and bottom seperately
cube root of 1 is 1
and of 1000?
10x10x10
10
so 1/10
yes?
heady smelt
yeah exactly
mystic saffron
u trying to find what times it self 3 times
heady smelt
is that not what we’re doing
heady smelt
3/30
mystic saffron
NO
heady smelt
so 1/10
heady smelt
1
heady smelt
300
mystic saffron
i give up
heady smelt
oh sry
mystic saffron
times 10
heady smelt
1000
heady smelt
YES
mystic saffron
yes?
heady smelt
MAAM
mystic saffron
s whats the answer
heady smelt
1:10
mystic saffron
to ur original question
heady smelt
1/10
heady smelt
thank you mimi
mystic saffron
do u understand why
heady smelt
ilysm bb
mystic saffron
ok good
heady smelt
k thx
now go flirt with @frail comet
he said he wants to go into ur dms
sped bb
.clsoe
.close
odd edgeBOT
summer river
I MOG EVERYONE
Mog?
odd edgeBOT
Available help channel!
flat mortar
odd edgeBOT
flat mortar
someone help me with 1
subtle depot
Recall that tan = sin/cos
subtle depot
U can multiply
tender skiff
i got this can’t do much
do you know the identity sin^2x+cos^2x =1
subtle depot
Recall how to subtract fractions
flat mortar
do you know the identity sin^2x+cos^2x =1
i have a flip book
how can i subtract 1 from sin^2?
tender skiff
so according to that identity what should 1-sin^2x equal? (use sin^2x+cos^2x =1)
granite citrus
Careful where you put your squares. Also don't forget to include theta with your trig functions
tender skiff
yes now use this to find 1-sin^2x
flat mortar
cosine^2?
subtle depot
You can only subtract because they share a denominator
flat mortar
tender skiff
no, where did the cos(theta) in the denominator go?
tender skiff
yes!
tender skiff
whenever you see a complex problem just try simplifying everything into the form of sin and cos
in most cases it makes the problem easier
so now simply the 2nd problem to terms of cos and sin
tender skiff
yes now simplify these terms more
flat mortar
i don’t know how to simplify
tender skiff
use this property of fraction
$\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}$
flat mortar
i forgot
clever fjordBOT
Intecules ∮
tender skiff
this one
tender skiff
yes , now try to use it here
flat mortar
tender skiff
exactly
now use the property again
sin^2x+cos^2x =1
so what should 1-cos^2x equal
flat mortar
is this full answee?
clever fjordBOT
Intecules ∮
tender skiff
yup
tender skiff
your welcome
tender skiff
i’m gonna be doing some more
if you dont have any more doubts then use can use .close
odd edgeBOT
@flat mortar Has your question been resolved?
analog tundra
I'm not sure how to do this I got -cos(x)/sin(x)^2
odd edgeBOT
I'm not sure how to do this I got -cos(x)/sin(x)^2
analog tundra
This is what I did
trim owl
This is what I did
treat it as a 1/sin(x) = csc x and find the derivative of that
tardy lagoon
You're answer is correct
torn lotus
I'm not sure how to do this I got -cos(x)/sin(x)^2
differentiation of $sin^-1(x) is 1/\sqroot1-x^2$
tardy lagoon
Just need to simplify
analog tundra
Just need to simplify
oh what do I simplify?
clever fjordBOT
swerriee
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
torn lotus
ah the root
tardy lagoon
Cos/sin = cot and 1/sin = csc
odd edgeBOT
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Available help channel!
stray rampart
I need help on 10
odd edgeBOT
I need help on 10
stray rampart
.close
odd edgeBOT
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Available help channel!
night night
odd edgeBOT
knotty swan
!status
odd edgeBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
i got the same answer without the root over the bottom by doing:
log(4)+log(a)=log(4a)
1/3log(a+1)= log(a+1)^3
wait
not it dosnt
there is my issue :)
.close
odd edgeBOT
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narrow gale
hey
wheat raptor
huh, havent really messed around with factorization methods of HCF algebraic expressions but i think ive got notes on how to do 3?
let me go check
trail void
okay so
wheat raptor
i guess with 2 you can just start to factorize
12x^3 - 96 is dotc
difference of two cubes
trail void
12x^3 - 96 is dotc
and taking 8 common from the first one?
