help-10

[teal turret]

Take a look at this

Simplify that

[fleet geyser]

So, raise each side with base e

And then manipulate

[teal turret]

Yes

And what do u get

[fleet geyser]

Circle formula nice

[teal turret]

Ye

[fleet geyser]

Okay so then for ln 3 = ….

Would I just be able to drop the LN?

[teal turret]

Same process

Ye by doing both sides e^, that’s what ur doing inadvertently by dropping from both sides

[fleet geyser]

Like this?

[teal turret]

Ye

[fleet geyser]

Nice

Thx

[teal turret]

Np

[fleet geyser]

.close

[obtuse pebble]

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[nimble igloo]

pls help, I don't remember the steps to solve this

[leaden ginkgo]

do you know how to take the derivative?

[nimble igloo]

oh, so the derivative would it be -3(2-x)^2 in this case?

[leaden ginkgo]

yes

[nimble igloo]

ty :D

then that becomes f'(x)?

[leaden ginkgo]

yes, first derivative

[prisma hatch]

how?

[marsh geyser]

sorry, wrong chat lol

[leaden ginkgo]

(3(2-x)^2) * (-1)

[prisma hatch]

,w d/dx (2-x)^3

[warm shale]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by Pro_Hecker#1307

[leaden ginkgo]

not wrong

[marsh geyser]

i didn't mean to write here

[prisma hatch]

[nimble igloo]

phew

wait

so how do l continue :(

[leaden ginkgo]

you set the derivative = 3 and then solve for x

[nimble igloo]

3=-3(2-x)^2? i think i am wrong

[leaden ginkgo]

that looks right

[obtuse pebble]

@nimble igloo Has your question been resolved?

[nimble igloo]

tried solving but still didn't get the answer

[ivory estuary]

i reckon there is no answer lol

[obtuse pebble]

@nimble igloo Has your question been resolved?

[nimble igloo]

seriously? I just wasted my time but thanks

[obtuse pebble]

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[ivory estuary]

naw, its just the shape of the resulting parabola, that's why

[obtuse pebble]

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[karmic granite]

Yoo I forgot what $R \rightarrow R$ means. Like what do they mean by this?

[warm shale]

Karsten

[royal shard]

Consider any function f

for example sin(x)

[karmic granite]

So it goes from any real number to any real number?

[royal shard]

Yes

It means that it is defined for all of R and maps its elements onto R

That does not mean it maps onto the complete R though

[karmic granite]

Then what do they mean by $R^n \rightarrow R$

[royal shard]

That would be surjectivity

[warm shale]

Karsten

[royal shard]

R^n is another vectorspace

Like

[wooden marten]

takes the nth dimensional reals to the reals

[karmic granite]

ohh so it maps it's answers to the real number line?

[royal shard]

The vector (x,y,z) is in R^3

Yeah

The norm is an example for R^n->R

[karmic granite]

So R^n -> R maps it's answers from the n^th domain to the real domain? So any vector (x1...xn) to just real number?

[royal shard]

You take a vector and get a scalar

[karmic granite]

ahhhh ok thanks

makes sence

that was it thanks

.close

[obtuse pebble]

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[fringe trench]

i cant solve both of them

<@&286206848099549185>

[timid silo]

Substitute x from one equation in the other and get the value of y

[fringe trench]

the next one please

[ivory estuary]

its about setting up the equations

you have a number of students and a number of chocolates such that
chocolates = 2 * students + 6
chocolates = 3 * students - 16

[fringe trench]

i couldnt do it either

[ivory estuary]

again, setting up the equation

you'll want to write

1. score in terms of correct and incorrect
2. correct and incorrect add to the total number of questions on the test
   can you construct the equations yourself?

[obtuse pebble]

@fringe trench Has your question been resolved?

[fringe trench]

Thanks last one

[ivory estuary]

can you show us what you think the answer is

[fringe trench]

18

i got by solving equation

[fringe trench]

this one please too

[ivory estuary]

alright

what is your thinking? what have you got so far

[fringe trench]

I tried to made one but wrong

[ivory estuary]

consider
guests = (guests per table) * (tables)

[ivory estuary]

as long as you know what went wrong 👍

[fringe trench]

I didnt understand properly

[fringe trench]

<@&286206848099549185>

[ivory estuary]

ok

take the first situation

the number of guests is such that if there are 6 guests per table then we need 3 tables more than we currently have

that is to say (the number of guests) = (6 guests per table) * (number of tables we have + 3)

can you do the other one

[fringe trench]

G=6x+3

Like this

Another one please

G=12x

Like this the another one

[ivory estuary]

Pretty close but not quite

First thing is remember the parentheses

Second thing is the second situation has three tables that will be unused, so
(number of guests) = (12 guests per table) * (total tables - unused tables)

[fringe trench]

kindly give me the equation please'

that way will be easier for me

like it seems which is what conneted is tough

[ivory estuary]

sorry i don’t quite understand this sentence

[fringe trench]

like i cant understand why you made parenthesis with( total table -3)

thats why i said

dont give eq

with like this give with variables

[ivory estuary]

sorry, but these channels are for giving guidance, not for answers

so i think it’s better if you yourself put the numbers in

1st equation. guests = (6 guests per table) * (tables we have + extra tables we need)
2nd equation. guests = (12 guests per table) * (tables we have - tables we don’t need)

[fringe trench]

thanks

[obtuse pebble]

@fringe trench Has your question been resolved?

