help-17

[vocal sleet]

@carmine ether Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[boreal wren]

indefinite integral of sin3θ

[boreal wren]

@=theta

-cos3@.@²/2 ?

[arctic mantle]

no

and just use t, it's easier

[vast shale]

!show

[vocal sleet]

Show your work, and if possible, explain where you are stuck.

[boreal wren]

I didn't write it in presentable form that's why I was only making a simple question out of it and asking here

[hybrid flicker]

how did you get cos(3t).t^2/2?

for sin(3t)

or did you mean t*sin(3t)

which is still wrong

[boreal wren]

sin=-cos

and

3@= 3 × @²/2

[hybrid flicker]

ok I get what you did but

1. it's sin(3t) not sin(t)
2. the integral of the product is not the product of the integrals

just a small counterexample

indefinite integral of 1

it's just x right?

or whatever the variable is

well

1 = 1*1

1 integrates to x

1 integrates to x

so 1 integrates to x*x = x^2?

[boreal wren]

sin3t integration= -cos3t × 3×t²/2

is this wrong

[hybrid flicker]

again you mean t * sin(3t)

and it's still wrong

[boreal wren]

?

[hybrid flicker]

the integral of the product is not the product of the integrals

[boreal wren]

where is product

[hybrid flicker]

wait

are you really

[boreal wren]

its only sin3t

[hybrid flicker]

ok so different problem

you're saying

[boreal wren]

what

[hybrid flicker]

f(g(x)) integrates to F(g(x))* G(x) right

which is wrong

just imagine

[boreal wren]

isn't there chain rule like differentiation in integration

first we see sin then we see theta

[hybrid flicker]

you can only reverse the chain rule, not create a new one

f(g(x)) differentiates to f'(g(x)) * g'(x)

which means

f'(g(x)) * g'(x) integrates to f(g(x))

however

nothing was said about integrating f'(g(x)) or f(g(x))

what you should do

is a variable change

[boreal wren]

is there shortcut to do without substitution

[hybrid flicker]

generally, no

unless you already know the integral

which here you don't

[boreal wren]

can u tell what I did wrong

[hybrid flicker]

.

you tried to create a semblance of "chain rule" for integration

which doesn't exist

[boreal wren]

so how do you quickly solve it

using a trick or smth

[hybrid flicker]

the point of substitution

[hybrid flicker]

is to figure out where the "f'(g(x)) * g'(x)" was in the expression

here you have sin(3t)

you know sin integrates to -cos

but in order to integrate sin(3t)

you need to multiply it by g'(x)

g(t) = 3t

g'(t) = 3

so sin(3t) * 3

[hybrid flicker]

is finally of this form

so it integrates to f(g(x))

or -cos(3t)

which means

dividing by 3 again

sin(3t) integrates to

-cos(3t)/3

this trick only worked because g'(x) was a constant

(using the linearity of the integral)

g'(x) is a constant when g(x) = ax + b

so whenever you have to integrate f(at+b)

it's gonna be 1/a * F(at+b)

[boreal wren]

int sin3t = int t.sin3t ?

[hybrid flicker]

no

they're different functions so they're gonna have different integrals

[boreal wren]

but you said this

[hybrid flicker]

is this t*sin(3t)?

[boreal wren]

oh sorry

I mean

int sin3t = int 3.sin3t

[hybrid flicker]

no, we multiplied by 3 to get the form "f'(g(x)) * g'(x)"

[hybrid flicker]

we had to undo it afterwards

int sin3t = 1/3 int 3.sin3t

[boreal wren]

can u ot talk in variables, I don't understand

[hybrid flicker]

ok

we have a chain rule for differentiation

we want to rewrite what we got

so we can "undo" chain rule

[boreal wren]

yes

[hybrid flicker]

so we multiplied by a constant to recognize that

and then divided by the same constant further down

and we were allowed to do that

because it's a constant

if it were a non constant function it wouldn't have worked

[boreal wren]

like

[hybrid flicker]

this is a very reduced example but

x^2 differentiates to 2x

i want to know $\int xdx$

[twin meteor]

rafilou2003

[hybrid flicker]

if I multiply by 2, I get 2x

which integrates to x^2

divide by 2 to "undo" what I did

and my original integral is x^2/2

(+ constant)

[boreal wren]

u are multiply 2 inside integral?

[boreal wren]

shouldn't it be x²/2

[hybrid flicker]

2x integrates to x^2

because x^2 differentiates to 2x

[boreal wren]

oh ok

[hybrid flicker]

as I said this is a really weird example

but just to show

I can multiply by some constant INSIDE the integral

as long as I divide by that same constant OUTSIDE

[boreal wren]

yes

[hybrid flicker]

$\int ...dx = \frac 1a \int a\cdot...dx$

[twin meteor]

rafilou2003

[hybrid flicker]

which is what we did in your question

$\int sin(3t)dt = \frac 13 \int 3sin(3t)dt$

[twin meteor]

rafilou2003

[hybrid flicker]

why did we do this

is because this form specifically

allows us to "undo" chain rule

[boreal wren]

isn't this way a lot of brain consuming

[arctic mantle]

no

it isn't

the constant inside can be moved outside and cancelled

[boreal wren]

no

I mean

I'll have to think first about what will the differentiation of inside function (3t) and then multiply divide by it

but here it's only 3t

but what if something bigger comes

[boreal wren]

what do I do after this

also.my teacher said that integrating sin will give zero if limit is of full cycle

it'll be because, sin3t and we put 2π/3=t, it'll become sin2π and it's integration will be zero?