narrow gale
im not getting
wheat raptor
yeah lets just make it
8(x^4-16) and 12(x^3-8)
trail void
and taking 8 common from the first one?
get's an expresion of x^4-16
then the expression becomes a form of a^2-b^2 = (a+b)(a-b)
trail void
from that again we get the a^2-b^2 as x^2-4 which can be more factorized
we get something like 8(x^4+4)(x-2)(x+2)
from the first one
from second take 12 common we get (x^3-8)
which can be also factorized using
a^3-b^3
wheat raptor
we get something like 8(x^4+4)(x-2)(x+2)
change the x^4 in x^4 + 4 to x^2
wheat raptor
8(x+2)(x-2)(x^2 + 4)
narrow gale
bruh who is correct
trail void
bruh who is correct
better try solving it-
its easy
have the formulas infront of you
and compare and try
its 8x^4-128
the first thing i try to do is think of a way to make the coeff of the variable term 1 and then i get futrther clues
try that way
narrow gale
Are you all asians, cuz for me math is not easy it was easy in grade 8
narrow gale
yes
wheat raptor
no i just teach it for a living
narrow gale
Az you are a teacher
narrow gale
teach me too
wheat raptor
Az you are a teacher
im a few months off my final teaching qualification 😓
where are you lost in the method puchey
what the goal is or a step we have made
narrow gale
oh Congrats in advance
trail void
like how to start
you've tried other questions like these?
from the examples section?
wheat raptor
'find the HCF of the following by factorization'
narrow gale
nope
narrow gale
ok
wheat raptor
the idea is to find the largest shared factor by factorizing
trail void
you will understand what is HCF and all
wheat raptor
so what we are actually doing now is just factorizing each one individually and seeing what we get
then we can look at em and find the HCF
narrow gale
ok making sense
wheat raptor
'find the HCF of the following by factorization' --> 'factorize each of these invidually, then look at the factors and discern the HCF'
so we started with 1. factorize 8x^4 - 128
trail void
like until i know , if you're given two numbers you just have to break them down to several numbers and check which highest number is matching with the other
narrow gale
okk..
wheat raptor
if i just told you to factorize 8x^4 - 128, what would you do
narrow gale
i wll break it one by one
wheat raptor
show me
narrow gale
2^7 = 128
wheat raptor
lets go back to 8x^4 - 128 since you can overcomplicate it by looking at 2^7
narrow gale
ok
wheat raptor
the first thing you ought to do its take out constant values
8 is divisible by 8
128 is divisible by 8
so factorize out 8
8(x^4 - 16)
narrow gale
ok i got it
wheat raptor
do you think we are done, or can we factorize more?
in other words, can 'x^4 - 16' be factorized?
narrow gale
i guess maybe we can factorize more
wheat raptor
what makes you think that?
wheat raptor
do you mean x squared?
narrow gale
I mean x^2
wheat raptor
well expand that bracket
2(x^2-8)
you get 2x^2 - 16
that is not the same as x^4 - 16
so no
narrow gale
ok
wheat raptor
here is what should make you think 'i can factorize this' :
x^4 is a square number
it is the square of x^2
16 is a square number
it is the square of 4
these are both square numbers and one is taken away from the other
narrow gale
ok
wheat raptor
this is known as 'difference of two squares' which is a very very very common trick math questions like to use
here is the trick
narrow gale
ok
wheat raptor
whenever you have two squared numbers like this you can change it into those brackets on the right
that is factorizing
narrow gale
isnt similar to perfecct square
wheat raptor
a little similar in that it is a thing you notice and have a rule for yes
narrow gale
whenever you have two squared numbers like this you can change it into those bra...
ok
narrow gale
ok
wheat raptor
similar to the q we did earlier
narrow gale
so (x^4 + 16)^2
narrow gale
x^2
narrow gale
4
wheat raptor
yep, now put these into (a+b)(a-b)
narrow gale
(x^2 + 4) (x^2 - 4)
narrow gale
yes
wheat raptor
now, can we factorize this more?