[fringe trench]

Yes

[ivory estuary]

press the green check mark to close this channel

[obtuse pebble]

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[jagged matrix]

hello

[jagged matrix]

i dont know how to start this w]question

i understand proof by induction, its the matrix element thats throwing me off, i dont know where to begin or how to solve this

[timid silo]

calculate A^n for small values of n

[obsidian isle]

hint: this matrix is upper tri

just prove it's true for A^1, then consider the product A A^n

[jagged matrix]

okie dokie, how does the product come into play?

[tardy epoch]

@jagged matrix do this for n=2,3

[timber fox]

so sorry to sidetrack but is this from the AH 2022 paper?

[jagged matrix]

yeah it is

[timber fox]

lol, it's the paper I sat last year

[jagged matrix]

💀

i think i understand roughly how to go further but i still dont understand the product of A A^n

sorry :,))

[timber fox]

set your base case for n=1 , get an expression for n=k, and then from considering n=k+1 we establish A * A^k, as a^x * a^y = a^(x+y) from rules of indicies

then multiply it out and see if you can manipulate it

[tardy epoch]

Calculate A^2 first

[timber fox]

why n=2?

[jagged matrix]

ohhh okie dokie this makes sense, i will give it a try then, thank you man :))

[jagged matrix]

ive done this, can you not do 1 since it'll just give the og A matrix?

[tardy epoch]

What does "do 1" mean

Show your matrix

A^2=?

[jagged matrix]

n=1

why cant you do that?

[timber fox]

it asks for all n in N \ {0} so I would presume starting at n=1 is the most obvious thing to do, even tho it doesn't 'return anything'

[timber fox]

u can

[tardy epoch]

Do what?

A=A doesn't show anything

[jagged matrix]

im so lost now :,))

[tardy epoch]

@jagged matrix if you can't do this, you need to ask for help learning matrix multiplication

[jagged matrix]

no i understand that, i dont understand why you can't start with subbing n=1

i apologise, im being kind of dumb ik :,))

[timber fox]

I would start at n=1 , the expression of A^n requires all n in N \ {0} to be tested

the expression A^n is not A either, so showing A^n = A when n = 1 is somewhat meaningful

[jagged matrix]

how so?

ive worked it through with n=2 :)) thank you for the help

[timber fox]

set LHS = A and RHS = A^n and show LHS = RHS when n =1

[jagged matrix]

yeah

[timber fox]

if you do it for n=2 I would argue you haven't shown it for all n in N

[jagged matrix]

i think i understand the n=1 thing now

but if you did it with n=2, wouldnt it just give the same result?

[timber fox]

yes you would assume n=k after showing n=2 works just fine, and then consider n=k+1 ,

[jagged matrix]

yay thats what ive done, its all worked out fine :))

[wild swallow]

could take base case as n=0

[jagged matrix]

thank you for the help, i very much appreciate it :))

[wild swallow]

you don't have to check n=2 for the induction

[timber fox]

you don't end up answering the question

if you do this btw ^

your conclusion would exclude 1 and therefore be wrong

hence why I said start at 1

[jagged matrix]

ohh that makes sense

[timber fox]

' true for n=k, and since proven true for n=k+1 and since n=2 was proven to be true, all n>=2 holds'

[jagged matrix]

0 isnt in the N set, is it?

[timber fox]

different book to book

[wild swallow]

there are finitely many n less than 2 so the marker can check it for themselves in finite time

[wild swallow]

doesn't matter, still holds for n=0 and that's all you need for induction

[jagged matrix]

okie dokie then

i understand 🙌 thank you :))

[timber fox]

when is your exam

[jagged matrix]

i appreciate the help, thank you :))

[timber fox]

[jagged matrix]

next wekk 💀

[timber fox]

@wild swallow there is extra lore here

[timber fox]

enjoy

the last question for that 2022 paper was fun

[wild swallow]

i read the whole thread already

[jagged matrix]

i will try my best to, doubt i will 😔

[timber fox]

You can do ittt

I have an exam tmrw also

instead of fun applications its full of pure maths that makes me want to die in a hole

[jagged matrix]

i hope :,)) im terrified man, hoping for the best tho

good luck bro

[timber fox]

you too my dude

[jagged matrix]

you got this, just think about the fun after the exams 🥰

[timber fox]

what fun my life is eternal suffering

[wild swallow]

why are you here instead of prepping for your exam

[jagged matrix]

me too :DD

[timber fox]

already passed the course from coursework

cant say the same for physics and astronomy in 2 weeks

[wild swallow]

[jagged matrix]

physics has a special place in hell man

[timber fox]

got a special place in everyones heart

[jagged matrix]

definetely, the place in my heart thats full of hatred :))

anyway, study for your exam man and good luck, thank you all for your help and have a nice life :))

.close

[obtuse pebble]

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[fallow plank]

need some help

[fallow plank]

Any ideas?