[vocal sleet]

@boreal wren Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[lyric shadow]

What's the difference between line, surface and volume integrals? As I understand it line integrals give the area under a line or curve like basic 2d integration. Surface integrals give a volume but don't volume integrals do that? What is going on?

[heavy yoke]

in 1-variable integration, the only choice of region to integrate over is an interval of the number line. however in 3D space, we can choose many types of regions to integrate. these include curves (1D), surfaces (2D), and solid regions (3D).

none of these necessarily give "area under a curve" in the same way, but they can still have other useful interpretations

[lyric shadow]

so what is the difference between line and surface integrals?

[heavy yoke]

a line integral integrates over a curve in space like a helix (first picture) and a surface integral integrates over a surface like a hyperbolic paraboloid (second picture)

[lyric shadow]

i thought line integrals integrated like this

so a line integral can only give an area under a curve?

and surface integrals can cover a surface then?

[heavy yoke]

that's an interpretation which only applies to curves in the xy-plane. but the curves can potentially be in 3d space as well

[lyric shadow]

have you got an example of a line integral and surface integral over the same shape?

[heavy yoke]

the regions that a line integral does are curves (think like a bent wire) and the regions that a surface integral does are surfaces (think like a bent piece of paper)

so they integrate fundamentally different things. the closest you could get is integrating over a surface and its boundary curve

[lyric shadow]

so theoretically they could both integrate the same area?

if for example we have a 3d cube, we could get the surface integral of a face and then define a path through a face that would then allow the line integral to give us that face's area

[heavy yoke]

the "area above the curve" interpretation only holds for curves in 2D space (visualized as the xy-plane in the image). it does not really apply to curves in 3D space

[lyric shadow]

i don't understand

[lyric shadow]

this is a 2d area curving around 3d space

[lyric shadow]

the only difference between the yellow area and this, is that in the yellow area you can always draw a straight line from both ends of the area

[heavy yoke]

that is a 2D curve (in the xy plane) with the value of a function being interpreted as the height

[lyric shadow]

oh i see

but you can do line integrals for 3d functions right?

[heavy yoke]

you can do a line integral of a function of (x,y,z) over a 3D curve, yes. you just don't get the "area over a curve" interpretation

[lyric shadow]

have you got an example that isn't the helix, cause that is kinda confusing

[heavy yoke]

lets say we have a piece of wire bent into a semicircle like the figure. we also have a function f(x,y,z) which is the mass density of the wire. integrating that function over the semicircle would give the mass of the wire

alternatively, if we integrate f(x,y,z) = 1, we would get the arc length

[vocal sleet]

@lyric shadow Has your question been resolved?

[lyric shadow]

so this is the line integral?

[heavy yoke]

that's the curve. the line integral is an integral using that curve as the region of integration, like how a 1D integral uses a portion of the number line as the region of integration

[lyric shadow]

sorry man, still not getting it

i think i'm going to see some videos to see if i can visualize it better

[vocal sleet]

@lyric shadow Has your question been resolved?

Channel closed

Closed by @lyric shadow

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[vale garden]

please i need the limit of this integral. i cant use convergence dominance or anything letting me to put the limit inside the integral

[vale garden]

i know that Wn>W(n+1) and that Wn>0 but i dont know if it will help

[full hatch]

You want to show that the integrand becomes infinitely small for n-> inf. Here the x = 0 is the problem. So I'd split the integral into two: One from 0 to epsilon/2 and one from epsilon/2 to π/2. Then you can find upper bounds those two integrals separately

[vale garden]

but the problem will remain with the one from 0 to epsilon/2 no ?

and why exactly epsilon/2 ?

[full hatch]

We want to bound the whole integral by epsilon (for any epsilon > 0)

[vale garden]

(what bound means please ?)

[full hatch]

Well, 0 < cos(x) < 1 in this interval. So you can find an upper bound for this one by just using this 🙂

[full hatch]

I'm sorry, but I don't understand?

[vale garden]

"then you can bound those..."

whats the meaning of bound

[full hatch]

Maybe this is actually a false translation by my side.

[dense eagle]

bound means like find inequalities for

so we're trying to show i.e.

-n <= W_n <= n or smth like that etc.

anyway that's what we mean by bound

[vale garden]

i dont know, i dont study maths in english

[vale garden]

aaah ok thanks

[full hatch]

Thx, I was already doubting my english skills. I don't either

[dense eagle]

(obviously not those inequalities, but like just finding some inequalities)

[vale garden]

so if i get it

i can separate the integral

and find something bigger than each integral ?

[full hatch]

Yes

[vale garden]

any idea about it ? it looks difficult something that exactly fits

like bigger and having a limit going to

[full hatch]

I basically already told you how to do the left one

[vale garden]

sorry i really dont find that

if you have no problem with that, could you just write on a paper just the beggining of what you would do after splitting the integral to make this easier ?

my english doesnt help but maths in universal

[full hatch]

Unfortunately, there is just one more line after the split

[vale garden]

which is 😭 ?

[full hatch]

But let's do the first integral from 0 to epsilon/2 together

[vale garden]

okay im here

[full hatch]

Here we can just use the fact that cos^n(x) =< 1 to bound this above by the integral of 1 over [0, epsilon/2]

[vale garden]

wait to be sure
the integral of 1 over 0, epsilon/2 is epsilon/2 ?

bcz the primitive of 1 is x

and with the borders we get epsilon/2 ?