in other words: 'can we factorize x^2 + 4' and 'can we factorize x^2 - 4'
narrow gale
i think no,
wheat raptor
lets look at it one at a time
narrow gale
or maybe yes
wheat raptor
x^2 + 4 you cannot factorize
x^2 and 4 are both square numbers
but it is square + square which is no good
narrow gale
yes
wheat raptor
so we lave it
x^2 - 4 however is square - square
so we can do the exact same trick
a^2 = x^2, b^2 = 4
now find a and b for these
and put it in the two brackets (a+b)(a-b)
narrow gale
(x+2)^2
wheat raptor
i was just asking you to substitute in what we found: (x^2 - 4) = (x+2)(x-2)
so 8(x^2 + 4)(x^2 - 4) = 8(x^2 + 4)(x+2)(x-2)
we replaced what we have found with the factorized version
and now we are at a similar question
can we factorize this more? the answer should be no as what we have that is new, (x+2) (x-2) are just x+something which is not factorisable
so it took a while but we have finished factorising one
narrow gale
okk
narrow gale
what is it
wheat raptor
first lets take out the numbers: 12x^3 - 96
12 is divisible by 12, 96 is divisible by 12
so 12(x^3 - 8)
now for x^3 - 8 you can apply a very similar rule but for cubic (power of three)
wheat raptor
yep
so you are just putting in x and 2 into these
i have to go soon, so i will help you out here:
narrow gale
k
wheat raptor
(x-2)(x^2 + 2x + 2^2)
= (x-2)(x^2 + 2x + 4)
so, overall:
12(x-2)(x^2 + 2x + 4)
this is fully factorized.
to explain why x^2 + 2x + 4 cannot be factorized takes too long
narrow gale
bro I have one question
wheat raptor
8(x^2 + 4)(x+2)(x-2) and 12(x-2)(x^2 + 2x + 4) are our final factorisations
wheat raptor
bro I have one question
go ahead
narrow gale
Why do we need to learn this, what is the use in real life
wheat raptor
you're in undergrad, right? presume you're just taking a math class?
there are two real answers
narrow gale
ok
wheat raptor
- being able to do this is pivotal for further mathematics and physics where there are actual important things to understand, physics is everywhere in engineering, construction, architecture, and fields such as quantum mechanics which are all actual things that people do
- more generally, this is just a 'are you smart enough to solve a problem given a set of rules'
the point is to develop your general problem solving skills and your thinking processes
rather than how important being able to factorize algebra is to the average person
wheat raptor
that i dont really have an answer for
there isnt really a good reason why they cant just give you all of them other than 'its an excuse to make you study'
in real life if you had to do these you would have access to everything
narrow gale
yeah right
wheat raptor
i guess theres a 'memorizing lets you do it quicker than having to look up ever little rule'
but it doesnt really matter
that aside: 8(x^2 + 4)(x+2)(x-2) and 12(x-2)(x^2 + 2x + 4) we have factorized these so we are very close to getting the HCF
narrow gale
and I suck at memorizing like I can only understand things
wheat raptor
the last thing you need to do is break 8 and 12, the numbers at the end, into prime number multiples
this is something you can do for any number, rewrite it as a product of prime numbers
it will always be unique and there is only one way to do it for every num mber
8 = 2 * 2 * 2
12 = 2 * 2 * 3
narrow gale
ok
wheat raptor
rewritten:
(2)(2)(2)(x^2 + 4)(x+2)(x-2)
(2)(2)(3)(x-2)(x^2 + 2x + 4)
the final step is write all the brackets that are in both of these
two (2)s are in both
and (x-2) is in both
nothing else is in both of them
so the answer is (2)(2)(x-2)
4(x-2)
narrow gale
okk
wheat raptor
overall, your method:
factorize both, including finding the prime factorization of the number outside
HCF is all brackets in both factorizations
only works if fully factorize so make sure to look out for difference of squares and cubes
i have to go now so sorry i cannot help with 3)
wheat raptor
try and look up a method for polynomial division and HCF though you will likely find some kind of help there or ask here about it
bye
narrow gale
bye
odd edgeBOT
@narrow gale Has your question been resolved?