[pine sail]

Need some help with what?

You need to find y(x)?

[fallow plank]

Well yea

[teal turret]

Try getting all terms with an x on one side

@fallow plank

[fallow plank]

i get 1 + 2tanx

[teal turret]

Ok now u have

$y’ = 1 + 2 \frac {\sin x}{\cos x}$

[warm shale]

Stephen

[teal turret]

Now integrate

U got it?

@fallow plank

[obtuse pebble]

@fallow plank Has your question been resolved?

[obtuse pebble]

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[crude ice]

how do i know how the graph is shaped and if the graph will be cut at exactly one point?

[crude ice]

if i had no graphing calculator what am i supposed to do

[timid silo]

use the algebraic definition of one-to-one

[crude ice]

what is that

[timid silo]

https://en.wikipedia.org/wiki/Injective_function#Definition

[obtuse pebble]

@crude ice Has your question been resolved?

[crude ice]

.close

[obtuse pebble]

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[lost moon]

ok i have a simple question

[real hound]

ok

[lost moon]

if the ducks and the cows has a total of 132 feets et 48 heads, how many cows are there?

[real hound]

System of linear equation?

[lost moon]

yeah

[obtuse pebble]

@lost moon Has your question been resolved?

[teal turret]

@lost moon

What’ve u fried

Tried

[obtuse pebble]

@lost moon Has your question been resolved?

[obtuse pebble]

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[obtuse pebble]

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[idle thunder]

could you post your work please

[obtuse pebble]

@rain hull Has your question been resolved?

[obtuse pebble]

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[bold garnet]

How do I do this example question: 3(2ab^)^

[bold garnet]

It's simplifying brackets with powers thing

But is has a number outside the bracket

[static beacon]

thing it has a few missing numbers buddy

[bold garnet]

What do you mean

[wooden marten]

we don't know what things are to the power of

[bold garnet]

It's this but my question has a number at the front outside the brackets

[wooden marten]

sorry about them

[wooden marten]

the number outside the parentheses distributes to the things inside

exponentiation distributes over multiplication

[bold garnet]

Show me an example question

[wooden marten]

4*(3^5)^6
exponentiation rules
5*6 = 30
=>
4*3^30

[bold garnet]

Could you tell me the answer to this question please

Click on the image btw

[wooden marten]

each thing in the parentheses gets taken to the power of 3

[bold garnet]

So u times it all by 3

[wooden marten]

no not multiplied

to the power of

[bold garnet]

Is this the answer

[wooden marten]

no

[wooden marten]

you need to take a to the 3 and 3 to the 3

and you don't do anything with the 4 on the outside of the parentheses

[bold garnet]

.close

[obtuse pebble]

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[lyric wind]

Are these two possible points correct?

(5, -1) (1, 3)

[static beacon]

are you not supposed to minus 1 from each side?

you have 1 + 8i

or am i trippin

[lyric wind]

Uhm so im supposed to form a equation from the real nums and the imaginary ones

Imaginary = imaginary

Real = real

[static beacon]

Yea

your 2nd line

you have 1 + 8i

how did the 1 become positive

[lyric wind]

Oh shit

-1+8i

[static beacon]

hehe

[lyric wind]

Noooooooo

[static beacon]

have fun doing it again

[lyric wind]

:{

.close

[obtuse pebble]

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[obtuse pebble]

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[winged shore]

help

[timid silo]

0.4 or 0.40

[winged shore]

Nope

its to 3dp

[clear condor]

the 2 dots on the top mean repeat?

[winged shore]

yes

[clear condor]

well expand it

like repeat the .47

[winged shore]

0.474

idk

[winged shore]

how

i very confused

and tired

[winged shore]

0.474?

<@&286206848099549185>

[old isle]

repeat both infinite

[rich plume]

The expansion goes 0.4747

[old isle]

infinitely*

[rich plume]

At end there is 7 so the number is rounded to 0.475

[winged shore]

what about this

[rich plume]

same way

0.2626

6>5 so its rounded to 0.263

[winged shore]

finally

thanks

.close

[obtuse pebble]

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[formal raft]

Estimate the area of the parallelogram.

[formal raft]

what

OH

its a bot nvm

[clear condor]

!status

[obtuse pebble]

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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[formal raft]

1

its very much 1

[clear condor]

maybe find the side lengths

[old isle]

it says estimate tho

thats a little unclear as to what you're meant to do

[formal raft]

hm

[old isle]

can you show other problems so i know what it means by estimate

[versed arch]

isn't that a square tho ?

[clear condor]

if it is

then you know how to find the area of a sqaure

[old isle]

if you don't know how to find distance between two points you can just find the width and height, multiply and divide by two to find the area

[formal raft]

the topic is Area On A Coordinate Plane

[clear condor]

?