[full hatch]

Yes

So we did the first one 🙂

The second one is your job. I don't want to give you the full solution. But you can have some hints if you want

[vale garden]

okay thanks, i will ping you if i dont find the solution

[full hatch]

Maybe ping helpers. I might already be sleeping then

[vocal sleet]

@vale garden Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[vocal sleet]

@strange zodiac Has your question been resolved?

[strange zodiac]

<@&286206848099549185>

[strange zodiac]

<@&286206848099549185>

[plush tinsel]

Can someone pls check this

<@&286206848099549185>

[strange zodiac]

help-8, help-10, and help-48 are open rooms

[plush tinsel]

Oh okay thank you

[strange zodiac]

<@&286206848099549185>

.close

[vocal sleet]

Channel closed

Closed by @strange zodiac

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[patent abyss]

I know its either option 1 or 3

[patent abyss]

but how do you get 4a-4 from this

or 5a-6

[vocal sleet]

@patent abyss Has your question been resolved?

[patent abyss]

<@&286206848099549185>

plz

[vocal sleet]

@patent abyss Has your question been resolved?

[patent abyss]

The function f is defined on all values of x. The function g is defined as g(x) = f (x + 1) – f (x), and the function h is defined as h(x) = g(x + 1) – g(x). If h(x) = –4 for all values of x, which of the following conclusions is correct?

[vocal sleet]

@patent abyss Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[crisp zenith]

Did she mean C - (R_ U {0})?

[crisp zenith]

the logarithm is defined for 0 right

[modern ledge]

ln(0)?

[crisp zenith]

isnt*

sorry

[crisp zenith]

yeah

its not defined right

[modern ledge]

they should have excluded 0 yesh

[crisp zenith]

oh ok

just wanted to make sure

Ok I assume I'd have to show log satisfies the cauchy reiman equations for those restrictions

but how do I write log as u(z) + iv(z)?

[modern ledge]

im not sure if u have to do that?

[modern ledge]

im still in highschool so i will use what methods im aware of, they might be tedious tho

[cobalt crypt]

could be possible that 0 is included in R_

[crisp zenith]

oh right

it can be included

[cobalt crypt]

depending on convention

[crisp zenith]

as long as

y is not 0

right

[modern ledge]

so we are interested in log(z) being analytic on what domain
we just need to show domain of z is s.t. |z|>0 and its in principle argument

i think thats the conditions at least for log(z) being analytic

[cobalt crypt]

do you have a definition of the principal branch of the logarithm?

how does your course define it

[crisp zenith]

oh wait

ok no nvm we exclude 0

we used to book definition let me grab it

[cobalt crypt]

because to show analyticity, you need to show that log(z) has a complex derivative

[cobalt crypt]

so depending on your definitions, this might either be completely trivial or require a long computation

[crisp zenith]

ohhh

logz = lnr + itheta

[cobalt crypt]

right

[crisp zenith]

forgot about that

[cobalt crypt]

you need to show that has a complex derivative away from the negative real axis

[crisp zenith]

can show it satisfies cr equations using (r,theta)

[cobalt crypt]

ye

that's the idea

[modern ledge]

oh i didnt even realise this was called cauchy riemann equation lol

learnt something new today

[vocal sleet]

@crisp zenith Has your question been resolved?

[crisp zenith]

Does this seem about right?

r is greater than 1 right?

Actually Greater than or equal to

.close

[vocal sleet]

Channel closed

Closed by @crisp zenith

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[stiff dove]

Im not sure its right

[sullen shoal]

When u put infinity

(x+5) positive

x-4 also positive

(x+4) also positive

But there is negative sign

So it will become negative infinity

[stiff dove]

where

[sullen shoal]

-3(

[stiff dove]

k

[sullen shoal]

At the start

Yep

[stiff dove]

so like basically,

its like (x+5) * +inf

[sullen shoal]

Mhm

[stiff dove]

+inf turns whatever - into +

[sullen shoal]

Mhm

I didn't get you

[stiff dove]

im thinking like how -5*-5=25

is like how that +inf works

.close

[vocal sleet]

Channel closed

Closed by @stiff dove

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[random widget]

Hey just a quick question but a directional derivative exists when all partials of a given function equal to zero, or in other words the limits of all partial derivatives equal to zero?

[vocal sleet]

@random widget Has your question been resolved?

[vocal sleet]

@random widget Has your question been resolved?

[random widget]

.close

[vocal sleet]

Channel closed

Closed by @random widget

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[heavy pollen]

Can someone explain how the integral is introduced in the final step

[heavy pollen]

i understand it's some sort of application of the mvt

<@&286206848099549185>

[vocal sleet]

@heavy pollen Has your question been resolved?

[heavy pollen]

:(

<@&286206848099549185>

[vocal sleet]

@heavy pollen Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[heavy pollen]

.reopen

[vocal sleet]

✅

[vocal sleet]

@heavy pollen Has your question been resolved?

[hexed needle]

good now?

[heavy pollen]

no

can you help pls

[vocal sleet]

@heavy pollen Has your question been resolved?

[heavy pollen]

<@&286206848099549185>

[vast shale]

What's the question

[vast shale]

Oh nvm

[flat whale]

Try understanding it for one dimensional x first

Gradient of f is just the derivative of f. What do you get when you integrate the second derivative of f?

[heavy pollen]

oh i'm dumb

.close

[vocal sleet]

Channel closed

Closed by @heavy pollen

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[frank ivy]

Math gods, i summon you 😭

[sacred terrace]

try listing out the powers of seven

[peak matrix]

sounds painful

[sacred terrace]

yeah but it could help

see if anything interesting occurs

[peak matrix]

oh, it's surprisingly not that bad

[inner osprey]

it’s not

powers of seven are known to behave nicely (mod 100)

[frank ivy]

powers of 7? upto ?