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desert marlin
odd edgeBOT
zinc glacier
get told off
desert marlin
Okay
mortal mirage
we did it chat
zinc glacier
:)
mortal mirage
.close
odd edgeBOT
desert marlin
Bro I was typing
zinc glacier
who said
untold jay
yeah thats the average help channel user
desert marlin
I couldnt type before it got closed
mortal mirage
yeah thats the average help channel user
average austinu channel
untold jay
he farmed the helpful role so he can ask shit questions
and we'd be like "nahhh hes one of us we'll help"
desert marlin
untold jay
Bro I was typing
average help channel user when confronted about how their post had no text
desert marlin
untold jay
only thing missing was some ridiculous rotation
odd edgeBOT
Available help channel!
clear scaffold
Sum of two natural number is 12 and sum of their squares is 74 . find the greater number
clear scaffold
I solved it using quadratic equation the answer is 7,5 for x
If I substitute in y it gives 5,7
How I consider which is greater X or Y
noble forge
is 7>5
west merlin
find the greatest number
clear scaffold
Which is greater X or Y
west merlin
the two numbers you found are 5 and 7. plugging the equation back in, it doesnt matter which x or y you used, what matters is the numbers that came out of it
clear scaffold
Ohh ok
west merlin
5 and 7 are your two natural numbers. find the greatest number
clear scaffold
7
west merlin
there you go
odd edgeBOT
@clear scaffold Has your question been resolved?
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clever violet
odd edgeBOT
clever violet
sorry but is this substituion?
what is it called when i set them equal to eachother
dull fossil
sorry but is this substituion?
it's just equating 2 similar expressions
both are equal to c so they're equal to each other
you can call it "elimination"
odd edgeBOT
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Available help channel!
worldly lava
Hello
mystic saffron
!da2a
worldly lava
I know my answer is incorrect but I'm trying to figure out how they answer key did it, if they are leaving out crucial info
odd edgeBOT
@worldly lava Has your question been resolved?
worldly lava
No
odd edgeBOT
@worldly lava Has your question been resolved?
candid mica
hello
candid mica
if i were to build 2 trapezoid on top of each other
what would the ratio of their areas be?
call area of trapezoid ABEF s
what would area of EFCD be in terms of S?
is there a way to express it?
i dont really know if the example is for all scenrios but i suppose it is
because my trapzoids are different
how can i find EF maybe using AB and CD will be more accuarate
would this expression be true?
AB/DC = DC/EF
eternal oyster
ABFE is similar to ABCD in the sense that they are just zoomed out or zoomed in copies of one another
lofty kettle
candid mica
you mean bases?
eternal oyster
The area is proportional to any length
eternal oyster
A way to answer that question yourself is as follows: can you think of two scenarios that have the same AB, EF and a different DC?
You will see that it depends on the separation between AB and EF and the separation between EF and DC
eternal oyster
If two shapes were similar they would have the same "shape" just scaled in or scaled down
Consider ABFE has a very different shape to EFCD
Bases being paralel does not imply similarity. All sides should be proportional and all angles should be equal.
eternal oyster
True
candid mica
dont you think ABFE has too small of a shape?
to fit into ABCD
you only stretched EF
not AB
eternal oyster
candid mica
your doing the same with the base CD just stretching the upper base to be smaller
eternal oyster
not AB
Look at these too pictures, do they seem similar? Clearly not, the sides are not in proportion
candid mica
which sides?
DC looks fine to me
its a shared side in both trapz
show me the same thing with the other trapezoid you claim to be similar
eternal oyster
Yes, that's the problem. The trapezoid to the left and the trapezoid to the right have the same base, but all the other sides are different. They have a different "shape"
One of them is flatter
Wait
You are right
They are both dissimilar
candid mica
would you agree DC > EF
eternal oyster
I was wrong, sorry. None of them are similar
candid mica
the same way EF > AB
candid mica
they are stretched opposite direction while all the other conditions are met
continuing FC and DE of course this should look like the original trapeziod
same way
just opposite direction
down to the other one
eternal oyster
Yeah, but similarity implies that all sides are scaled by the same factor
You can't stretch one side by one factor and another by another factor
I think there aren't any similar trapezoids in this picture
candid mica
you are not just stretching one side, you are also making EF to be DC and you add FC to BF forming BC and same for AD
that act is what created the similarity
when i say strectch i dont mean just the base
but the other one stays still
eternal oyster
What are you claiming again? Do you claim two trapezoids are similar?