[old isle]

as in find the width on the x axis and height on the y axis of the shape and divide by two

[clear condor]

oh

true

[formal raft]

Ima attempt to do that

I suppose the formula is: A = B * H where B is the base, H is the height, and * means multiply. The base and height of a parallelogram must be perpendicular.

[fierce lagoon]

Hm do you know vectors

That's the easiest way I see without having to find each side

[formal raft]

no

[old isle]

you can just find the width in x and height in y and multiply x and y and then divide by 2

[formal raft]

Using the distance formula i can find that one side is equal to 6sqrt(2). Thus you can find that the area is 72 by squaring 6sqrt(2)

i think i have it ty

ty everyone

.close

[obtuse pebble]

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[coral axle]

Kim throws a ball 100 ft. If she has an initial velocity of 43 ft/sec, with an initial height of 12 feet.

How far will she throw the ball?

[coral axle]

i really need help with that

[obtuse pebble]

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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[coral axle]

I don't know where to begin

[nocturne minnow]

Do you know your kinematic equations?

[coral axle]

uhh nope

i know vertical motion models like h=-16t^2+vt+s

hello?

[old isle]

is there an initial angle given

[coral axle]

nope

help?

[obtuse pebble]

@coral axle Has your question been resolved?

[frigid prism]

no

[frigid prism]

then do that

this is purely application of those equations

unless this is calculus, in which case use that single equation

but you also said she throws the ball 100 ft

[obtuse pebble]

@coral axle Has your question been resolved?

[coral axle]

it isnt calculus

its intermediate algebra

[coral axle]

well

i guess no help after 2 hours..

.close

[obtuse pebble]

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[hard minnow]

Question 7 asks me to use lhopital…am I using it rightly to solve for number 7?

[civic zealot]

you didn't use lhopital

[vernal hill]

For that i think all you'd need to change on the first derivative is n^2 to 2n, then the second derivative is 2

Everything else remains the same

[hard minnow]

Would I have to in order to get the answer of 3/5 though…?

Seems like I’ve gotten it without it

[vernal hill]

So you'd take the limit to infinity of 3e^n/(5e^n+2) and its 3/5

[civic zealot]

what you did is fine, sure.

but it isn't lhopital. So you didn't correctly use lhopital....

[vernal hill]

^

Yea the method you did works its just a bit more confusing

Lhopital would honestly be a bit easier there imo

[hard minnow]

May I ask an unrelated question?

[civic zealot]

go for it

[hard minnow]

This is a question on income streams..I'm a bit confused on this..

I'm trying to solve it based off of my notes..

Like for here, how did they get 1000e(^0.01t)e^(-0.04t)dt...

I can understand how they got e^(0.04)(5) integral from 0 to 5...but everything after is just a blur..

Anyone?

[civic zealot]

it's just $\int_0^x f(t)e^{-rt}\ dt$.\
In your example, f(t) = 3650. In the problem $f(t) = 1000e^{.01t}$.

[warm shale]

Zybikron

[hard minnow]

Snap I wasn't paying attention, you're right. We have 1000e(-0.01t)e^(-0.04)dt, but for the next expression, how do we get 1000e^0.2 and e^-0.03t?

[civic zealot]

0.04(5) = 0.2

add the exponents in the two e^() inside your integral

[obtuse pebble]

@hard minnow Has your question been resolved?

[hard minnow]

Sort of confused, the integral of e^(-0.04t) is e(-0.04t)/-0.04?

[obtuse pebble]

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[fathom mural]

is this correct?

[plain stag]

looks good to me

[fathom mural]

thanks

mind checking a few more?

are both correct

[obtuse pebble]

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[signal crater]

Does the 0 vector always map to a 0 vector in any linear transformation?

[forest sinew]

can you prove it

whatd you try

[signal crater]

Uh

Well

I was watching a video on image and kernel

That's where the whole thing came up

[grizzled shore]

what does it mean to be a linear transformation?

[forest sinew]

[signal crater]

A linear transformation

Is something that takes

A vector space

And gives back another vector space

Under the conditions that

[grizzled shore]

defined over the same field*

both vector spaces have to be defined over the same field

[signal crater]

For our course we refer to only real fields

The Real field*

[grizzled shore]

ok continue

[signal crater]

The conditions are that

The transformation is linear

[grizzled shore]

what does that mean

[signal crater]

Obeying scalar multiplication

And vector addition

[forest sinew]

yea write the equations

how is the transformation defined

then see if you can finagle a 0 in

[signal crater]

In the video

[grizzled shore]

ok, and what does it mean for something to be a vector space? how does it relate to scalar multiplication?

[signal crater]

It was written like a matrix

[forest sinew]

right

define the transformation $Ax = T(x)$

[warm shale]

jan Niku

[forest sinew]

now express those two things you said using T(x)

write two equations

[signal crater]

I don't get what ur saying

[grizzled shore]

the transformation is a function

[forest sinew]

you said scalar multiplication and vector addition

[signal crater]

Yes

Yea

[grizzled shore]

it takes in some input (from input vector space) and returns 1 output (output vector space)

[signal crater]

The matrix of linear transformation

[grizzled shore]

we can define the matrix A that does the transformation

as a function

T

that's why he wrote Ax = T(x)

where A is your transformation, x in the input

but rewritten in a function notation, where T(x) is the transformation of input x

[signal crater]

x is the input vector?