[peak matrix]

I'd expect the cycle to be much longer

[peak matrix]

just modulo 100

until you get back to 07

[flat whale]

or 1

,calc 7^4

[twin meteor]

Result:

2401

[peak matrix]

yes, 07 comes after that

[flat whale]

fuck

,calc 7 * 1

[twin meteor]

Result:

7

[flat whale]

you're right

[frank ivy]

yeah i saw the solution, its 7⁴ but i don't get it why

[peak matrix]

Do you know modulo?

[sacred terrace]

u have to make it approachable

so if you start with something that seems natural

[frank ivy]

can't seen to remember

[sacred terrace]

then after you find a pattern/ interesting observation, build a function to put it into an expression

that helps more

[frank ivy]

no, i dont remeber

[peak matrix]

Yeah, the pattern is often not so simple

[peak matrix]

alright, what are the last 2 digits of 7^1?

[sacred terrace]

but now they know to look for something similar in other numbers

[frank ivy]

07?

[peak matrix]

Yeah, what about 7^2

[frank ivy]

49

[peak matrix]

7^3?

[frank ivy]

343

[peak matrix]

yes, and last 2 digits are 43

[frank ivy]

oh sorry, i panicked

[peak matrix]

now if we wanted to compute last 2 digits of 7^4, it would suffice to do 7*43 and take the last 2 digits of that

[peak matrix]

np, you dont have to panick btw. Take your time

[peak matrix]

,calc 7*43

[twin meteor]

Result:

301

[brittle cipher]

I used euler's theorem, since phi(100)=40 and 2008 mod 40 is 8, that means you can square 7 three times.
7^2 = 49 mod 100
(50-1)^2 = 1 mod 100
1^2=1 mod 100

[peak matrix]

nice approach, but i think the OP doesnt know modulo

[brittle cipher]

they should learn

[frank ivy]

is this a number system concept?

modulo?

[peak matrix]

sorta, when working under modulo 100, numbers like 343, 43, or even 53143 are considered to be "the same"

so we only focus on last 2 digits when doing adding them, multiplying them etc

and there are some nice theorems that simplify work with modulos

[frank ivy]

Ok, i will learn

should i check yt

[peak matrix]

you dont necessarily need it here though

[brittle cipher]

yeah this is number theory, all of these are about computing remainders, you really should know modulo to be working these

[peak matrix]

sure, it's called "modular arithmetic" or in general, "number theory", which also has many concept outside of modular arithmetic

[peak matrix]

this particular task can be done without it though

[frank ivy]

thanks, i will check

[frank ivy]

u mean an easier alternative?

[peak matrix]

the way without modulos is essentially just rewording an approach with modulos in a way that it doesnt use modulos

so just to finish the solution, we have:
7^1 = 7 ----- 07
7^2 = 49 ----- 49
7^3 = 343 ----- 43
7^4 = 2401 ----- 01

The key note here is that to compute last 2 digits of following powers, it suffices to take the last 2 digits of the previous power and multiply them by 7.

So e.g. instead of doing 2401 * 7 to compute 7^5, it suffices to compute 01 * 7 = 07

and so last 2 digits of 7^5 will be 07

but then, last 2 digits of 7^6 will be 49

and the cycle will repeat

07 -> 49 -> 43 -> 01 -> 07 -> 49 -> 43 -> 01 -> 07...

[frank ivy]

sheeeeshh

so it only works with 7?

or any other number?

[peak matrix]

similar approach would work for all numbers, but the cycle would usually be much much much longer

it could have up to 100 terms before it starts repeating

[frank ivy]

but 7 is different?

his cycle repeats after 4 numbers

[peak matrix]

7 has a very short cycle, yes

[frank ivy]

poor 7

[peak matrix]

there might be some other numbers with short cycles

the length of this cycle of n is called "the order of n" in number theory

[frank ivy]

Thanks kind stranger, for wasting your time on ne, may peace be upon you

.close

[vocal sleet]

Channel closed

Closed by @frank ivy

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[woeful plank]

I need help in finding what this equation I just came up with is called, I’m pretty sure it’s probably been thought of before because it’s really simple but I can’t find anything about it online. The top equation in the first picture is defining the variables for the equations formatted like the ones in the questions in the second picture. The second equation of the first picture is defining how to calculate the (x+…)^2+(y+…)^2 equations as shown in the second picture written in pencil. If my writing is unclear just ask me to type it out for you. Copy of my last question because it was closed with no resolution.

[vocal sleet]

@woeful plank Has your question been resolved?

[vocal sleet]

@woeful plank Has your question been resolved?

[hallow crescent]

oh thats a pretty cool derivation

i did that also for fun

idk if it has a name

i dont think so

but its pretty common

but you made a mistake

its not equal to the absolute value

cause you cant have negative radius

in the case of negative radius, the equation isnt a circle

@woeful plank

[woeful plank]

Yah, idk why I put that. I’m too tired lol

[hallow crescent]

hahah all good

[vocal sleet]

@woeful plank Has your question been resolved?