You claim EFCD is similar to ABCD?
candid mica
yes as they claim the condition AB || EF and DC
candid mica
You claim EFCD is similar to ABCD?
no
candid mica
i claim that ABFE ~ EFCD
eternal oyster
I don't think that's the case, what is your argument for ABFE ~ EFCD?
eternal oyster
That's not a good enough reason. There is no rule that says two trapezoids having paralel sides must be similar
Similarity means that you stretch all sides of a thing by the same factor and you get the other thing
Look at this, the base of the trapezoid ABFE must be stretched by a factor higher than 1 to get the trapezoid EFCD
But the side AE must be stretched by a factor lower than 1 to get the side ED
Therefore, you can't stretch all sides of ABFE by the same factor to get EFCD
To produce a similar shape, you need to scale by the same factor
Like this
candid mica
i think something that will help you understnad this is maybe dividing the original figure into triangles and proving they are similar and from there working on sides who are proportional. you can then infer they are similar by the definition of similar polygons
while i do want to understand this with you i dont really have time so im very sorry but i still do stand behind the fact they are similar asa what i stated
thanks for ur time
.close
odd edgeBOT
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lament stone
Max is challenged by his friends to find the height of a pizza slice. His friends give Max a chart that would help Max find the height. What is the height of the pizza?
lament stone
You're gonna have to wait until another help channel opens. Sorry about that.
rapid geyser
abt how long?
lament stone
abt how long?
Don't know, but you could post that into the help-forum.
noble briar
Max is challenged by his friends to find the height of a pizza slice. His friend...
is the pizza isosceles?
lament stone
is the pizza isosceles?
That information isn't given.
odd edgeBOT
@lament stone Has your question been resolved?
lament stone
<@&286206848099549185>
lament stone
<@&286206848099549185>
lament stone
Yes
I made quite a bit of progress, but I ran into this issue; angle x doesn't exist?
solar kestrel
Ah yeah, ur past the point where I can help unfortunately,good luck tho! :)
lament stone
<@&286206848099549185>
lament stone
<@&286206848099549185>
YES, I FIGURED IT OUT. Thank you for all for helping.
.close
odd edgeBOT
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Available help channel!
timber geode
,tex
Let
\begin{flalign*}
& S_1 = \left{ x \in \mathbb{R}^4 \mid 2x_1 - x_2 + x_3 - x_4 = 0 ; \ x_1 - 3x_3 + x_4 = 0 \right}; & \
& S_2 = \left{ x \in \mathbb{R}^4 \mid 2x_1 - x_2 - x_3 + x_4 = 0 \right}; & \
& T_1 = \left\langle (1,0,1), (0,1,1) \right\rangle; & \
& T_2 = \left\langle (2,1,3), (0,0,1) \right\rangle. &
\end{flalign*}
Find a linear transformation ( f : \mathbb{R}^4 \rightarrow \mathbb{R}^3 ) that simultaneously satisfies:
\begin{flalign*}
& f(S_1) \subseteq T_1; & \
& f(S_2) \subseteq T_2; & \
& \dim \mathrm{Nu} \ f = 1. &
\end{flalign*}
clever fjordBOT
renato
timber geode
how do I start this badboy?
odd edgeBOT
@timber geode Has your question been resolved?
odd edgeBOT
@timber geode Has your question been resolved?
odd edgeBOT
@timber geode Has your question been resolved?
cobalt forge
,tex
Let
\begin{flalign*}
& S_1 = \left\{ x \in \mathbb{R}^4 \mid 2x_1 - x_2 + ...
S1 and S2 are spans of vector sets in R^4 (two vectors in the case of S1, three in the case of S2); I’d recommend starting by finding them (solve the systems of equations for each set and parametrize the solution set)
timber geode
S1 and S2 are spans of vector sets in **R**^4 (two vectors in the case of S1, th...
three in the case of S2?