[grizzled shore]

yep

[forest sinew]

yea

to me this is overly granular but maybe its important

[signal crater]

Ah ok

[forest sinew]

linearity has these properties you will see over and over

[signal crater]

So what's Ax

[grizzled shore]

some transformation

[signal crater]

It definitely can't be matrix multiplication

[little quartz]

yes,T(0) = T(0+0) = T(0) + T(0),so T(0) = 0

[forest sinew]

its matrix multiplication

[signal crater]

How can u use matrix multiplication here

[grizzled shore]

u can shorten this into T(cu + v) = cT(u) + T(v) as well :p

a 1 liner banger

[signal crater]

Yes

[forest sinew]

were not using it

were abstracting it

[signal crater]

Ok wait so

Like

Let's say

We have a transformation

That goes from R3 to R2

[grizzled shore]

T: R^3 -> R^2

[signal crater]

How to write vertical matrices

Can I send images here

Wait I'll send image

[forest sinew]

u mean vectors?

no u have to learn latex

[grizzled shore]

well vertical matrices are vectors

[signal crater]

Yea vectors

[grizzled shore]

i swear latex vectors are a hassle to write

[signal crater]

Say u have this

So is this matrix the same as A

[grizzled shore]

ok hold up

L(v) is not that

[signal crater]

v1 v2 V3 are the components of vector in R3

[signal crater]

Oh

[grizzled shore]

L(v) has 2 parts to it

you need the A and the x to make Ax = L(x)

[signal crater]

L(v) is the same as T(v) right

[grizzled shore]

or A and v to make Av = L(v) in this case

yes

[signal crater]

Oh ok

Ahh right

So we will have to extract A out of it

Is this right?

[grizzled shore]

sure

[signal crater]

Ok

[grizzled shore]

but that's not really that relevant to your original question

[signal crater]

Yes

My question was about how we can put a zero vector

And somehow end up with

Non zero vector

In a linear transform

[grizzled shore]

look at the definition of a linear transformation

[forest sinew]

oop

[grizzled shore]

this one specifically

[little quartz]

here

[grizzled shore]

and also what it means to be a vector space

[forest sinew]

shouldnt just give out answers

there are obviously gaps leading up to this

[signal crater]

So is T(v) the same as A product with v

[little quartz]

[grizzled shore]

kinda but that's not really the point of the exercise i dont think

[signal crater]

Yea I am convinced it's true but idk how to logically build up to here

[grizzled shore]

you shouldn't think of it in terms of matrices

[forest sinew]

just think of it as a function

with special properties

[signal crater]

Ok it's a function

But then

[grizzled shore]

this is super important

you really need to understand this first

[signal crater]

Ok

[forest sinew]

if you have $f(x+y)$, its the same as $f(x)+f(y)$

[warm shale]

jan Niku

[signal crater]

Right

[grizzled shore]

^^that's what the picture is saying

[forest sinew]

well wait arent we missing the one where f(0)=0

[grizzled shore]

that's part of the vector space def

[forest sinew]

yea\

[little quartz]

a linear transform is about length and angle,so '0' won't change

[forest sinew]

but we havent pasted it

[grizzled shore]

so it's actually built into the linear transformation def

[signal crater]

But a transformation is only defined from one vector space to another tight

[forest sinew]

i mean we dont have it in a nice image

[signal crater]

Right

[grizzled shore]

because the linear transformation def requires vector spaces

well

go look at vector space def i guess

[signal crater]

Vector space is a set of vectors which satisfy the 8 axipms

Set of any elements

[grizzled shore]

yeah and one of them has to do with the 0 vector

one of the 8 axioms

[signal crater]

Yes it has to have a 0 vector

[grizzled shore]

do you remember which one

[signal crater]

The additive inverse exists

Additive identity*

[grizzled shore]

nah the 0 vector one is the important one here

[signal crater]

Or what u call kt

[signal crater]

What do u mean

[grizzled shore]

well your input is a vector space right?

[signal crater]

Yes

[grizzled shore]

it by definition of being a vector space must contain the 0 vector

[signal crater]

Yes

[grizzled shore]

ok so if v = 0 exists, what happens to T(0)

[signal crater]

Ohhh

Ok

So u can say

[grizzled shore]

remember that the linear transformation T satisfies those 2 conditions earlier

[signal crater]

T(0+0) = T(0)+T(0)

[grizzled shore]

yes

and also?