Channel closed

Closed by @woeful plank

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[formal meteor]

bro how did they simplify the numerator into a 12 😭

[formal meteor]

you do the distributive on the top and it's -20 + 4h + 20 so like how is 4 getting tripled

does it have to do with the weird combining of the denominators

[lone linden]

The numerator is equal to $$\frac{4(-5+h)(-5+3)}{(-5+h)+3} -\frac{4(-5)((-5+h)+3)}{(-5+h+3)(-5+3)}$$

[twin meteor]

Civil Service Pigeon

[lone linden]

,w 4(-5+h)(-5+3)-4(-5)((-5+h)+3)

[twin meteor]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by civil_service_pigeon

[formal meteor]

ive been staring at this the whole time and i have no idea how you got there

i mean im bad at simplification so that's to be expected but i genuinely am at a loss

i dont have wolfram alpha pro either so i cant see their step by step

ive seen multiplying a fraction by another fraction with same numerator and denominator, but ive never seen multiplying two fractions' numerators by each other's denominators

no wait that is what you did to the right fraction

so i get the right fraction cause you're trying to match the denominator of the left fraction... but you dont change the left fraction's denominator? and there's a random (-5 + 3) being added to the left fraction's numerator?

<@&286206848099549185> can someone pretty please hold my hand through this i dont know how to simplify very well

[woeful plank]

.reopen

[vocal sleet]

@formal meteor Has your question been resolved?

[jovial bloom]

still working on this one?

[formal meteor]

yes

[jovial bloom]

ok

well

do you know how to subtract/add two fractions together if their denominators are different?

[formal meteor]

you multiply both of them by a fraction of each other's denominators in the numerator and denominator

so you get a common denominator and it becomes easy right

[jovial bloom]

yeah

so try doing that with the fractions in that numerator

[formal meteor]

something like that right

sorry it's kinda messy it's a bit hard to avoid with these things

@jovial bloom what do i do now i still dont see the light

[jovial bloom]

expand the numerators

that's something you can try

[formal meteor]

alrighty

[formal meteor]

i havent done the right side yet cause it's really weird looking but is this like what it should look like?

the denominator of the right fraction would also be 2h + 4 cause they're the same and all that, it's just the numerator i dont really know about

[jovial bloom]

wait

[jovial bloom]

why is the first line of this different to the question?

[formal meteor]

huh it looks the same, i just copied the numerator part

[jovial bloom]

why have you written 4(-5h + h) instead of 4(-5 + h) in the first numerator?

[formal meteor]

OOPS

MY BAD

alright wonderful job me lol

[jovial bloom]

also you should be simplifying like terms when you can

-2 is quicker to write than (-5+3)

[formal meteor]

i figured id need to keep it like that for distributive property shenanigans

[jovial bloom]

why?

would it just make it longer to simplify?

[formal meteor]

because my simplification skills are lacking so sometimes my instincts arent good

currently here

at the start i dont really see why i cant just make the right fraction a whole number and put it in the left numerator and make it one fraction

but the guy who posted the vague answer at the start didnt so idk

is this anything

[jovial bloom]

ok but 40 - 40 isn't 80

[formal meteor]

it's 2 positive 40s

or am i crazy

i definitely wanted those 40s to simplify out LOL

[solar minnow]

You forgot to distribute the negative

[formal meteor]

which

[solar minnow]

That’s just the top portion but hopefully it’s clear

The whole right side is being subtracted

Then it should come out to 12h

[formal meteor]

oh shit the minus sign between the two fractions gets distributed??

[solar minnow]

Yea

[formal meteor]

that always messes me up i still dont fully have a grip on that one

[solar minnow]

Yeah it’s easy to forget that one

But that’s why I use the brackets

[formal meteor]

does that always apply when subtracting fractoins

[solar minnow]

Yea

[formal meteor]

alright good to know

both you and ojamamask thanks for the help

[solar minnow]

Np

[formal meteor]

being in calc 1 and not knowing basic simplification is embarassing but we live we grow we move

that's all adios

.close

[vocal sleet]

Channel closed

Closed by @formal meteor

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[tame summit]

I know I have to use the properties of orthonormal sets (orthogonality and normality) to show that it is LI. How would I use properties of orthonormal bases to show that the subset B is LI using the operations defined on the inner product space?

[dense eagle]

sps that av_1 + etc. = 0

now try dotting that with v_1 and seeing what u get

[vast shale]

hi im having trouble with this one question its for homework its about finding the slope

[plush garnet]

rise over run

use a diff channel tho

[vast shale]

oh kk

[tame summit]

guys im still needing help

[vast shale]

which channel do i use

[plush garnet]

@vast shale #help-41

[vast shale]

kk tysm

[plush garnet]

@vast shale nvm

use one of the available ones

[vast shale]

uhhh kk ty

how would i know if its available

[plush garnet]

two categories , math help available math help occupied

use a channel in the math help available one

sorry jar jar I got no clue

[vast shale]

ohh kk

[tame summit]

So i take the dot product of <sum(av),v_1> and get ||av_1||=|a|||v_1||=a right?

what happened when i tried to type in the norm operator?

[dense eagle]

yeah

[dense eagle]

|| || are spoilers in discord

[dense eagle]

then since 0 dot anything = 0

[tame summit]

ohhhh okay.

[dense eagle]

we know a=0

and similarly everything else = 0 so linearly ind.

[tame summit]

because sum(av)=0?

[dense eagle]

yh (cus we wanna show av_1 + bv_2 + etc. = 0 => a = b = etc. = 0)

[tame summit]

so <sum(av),v>=<a,v>=<0,v> implies that all a_i=0 is what we wanna get to?

[vocal sleet]

@tame summit Has your question been resolved?