@cobalt forge
@cobalt forge
timber geode
what do I need to do
cobalt forge
Solve the equations, like a regular system of equations
timber geode
,,2x_1 - x_2 + x_3 - x_4 = 0 \ x_1 -3x_3 + x_4 = 0
cobalt forge
Not all of them together
Each set presents its own system that must be solved separately
clever fjordBOT
renato
timber geode
,w 2x1-x2+x3-x4 = 0, x1 - 3x3 + x4 = 0
clever fjordBOT
clever fjordBOT
cobalt forge
So what are two vectors in R^4 that together span that null space?
timber geode
,, \textbf{span} = \left{\begin{pmatrix} 3 - y \ 9x-3 \end{pmatrix}\right}
clever fjordBOT
renato
clever fjordBOT
timber geode
,, x_1 + 0x_2 - 3x_3 + 1x_4 = 0 \ 0x_1 + 1x_2 - 9x_3 + 3x_4 = 0
clever fjordBOT
renato
timber geode
,, x_1 + - 3x_3 + 1x_4 = 0 \ 1x_2 - 9x_3 + 3x_4 = 0
clever fjordBOT
renato
timber geode
,, x_1= 3x_3 - 1x_4 \ 1x_2 = 9x_3 - 3x_4 \ x_3 = x_3 \ x_4 = x_4
clever fjordBOT
renato
timber geode
,, x_1= 3s - t \ x_2 = 9s - 3t \ x_3 = s \ x_4 = t
clever fjordBOT
renato
timber geode
,, \textbf{span} = \left{\begin{pmatrix} 3 \ 9 \1 \0 \end{pmatrix}, \begin{pmatrix} -1 \ -3 \0 \1 \end{pmatrix} \right}
clever fjordBOT
renato
timber geode
ye
cobalt forge
Now for the next one
clever fjordBOT
renato
timber geode
how do I do that
cobalt forge
First solve for x1 in the equation
timber geode
2x1 - x2 - x3 + x4 = 0
2x1 = x2 + x3 - x4
x1 = x2/2 + x3/2 -x4/2
x2 = x2
x3 = x3
x4 = x4
x2 = s, x3 = t, x4 = v
x1 = s/2 + t/2 -v/2, x2 = s, x3 = t, x4 = v
,,2x_1 = s + t - v, \ 2x_2 = 2s, \2x_3 = 2t, \2x_4 = 2v
cobalt forge
Don’t forget -v for 2x1
s+t-v
Also if you’re multiplying, you have to multiply all of the equations
Since it is essentially a change of variable
So x2 = 2s, x3 = 2t, x4 = 2v
clever fjordBOT
renato
cobalt forge
Hold up
Do not multiply both sides of the three bottom equations
That does nothing
timber geode
Also if you’re multiplying, you have to multiply all of the equations
.
please elaborate.
cobalt forge
Think of it this way: when you multiply the RHS terms of the top equation by 2, you’re really multiplying the vectors that span the solution set by 2. So you have to multiply all the mentions of the variables you multiplied.
It’s essentially a change of variable: s/2 -> s, t/2 -> t, v/2 -> v.
clever fjordBOT
renato
timber geode
better now?
cobalt forge
This won’t really get you a set of vectors…
You need to solve for x1, x2, x3, and x4
For now keep the fractions in the top equation, then you can remove them once you have vectors and it’ll make more sense.
timber geode
,,x_1 = \frac{s}{2} + \frac{t}{2} - \frac{v}{2}, \ x_2 = s, \x_3 = t, \x_4 = v
clever fjordBOT
renato
timber geode
@cobalt forge
cobalt forge
So now what are the vectors corresponding to this parametrization
timber geode
,, \textbf{span} = \left{ \begin{pmatrix} \frac{1}{2} \ 1 \ 0 \0 \end{pmatrix}, \begin{pmatrix} \frac{1}{2} \ 0 \ 1 \0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{2} \ 0 \ 0 \1 \end{pmatrix}\right}
clever fjordBOT
renato
cobalt forge
Yes
And if we want to eliminate the fractions, we can multiply each vector by 2 since span would remain the same
timber geode
,, \textbf{span} = \left{ \begin{pmatrix} \frac{1}{2} \ 1 \ 0 \0 \end{pmatrix}, \begin{pmatrix} \frac{1}{2} \ 0 \ 1 \0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{2} \ 0 \ 0 \1 \end{pmatrix}\right} = \left{ \begin{pmatrix} 1 \ 2 \ 0 \0 \end{pmatrix}, \begin{pmatrix} 1 \ 0 \ 2 \0 \end{pmatrix}, \begin{pmatrix} -1 \ 0 \ 0 \2 \end{pmatrix}\right}
clever fjordBOT
renato
timber geode
And if we want to eliminate the fractions, we can multiply each vector by 2 sinc...