[signal crater]

T(c*0)=T(0)= cat(0)

cT (0)

[grizzled shore]

not only 0 + 0, it should work for 0 + a for any a in the input vector space

[signal crater]

So (c-1)T(0)= 0

[signal crater]

Ah yes

[little quartz]

how do u get it

[signal crater]

So does a linear transformation take in a whole vector space or just individual vectors

[signal crater]

Partially, and i have a few follow up questions

[grizzled shore]

it takes in individual vectors of a vector space

[signal crater]

Right

So now

[grizzled shore]

well it can transform the entire space

since it can take any vector of the input space

[signal crater]

We have this set of vectors inside the vector space which a linear transformation takes in

[signal crater]

Right

[signal crater]

Which gives out 0

Vector

[grizzled shore]

right

[signal crater]

Which is called kernel

[grizzled shore]

yes

kernel of the transformation*

[signal crater]

Now they also said something about the kernel being the same as null spave

Space

[grizzled shore]

kernel = null from what i know

[signal crater]

How r the two things related

[grizzled shore]

it's just a different name for the same concept

i think kernel might be a bit more generalised for other ideas in math as well

but im not too sure

but in the context of matrices it should be meaning the same thing

[signal crater]

Oh

So when u have a

Linear system

Ax=0

Homogenous

Is the Ax on the left side

The same as the Ax you mentioned

When u said

Ax = T(x)

[grizzled shore]

yes

[signal crater]

So Ax = 0 is basically the same as saying the a linear transformation on a vector space giving the 0 vector?

All x for which a linear transformation

Gives 0 vector

So essentially yielding the null spave

[grizzled shore]

well kinda

[signal crater]

Which is the same thing as kernel

[grizzled shore]

if you solve for all x that satisfies Ax = 0, that set of x will be the kernel/null space of A

[signal crater]

So a system of linear equations is a linear transformation ?

[signal crater]

Oh yes

[signal crater]

I'm not able to connect those two ideas here

U there?

[grizzled shore]

im not entirely sure if it's exactly the same tbh

[signal crater]

Cuz Ax=B is the general system of equations equation right

Where x is essentially a vector

From Rn

[grizzled shore]

b is also a vector (try to use lower case)

[signal crater]

Ok

Ohh

Right

[grizzled shore]

upper case suggests its a matrix

[signal crater]

Ax=b

And b can have a different dimension from x?

[grizzled shore]

yes

[signal crater]

Oh

When does this happen

[grizzled shore]

a transformation from R^n to R^m

n does not have to equal m

[signal crater]

Exactly

So is there a system of linear equations

Where

x is from Rn and b is from Rm

m≠n

?

[grizzled shore]

i mean, let me think

hmm i suppose so

[signal crater]

Yea

So essentially any linear system of equations is basically equating a certain vector with constant elements to a linear transformation on a vector with variable elements?

I think I'm close to getting the answer here

[grizzled shore]

im not sure what you mean do you think you can rephrase it

[signal crater]

Like if u have a

Linear system of equations

You can think of all the constant elements as

A vector from Rm

The variables as a vector from Rn

And the matrix of coefficients as a matrix A

And given that Ax is the same as T(x)

We r Essentially asking

What vector from Rn

Would be required

So that after the linear transformation is applied on Rn

We get the vector of constants from Rm

Rn Rm or even Rpxq perhapns

[grizzled shore]

what do you mean by constant elements

[signal crater]

I guess that's redundant and makes no sense

I just wrote it

But ofc all will be constant

I mean variable elements when

The vector isn't yet defined

And we are trying to find that vector

[grizzled shore]

i dont even know what is a variable element

i dont really understand what you're referring to when you say "constant" and "variable" element

[signal crater]

By elements i mean the

Things in the vector Rn

[grizzled shore]

things?

[signal crater]

v1 v2... vn

[grizzled shore]

you mean the components of a vector from R^n

[signal crater]

In a vector v from Rn

Ye components

[grizzled shore]

right ok

ok what's the difference between constant and variable components of said vector

[signal crater]

Welp

As I said

I was mistaken

Well yk

I was thinking like

[grizzled shore]

ok what did you mean then, if you can rephrase it now

[signal crater]

When we have ax + by = c and dx + ey = f

a b
d e

Is the matrix A

[grizzled shore]

you mean
a b
d e

[signal crater]

Yes

x
y

is the matrix X

x

[grizzled shore]

is the vector v

[signal crater]

And
c
f

Is the same as b

Yea vector v

So I referred to b as the constant matrix

Cuz c and f are constants

And v as variable cu x and y r variable

[grizzled shore]

this is equivalent to asking given Ax = b
where A =
a b
d e
and b = (c, f)

what values of x is the solution to this system?

[signal crater]

Exactly

So after this particular linear transformation has been applied on Rn from which x belongs

What vector x from Rn

Would be transformed into

The vector b

[grizzled shore]

in Rm yes

[signal crater]

Right?

Yes

[grizzled shore]

yes

[signal crater]

Ahh makes sense now

[grizzled shore]

asking Ax = 0 is where b = 0

[signal crater]

So again kernel will always contain the 0 vector?

[signal crater]

WhatM

What?