Channel closed

Closed by @tame summit

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[wooden badge]

,rotate

[wooden badge]

idk how to get the equation to solve for derivative

[vast shale]

use pythagoras

[vast shale]

.

[plush garnet]

shouldn't the equation be

DT/dt = DT/dx * dx/dt

[mint lily]

Rate of walk by man, dx/dt = 5s
15/y = 6/(y - x)
upon solvin
3y = 5x
3(dy/dt) = 5(dx/dt)
dy/dt = (5/3)(dx/dt)
dy/dt = (5/3)(5)
dy/dt = 30/3
dy/dt = 10 ft/s
rate

[wooden badge]

sorry where did u get 15?

[mint lily]

ive recently done the similar question
take whats given in ur question

pretty much everything's same

hope this would have helped

[wooden badge]

nvm i got it

.close

[vocal sleet]

Channel closed

Closed by @wooden badge

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[lethal shore]

In order to calculate the Correlation, Standard Deviation, and Covarriance, could someone possibly varify the Answers step by step by calculating each variable correctly?

[vocal sleet]

@lethal shore Has your question been resolved?

[vocal sleet]

@lethal shore Has your question been resolved?

[lethal shore]

nevermind, i got it : 0

[fresh obsidian]

[lethal shore]

.close

[vocal sleet]

Channel closed

Closed by @lethal shore

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[radiant bone]

I tried to set $\mathbf{a} \cdot \mathbf{b}=4\sqrt{26}$

and then i would get 5x-z=4sqrt26

but then I get stuck so I feel like I messed up somewhere else

[twin meteor]

za

[radiant bone]

is there an other way I should approach this

nvm that was super easy woops

.close

[vocal sleet]

Channel closed

Closed by @radiant bone

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[jolly kite]

Hi where to ask question related to permutation?

[full ferry]

you could just ask it here

[jolly kite]

Find the total number of arrangements of the letters in the word "INDEPENDENCE".
Out of these, how many arrangements are there where the vowels never occur together? please provide sample as outcome only.

[vocal sleet]

@jolly kite Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[orchid walrus]

can i prove bernoullis inequality with the binomial theorem

[orchid walrus]

im up to $(1+x)^n = 1 + nx + \sum_{k=2}^n \binom{n}{k} (1)^{n-k} x^k$

[twin meteor]

syecko

[orchid walrus]

since x > -1 i have -1<x<0 and x>=0

i cant find a way to prove the sum is positive for all x > -1 though

[vast shale]

x>-1 so you have x>=0 ?
how's that ?

[orchid walrus]

nono thats not what i meant

i meant or like theres two cases, its negative in the first instance and postiive in the second right

[vast shale]

you don't need two cases

x>-1 so 1+x>0

[orchid walrus]

mhm

i need to manipulate the right though

induction seems easy but also why can we use base case x = -1 if x > -1

i mean inuitively its obvious

just didnt seem rigorous

so i wanted to use the binomial theorem

[vast shale]

you have to induct on n

so what do you mean by base case x=-1 ?

[orchid walrus]

OHH im stupid

lol

[vast shale]

x can take any real value greater than -1, not just naturals

you can't induct on that

[orchid walrus]

yea oops

[vast shale]

only on the naturals or integers

[orchid walrus]

for nonnegative why wouldnt the base case be 0

it would be right

n = 0

[vast shale]

yea it can be

or 1

[orchid walrus]

why or

[vast shale]

doesn't make much difference

you can even write both

[orchid walrus]

wait in induction can you use any base case

so long as it satisfies the condition on n

like can i use n = 100 for instance

obviously 0 and 1 are nicer because theyre simpler

[vast shale]

whatever base case you use, you have proved the statement for cases larger than that
if you use base case n=100, you have shown your result to be true for n>=100 and not for 1<n<100

[orchid walrus]

ahh so then i should use n = 0

[vast shale]

yea

[orchid walrus]

because if i only show n = 1 then it doesnt show for n = 0 in the induction step

ok thank you

.close

[vocal sleet]

Channel closed

Closed by @orchid walrus

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[flat whale]

Oh are you 8 years old yet

[lost yarrow]

.close

[vocal sleet]

Channel closed

Closed by @lost yarrow

Use .reopen if this was a mistake.

[lost yarrow]

troll or underage

doesnt make a difference to use either way

we have to ban

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[delicate veldt]

hey can someone please explain me how to get the answer for the circled one

[tawny hollow]

Idk what I’m missing but that’s not possible

[vocal sleet]

@delicate veldt Has your question been resolved?

[tawny hollow]

So first off since the RHS is non-negative, the only x’s such that is true for the LHS is when |x| <= 2/3.

so let’s separate the equations into cases.

If -2/3 <= x < 0 and lambda x - 2/3 < 0

Then if lambda => 0 we have that -2/3 <= x < 0, if lambda < 0 then we have that x is inbetween 2/(3 lambda) < x < 0.