ok. what now?
cobalt forge
So now we have S1 and S2
And now it’s a question of mapping vector spans to vector spans, which is ultimately a question of mapping vectors to vectors
This is where the fun begins
Now S1 -> T1 is a bit simpler because the vector spans have the same dimension
clever fjordBOT
renato
cobalt forge
Here, you need to map the vectors that span S1 onto the vectors that span T1
So for now we can just assign what vectors will map where
cobalt forge
Alright I have to go for like 10 minutes, sorry; I’ll be back
In the meantime think about which vectors in R^4 need to map to which vectors in R^3
alr I’m back
Sorry about that
@timber geode
cobalt forge
Let’s see
You need to map certain spans in R^4 onto certain spans in R^3
We can use the nice fact that if a vector u is in span(v), f(u) is in span(f(v))
Since f is linear
cobalt forge
It’ll be a 3x4 transformation matrix
cobalt forge
We can use the nice fact that if a vector **u** is in span(**v**), f(**u**) is i...
This is a consequence of linearity
clever fjordBOT
renato
timber geode
how would a 3x4 transformation matrix do the trick tho?
timber geode
This is a consequence of linearity
linearity of linear maps?
cobalt forge
Yes
timber geode
xd
cobalt forge
how would a 3x4 transformation matrix do the trick tho?
What dimension vector can you right-multiply with a 3x4 matrix?
Column vector that is
timber geode
1x3 * 3x4
timber geode
3x4 * 4x1
cobalt forge
And what dimension vector will that give you
timber geode
4
timber geode
4x4
cobalt forge
You take the outer dimensions
timber geode
I think.
cobalt forge
It would be 3x1
timber geode
mmm
cobalt forge
Remember that when you do matrix multiplication by a vector you’re essentially taking a linear combination, with coefficients determined by the vector entries, of the matrix’s columns
So 3 dimensional columns give a 3 dimensional linear combination
Do you follow that?
timber geode
i think yes
cobalt forge
So that’s how you get from R4 to R3
Now I hate to say it but it’s getting really late for me and I am tired as hell, so I’m going to go before I fall asleep :/
<@&286206848099549185> could someone else take over with this fellow
timber geode
gn hat
mortal mirage
what is the current question @timber geode
timber geode
how to find the linear transformation
mortal mirage
can you be more specific
timber geode
I dont know where to start
desert marlin
where to start what
clever fjordBOT
renato
timber geode
finding f, the linear map from R⁴ to R³
timber geode
we were finding a basis with the vectors from both the first and second set
I think
desert marlin
**renato**
you should be able to find 2 basis vectors that span S1, and 1 basis vector that spans S2.
If you define a linear transformation that maps the basis vectors of S1 to the basiss vectors of T1, and likewise with T2 then you've atleast satified the first two conditions
and, you can verify the 3rd condition using rank nullity
timber geode
wait a second. . .
2 basis vectors that span S1, right.
but 1 basis vector that spans S2?
@desert marlin
desert marlin
Yeah I looked at it wrong
timber geode
?
desert marlin
I haven't checked if those are the specific 3 that span, but I looked at it wrong when I said 1 vector spans, there should be 3
timber geode
right.
desert marlin
so if you have a 3D and a 2D subspace of R4
they must intersect right?
and that intersection must be mapped to both T1 and T2
by f
timber geode
If you define a linear transformation that maps the basis vectors of S1 to the basiss vectors of T1, and likewise with T2 then you've atleast satified the first two conditions
but how do I go from a fourth dimensional vector to a three dimensional vector form
desert marlin
here's an example map
f:R4->R3
f(x,y,z,w)=(x,y,z)
timber geode
timber geode
they must intersect right?