[grizzled shore]

well normally you ask what x satisfies Ax = b for a particular A and b

[signal crater]

Yes

[grizzled shore]

the kernel is a special case of this

[signal crater]

Yes

[grizzled shore]

you're asking when is this true, for b = 0

[signal crater]

Where b is 0

Exactly the x which is the solution is

The kernel

[grizzled shore]

right

which input vectors map to 0

[signal crater]

So x= 0 will always be a solution to a homogeneous set ofequations

[grizzled shore]

you should be able to see that since x = 0 must be in the input space (otherwise it wouldn't come from a vector space)

[signal crater]

Ofc

[signal crater]

Rightt

[grizzled shore]

and since it's linear

using the definition of linear transformation

the 0 vector must map to 0

so yes, it is the trivial solution to any linear system

[signal crater]

Yes so it all comes from the fact that any homogeneous system has to have a trivial solution

[grizzled shore]

yeah

if u have 0 of every component

[signal crater]

cuz 0x1+0x2+....+0xn will always be 0

[grizzled shore]

you have 0 in total of anything

yep

[signal crater]

Yuss

Thanks

Thanks a lot

I might have to open a help channel again p soon about similar stuff

[grizzled shore]

don't worry i did something similar a few weeks back

glad i can help you now

[signal crater]

Thanks

. close

.close

[obtuse pebble]

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[marble hamlet]

Did I go about this the right way?

[upbeat plinth]

@marble hamlet yes

[marble hamlet]

this one is the same process just written differently?

sorry my class did an example with the inverse matrix and I'm getting confused which process to use for each problem

[obtuse pebble]

@marble hamlet Has your question been resolved?

[obtuse pebble]

@marble hamlet Has your question been resolved?

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[torpid bane]

Can someone just help me make sure this is correct

Basic statistics btw^

<@&286206848099549185>

<@&286206848099549185>

I think my answers correct I just don’t understand it

.close

[obtuse pebble]

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[rich patrol]

When doing a problem like this should I draw out to find the bounds or is there an algebraic technique for it?

[dark stirrup]

In this specific problem, the bounds of the integral scream polar

For the bounds of the integral, not that x²+y²=1

[rich patrol]

Oh yeah totallyI understand that it literally says

But would I need to draw out the bounds to get the polar coordinates bounds

Or is there a way of algabraically determining them?

Without visualizing

[bitter cobalt]

hi

[rich patrol]

In order to solve I had to visualize a circle, is there a way just algebraically?

Draw out the bounds in xy plane

[dark stirrup]

I see

Cylindrical to Euclidean is defined by the following:

$$x=\rho \cos{\phi}\newline
y = \rho \sin{\phi}\newline
z=z$$

[ruby path]

$$ at the end of every row

[dark stirrup]

save me

[ruby path]

$$x=\rho \cos{\phi}$$
$$y = \rho \sin{\phi}$$
$$z=z$$

[warm shale]

NEONPerseus

[dark stirrup]

@rich patrol you can reverse this to solve for Euclidean to cylindrical

$$\rho=\sqrt{x^2+y^2}$$
$$\phi=\arctan(\frac{y}{x})$$
$$z=z$$

[warm shale]

SWR

[obtuse pebble]

@rich patrol Has your question been resolved?

[rich patrol]

Ahhhh I see thanks that makes a lot of sense

[obtuse pebble]

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[clear kraken]

what does this notation mean?

mod(a,2)

an ordered pair?

I am only familiar for some integer a: a mod n

[real hound]

I believe it is saying a is the input of f(a)

2 is the modulus

[clear kraken]

is this notation standard?

[real hound]

it works in desmos

so i guess

[clear kraken]

hmm

[obtuse pebble]

@clear kraken Has your question been resolved?

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@sleek pollen Has your question been resolved?

[obtuse pebble]

@sleek pollen Has your question been resolved?

[sleek pollen]

Can someone help me

[rugged kite]

try to write the relation that defines a+bi

[sleek pollen]

i dont understand

[rugged kite]

how is a+bi defined ?

[sleek pollen]

i just dont know

thats all the infomation

[rugged kite]

i'm asking the question because it's given in the statement

it's a square root of x+yi

[sleek pollen]

ok

how am i meant to expess 2ab as x any y

i just dont have a clue what that means

[rugged kite]

What does being a square root mean ?

[sleek pollen]

you can equal a negative

am i confusing square root with something else

[rugged kite]

Yeah I think so

a square root of a given number A is a number B such that B²=A

it's an ""inverse"" (using that term loosely because it's not unique) of squaring

[sleek pollen]

yep that makes sense

so (a+bi)^2 =x+yi

[rugged kite]

if you expand that square you'll see some stuff appearing

[sleek pollen]

ok

one sec

(a^2-b^2)+2abi=x+yi

[rugged kite]

indeed, now you can deduce A and C

Recall that two complex numbers are equal if and only if their real and imaginary parts are equal

[sleek pollen]

so 2ab = x+yi/(a^2-b^2)i

[rugged kite]

No

Read what I just said

[sleek pollen]

2ab = x+yi+(b^2-a^2)/i

i dont know

does 2ab = y

[rugged kite]

yes

(a^2-b^2)+2abi and x+yi are equal

so their real parts are equal : a²-b² = x

and their imaginary parts are equal : 2ab = y

[sleek pollen]

yep that makes sense

so how do i rearange to make a^2 + b^2

[rugged kite]

now, you don't rearrange, you're gonna have to use another concept

when you have a complex number a+bi, do you know any operation that makes the quantity a²+b² appear ?