That is 2 +3x = 2/3 - lambda x

This means we that x = (2/3-2)/(3+lambda) for x’s and lambdas that satisfy the condition above

By careful inspection you’ll notice that there is never such a lambda that we wanted to find, infact by symmetry something almost identical is seen for the other case. So something is either wrong or missing in your question

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[hallow lark]

For 3c I got cos^-1(7/(squareroot of 364)) = theta. The answer however is (second screenshot)

[agile lion]

i understand the quotient rule but dont understand where the 2x comes from

[agile lion]

can u go somewhere else this is my ticket :3

[hallow lark]

Oh apologies

[agile lion]

all good

@hallow lark can u delete ur things

[sharp lynx]

chain rule

compute the derivative of the denominator, you should also get a 2x bit

[agile lion]

oh

what part of the problem tells me that I need to do chainrule?

because

[sharp lynx]

the part where you have to compute a derivative that requires it

[agile lion]

to me it looks like i could use only quotient rule to get the der

like if i was given only this, what tells me i need more than just quotientrule

to find the derivatevew

[clever seal]

what's your question

[sharp lynx]

i don't know what "tell" you'd accept, but it would just come as a consequence of computing v'

[agile lion]

hold on i could have explained that better

when looking at the problem, what tells me "oh i need to use quotientrule AND chhainrule"

is that better

😭

[sharp lynx]

again this is really all i got

if the numerator and/or denominator requires the chain rule in its derivative computation, then you know you'll need it at some point(s)

[cunning plaza]

quotient rule: function of x on both sides of a fraction
chain rule: a composition of functions somewhere

[agile lion]

what separates a problem that can be sovled using only quotient and a problem that needs quotient+chain

[sharp lynx]

see my previous answer

[agile lion]

i read this like 5 times and still dont get it,

when/how would i know whether the numerator or denom "requires" chain rule?

[cunning plaza]

you identify a composition of functions

ie. sin(cos(x)). The inner term is cos, outer is sin, thus you have to use chain rule

in this case (x^2+4)^1/3, the inner term is x^2+4 (clearly demarcated by the brackets), the outer term is ^1/3, thus chain rule

[agile lion]

so if either numer/denom is a "composition" of fractions, id need to use the chainrule?

ex: both numer/denom are compositions, id need to chainrule twice, once for numer and once for denom

is that last part correct

[cunning plaza]

yup

[agile lion]

ok

thank you

[vocal sleet]

@agile lion Has your question been resolved?

[vocal sleet]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[loud chasm]

wooooooooo I feel so good about the midterm I have today

[loud chasm]

Its crazy how good you feel when you study at 3 a.m

you know its wierd because I tried to sleep, but I couldn't so I just did more math

Why am I enjoying this 😭

I might have turned into a nerd

but its ok, because Im feeling good

[bronze osprey]

why did you open a new hepl channel to tell us this..

[loud chasm]

idk

[clear canyon]

Is there an actual question

[loud chasm]

nope

[bronze osprey]

close it then

[clear canyon]

Then close it

[loud chasm]

have a good day 🙂

close!

[clear canyon]

Oh bruh

[bronze osprey]

.close

[vocal sleet]

Channel closed

Closed by @bronze osprey

Use .reopen if this was a mistake.

[loud chasm]

!close

[brisk moss]

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[delicate veldt]

ok. thank you for the reply

[delicate veldt]

.close

[vocal sleet]

Channel closed

Closed by @delicate veldt

Use .reopen if this was a mistake.

[vocal sleet]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[peak matrix]

!status

[vocal sleet]

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

[fossil pagoda]

1

[peak matrix]

Alright

let's start with A

do you know what kind of vectors will be there?

[fossil pagoda]

nope

u and v right?

[peak matrix]

it's a set of all vectors x, such that x = (a, 0) for some a in Z

do you know what Z is?

[fossil pagoda]

not sure no

[peak matrix]

Z is a set of all integers

[fossil pagoda]

ohhh right I forgot

[peak matrix]

Z = {... -3, -2, -1, 0, 1, 2, 3 ...}

[fossil pagoda]

gotcha

and R would be real numbers

etc

[peak matrix]

so if a is in Z, it means that a is some integer

[fossil pagoda]

ok gotcha

[peak matrix]

btw, i will be writting $(a, 0)$ instead of $[
\begin{bmatrix}
a \
0
\end{bmatrix}
]
$

[twin meteor]

MæthIsAlwaysRight
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[fossil pagoda]

sure

[peak matrix]

Okay, so A is a set of all vectors x, s.t. x = (a, 0) for some integer a

that means that e.g. (1, 0) is in A

and (-4, 0) is in A

and (104214124, 0) is in A

and even (0, 0) is in A

I don't exactly know how they want you to draw this, you are probably just supposed to draw that set of points in plane

can you do that?

And do you get what im saying? If not, stop me and ask at any time

[fossil pagoda]

I understand

you're just saying any of those integers can be a

[peak matrix]

Yes, and the set A consists of vectors in form (a, 0) where a is an integer

so (1,0); (-141, 0); (0, 0); etc are all in A

the problem now asks you to draw the set A

probably by sketching that set of points on a plane

Do you think you can do that?

[fossil pagoda]

so just sketching any points?

[peak matrix]

Yes, sketch a sample of the points in A

[fossil pagoda]

its just a line along the y axis right?

if so just sketched it

[peak matrix]

along the x-axis, and its not exactly a line

[fossil pagoda]

woops yeah got that backwards

[peak matrix]

it contains only the integer ponits

[fossil pagoda]

yes

I understand

[peak matrix]

So the next task is identifying whether this is a subspace (presumably of R^2, under standard operations)

do you know what conditions does it have to satisfy in order to be a subspace?

[fossil pagoda]

uh I think

something to do with vectors u and v?

[peak matrix]

yes, it has something to do with vectors in A

there are 3 conditions

1. it must contain the 0 vector, that is (0, 0)

2. it must be closed under vector addition, which means that when we add 2 vectors in A, we get another vector in A

3. it must be closed under scalar multiplication, which means that when we multiply a vector in A by some scalar, we get another vector in A

these conditions can actually be summarized in one, and that is:

A is closed under linear combinations of vectors in A, that means that any linear combination of vectors in A is also in A

does that look familiar? Or is it completely new to you?