I think so
timber geode
and that intersection must be mapped to both T1 and T2
mmm
desert marlin
well at the very least all subspaces intersect at 0
timber geode
how do I $f(S_1) \subseteq T_1$
clever fjordBOT
renato
desert marlin
no we don't need that
we need that f(S1) is contained in T1
we need to define an f, and like a good way atleast to start testing out some functions is you can just define f on the basis of S1
so like, f(S_b1)=Tb_1
the first basis vector of S, maps to the first basis vector of T
and do similarly with the second
not saying this is an f that works, but it is a good way to get started trying things
timber geode
,w rref {{3x_1, 9x_2, 1x_3, 0x_4, 0}}
clever fjordBOT
timber geode
,w {3,9,1,0}*x = {1,0,1}
clever fjordBOT
timber geode
,w solve A*{3,9,1,0} = a{1,0,1} + b{0,1,1}
clever fjordBOT
odd edgeBOT
@timber geode Has your question been resolved?
hushed oak
how are the half angle formulas for tangent derived?
hushed oak
.close
odd edgeBOT
odd edgeBOT
Available help channel!
coral bison
So I am honestly not sure if this is the correct channel for this so please redirect me if I am wrong. But here we go. One little disclaimer, I am from the Netherlands so my school system was probably a bit different than from other countries. In secondary school I did VWO which is like "university preparatory education" roughly translated. In this I followed Mathematics B is what we called it. I never really was good at math, but I do want to get better at it and learn atleast the simple basics that I should have learned in school, because I need a certificate that I can do the things needed for a VWO Math diploma to apply to the university I want to go to. As I am not in school anymore I don't have acces to the books or whatever we used to use. Does anyone have some tips on how I can start to learn the basics/what I need to know, like what resources to use etc? Any help is appreciated!
teal mirage
Hi, I'd say a good way to check what you'd need to know is to take a look at final exams of previous years, you can find these for free on government websites.
You can also find the 'Syllabus' there which outlines (albeit a bit vaguely) the exact topics you need to know
odd edgeBOT
@coral bison Has your question been resolved?
coral bison
You can also find the 'Syllabus' there which outlines (albeit a bit vaguely) the...
That’s probably a good idea yeah, once I done that is it just as simple as googling the terms and trying to do learn them? or are there resources i should avoid
odd edgeBOT
odd edgeBOT
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vague citrus
Can someone tell me if my reasoning is correct?
odd edgeBOT
Can someone tell me if my reasoning is correct?
@vague citrus Has your question been resolved?
odd edgeBOT
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crystal crag
is it possible to solve x?
odd edgeBOT
is it possible to solve x?
crystal crag
i know the value of x through a different formula but i tried messing around and came across this but im not familiar with solving upper limit given other values, if its even possible as well
forest sky
you can use the formula:
[ \sum_{k = 1}^x k^2 = \frac{x(x+1)(2x+1)}{6} = {\frac{2x^3 + 3x^2 + x}{6} ]
then, it becomes a matter of finding the root of a cubic polynomial
clever fjordBOT
cloud
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crystal crag
how are they related? just curious bc i have no clue how the sigma notatino equation became that
odd edgeBOT
@crystal crag Has your question been resolved?
crystal crag
can someone tell me
odd edgeBOT
Available help channel!
pastel garden
is this correct?
odd edgeBOT
is this correct?
pastel garden
the answer is supposed to be π/2 only
have I done some silly calculation mistake, i can't find it
quasi sparrow
that's the wrong integral of tangent
odd edgeBOT
@pastel garden Has your question been resolved?
forest sky
if you are doing integration by parts, one part of the original should be u and the other part should be v'
odd edgeBOT
odd edgeBOT
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mental ridge
How would i go about proving that the interior angles of any shape with N points is the same
mental ridge
Like all quadrilaterals have angles that add up to 360, all triangles add up to 180
Just a general proof
nocturne belfry
How would i go about proving that the interior angles of any shape with N points...
theres actually a basic proof on khan academy for triangles
well, as basic as these things can be
mental ridge
Yeah but i mean like a proof that for any shape with N points, the interior angles add up to the same value
nocturne belfry
oh, in general?
mental ridge
Yeah
nocturne belfry
hmm id have to search
mental ridge
Not necessarily finding the actual angle
But finding that there is some angle which all shapes with N points add up to
fluid tundra
How would i go about proving that the interior angles of any shape with N points...
what happens when you draw a diagonal through a quadrilateral for example
mental ridge
what happens when you draw a diagonal through a quadrilateral for example
well, it turns into two triangles
oh yeah induction
sorry i forgot
ive actually even done this same exact proof before, using the fact that any shape can be triangled for lack of a better word
fluid tundra
interesting
odd edgeBOT