[sleek pollen]

time by conjouget

whatever the spelling lolk

[rugged kite]

conjuguate

what that does is taking the norm (squared)

[sleek pollen]

i guess

[rugged kite]

|a+bi|² = a²+b², that is true

[sleek pollen]

yep

[rugged kite]

edit, don't forget the square

[sleek pollen]

oh

no

(a+bi)(a-bi)

=a^2+b^2

[rugged kite]

it ends up being the same thing

[sleek pollen]

ok

[rugged kite]

you can define $|z| = \sqrt{z\bar{z}}$ if you want to (in fact, that's a usual definition)

[warm shale]

Syst3ms

[sleek pollen]

yep im aware of that

[rugged kite]

anyway, can you relate the norm of a+bi to the norm of x+yi ?

[sleek pollen]

i asme

assume

[rugged kite]

you know (a+bi)²=x+yi

and you wanna talk about the norm of these elements

what should you do to the equation ?

this is the only relation you have, so if you have to do anything with a norm it must be on this relation

[sleek pollen]

im not quite sure

[rugged kite]

do you understand what "taking the norm of the equation" would be ?

[sleek pollen]

square each component

sum up and root

[rugged kite]

what about |a+bi|² = |x+yi|

do you agree with this ?

the norm is an operation like another (like squaring, like inverses), so you can apply it to both sides of an equation

If two things are the same and you do the same thing to them, they remain the same. It sounds obvious, but that's the only thing we're doing here.

Note that I implicitly used the fact that |A²| = |A|² for any complex number A

[sleek pollen]

im so confused

can you make a start

do the first line

please

[rugged kite]

So we have (a+bi)² = x+yi, ok ?

[sleek pollen]

yep

[rugged kite]

The norm |·| is an operation like any other, so if we apply it on both sides the equation will still be true, ok ?

[sleek pollen]

yep

makes sense'

[rugged kite]

So then |(a+bi)²| = |x+yi|

[sleek pollen]

sure

[rugged kite]

The norm works well with powers (because it's multiplicative), so |(a+bi)²| = |a+bi|²

[sleek pollen]

ok

[rugged kite]

so putting it all together, |a+bi|² = |x+yi|

[sleek pollen]

yep

[rugged kite]

Now write each side of the equation in terms of a,b and x,y

[sleek pollen]

isnt that already the case

[rugged kite]

ok sure

but what's the other way to write |a+bi|² that you told me earlier ?

[sleek pollen]

a+bi|² = a²+b²

[rugged kite]

yup

and what of |x+yi| ?

[sleek pollen]

|x+yi|= a²+b²

[rugged kite]

yes, but forgetting about that equation, what's the general formula for the norm of |A+Bi| ?

(hint : you know what happens when you square it, so just take the square root of that)

[sleek pollen]

of |x+yi| ?

[rugged kite]

ye

[sleek pollen]

sqrt(|x+yi|)

?

[rugged kite]

okay, with a picture

what i was trying to lead you to is that |x+yi| = sqrt(|x+yi|²) = sqrt(x²+y²)

[sleek pollen]

ok

im sorry for being so slow

thanks for the patience

[rugged kite]

it ok

[sleek pollen]

im good now

thanks for the help

.close

[obtuse pebble]

Channel closed

Closed by @sleek pollen

Use .reopen if this was a mistake.

[obtuse pebble]

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[wanton oar]

Can someone explain to me how you get from the red step to the blue step? Like what its doing

[wanton oar]

I understand whats happening underlined in red

but I dont understand how they are converting it to a summation rule

[timid silo]

they are pulling out a factor of 1/n^2 first

then writing it as a sum

[wanton oar]

why are they pulling out 1/n^2?

[timid silo]

the sum should probably read $\sum_{i=1}^ni^2$

[warm shale]

Toby

[timid silo]

because its a factor of all the summands

[wanton oar]

isnt 1/n also a factor

[kind hawk]

sure. but you can factor out more

so why wouldn't you do that

[wanton oar]

I see okay

so their factoring out 1/n^2 from all the sums

and then after factoring the sums would read like [1+2+3...infinity] ?

[timid silo]

it would be [1^2 + 2^2 + ... + n^2]

[timid silo]

which is this

[wanton oar]

oh right its i^2

Im just having a hard time figuring out how (1/n)^2 factors out (2/n)^2 and etc algebreically

and leaves it as 1^2, 2^2 etc

[timid silo]

$\left(\frac2n\right)^2=\frac{2^2}{n^2}=\frac1{n^2}\cdot 2^2$

[warm shale]

Toby

[wanton oar]

ahh okay

I see

thank you that helped

[timid silo]

np

[wanton oar]

here they subbed in n=5, does it matter what n is? like its just some arbitrary number right

[timid silo]

dunno about that, you would need to show the original question

[wanton oar]

its split into 5 parts actually on the x axis so im guessing thats actually where the n = 5 comes from

[timid silo]

yeah, that makes sense then