[fossil pagoda]

its new but makes sense

[peak matrix]

hmm, alright. is there any other definition in your book / course?

if so, could you post it here

[peak matrix]

Ah, okay

yes, condition
(i) is actually closure under vector addition

it means that whenever we take 2 vectors in A and add them, we get another vector in A

condition
(ii) is closure under scalar multiplication

whenever we multiply vector in A by some scalar, we get another vector in A

(And this condition actually implies the 0-vector condition, since it says that 0 * v = 0 must be in A as well)

anyway, let's now verify those conditions for A

[fossil pagoda]

that all makes sense to me

[peak matrix]

is A closed under vector addition?

does condition (i) hold?

when I give you 2 vectors in A, say
(a, 0) and (b, 0), where a and b are integers, what will be their sum?

what will (a, 0) + (b, 0) be?

[fossil pagoda]

if its 2,0 and 3,0 it would be 5,0 right?

which is a vector in A

two integers added together gives another integer?

[peak matrix]

Yes

[peak matrix]

Also yes

to prove it generally,
(a, 0) + (b, 0) = (a+b, 0) and since a + b is also an integer, the result will be in A

now condition (ii)

if you multiply any vector in A by any scalar, will the result always be in A?

[fossil pagoda]

no

[peak matrix]

correct, can you give an example?

[fossil pagoda]

a,0 lets say is 1,0

scalar is 1,1

nvm

Im a bit confused now cause

[peak matrix]

as in 1.1?

[fossil pagoda]

(1,1)?

well

[peak matrix]

scalar is just a real number

[fossil pagoda]

right

any real number

times that tho

5(1,0)

[peak matrix]

This example actually works

[fossil pagoda]

gives 5,0

yeah

[peak matrix]

yes, which is in A

[fossil pagoda]

right

[peak matrix]

but 1.1 * (1, 0) = (1.1, 0) which is not in A

was that your example?

[fossil pagoda]

it wasnt but it ironically works lol

[peak matrix]

ah i misunderstood it then lol

[fossil pagoda]

I get it now

the goal is to not use a integer scalar

[peak matrix]

yes, if you dont use an integer scalar on (1, 0), the result will definitely not be in A

[peak matrix]

so you can use this or similar as an example

since you will have to explain why A isn't a subspace

which is because it fails to satisfy condition ii), because [insert ur example here]

Can we move to B now?

[fossil pagoda]

yes

perfect

[peak matrix]

alright

[peak matrix]

so how would you draw this one?

what vectors are even in B?

and more importantly, what vectors are not in B?

[fossil pagoda]

[0,0]

does not satisfy condition 1

not a subspace

[peak matrix]

Yes, that's the only vector not in B

[peak matrix]

correct

[fossil pagoda]

can it even be drawn?

[peak matrix]

if we use your book's definition, which condition does it not satisfy?

[fossil pagoda]

"subspaces give a mathematically precise definition of a flat space through the origin"

[peak matrix]

You could probably just draw over the whole plane, and draw an empty circle at (0, 0) to show that it's not there

[peak matrix]

okay, that works lol

[peak matrix]

it also fails to satisfy condition (ii) here

because e.g. (1, 1) is in B, but 0 * (1, 1) is not in B

[fossil pagoda]

right

[peak matrix]

okay, so now C)

this one looks pretty similar to A, except t is now an y-coordinate and t can also be non-integer

[fossil pagoda]

right

[peak matrix]

do you think it satisfies condition i)?

[fossil pagoda]

yes

[peak matrix]

why? can you prove it?

[fossil pagoda]

0,1+1,1=1,2 which makes t a real number still

[peak matrix]

this is a particular example

[peak matrix]

also, (1, 1) isnt even in C

[fossil pagoda]

its not?

[peak matrix]

It's not.
C is the set of all vectors x, where x = (0, t) for some real t

is (1, 1) of the form (0, t)?

[fossil pagoda]

right no it isnt

so lets say added by (0,1)

=(0,2) which is R

[peak matrix]

*which is in C

[fossil pagoda]

yes

[peak matrix]

but what if there were some other vectors in C, for which the condition would fail?

You need to be more general when proving stuff

this only shows a particular example

2*(1, 0) was also in A, but A actually failed to satisfy condition ii)

what is the general form of a vector in C?

how does a general vector in C look like

[fossil pagoda]

It looks like a line on the y axis

[peak matrix]

yes, that's how you would draw it

and the vectors will be of form (0, a) for some real a

so take 2 general vectors in C, then you can write them as (0, a) and (0, b) for some reals a and b

what will (0, a) + (0, b) be?

[fossil pagoda]

(0,10) and (0,421)?

=

(0,431)

[peak matrix]

that's another example

there are infinitely many more

you can't list them out all unfortunately

and if you don't, it wont be a proof

so you will have to work generally here

[fossil pagoda]

gotcha

[peak matrix]

I'll show you how to do the proof for condition i), and you will then try to do a proof for condition ii)

So condition i) tells us, that we need to verify that for all vectors u and v in C, u + v is also in C

so we start by taking some vectors u and v in C

this tells us, that u and v will be of form (0, t) for some real t

so let's say that u = (0, a) and v = (0, b)

then u + v = (0, a) + (0, b) = (0, a + b)

and since a and b are real numbers, a + b must be real number as well

and so (0, a+b) must also be in C

which means that u + v is in C

[fossil pagoda]

yes

[peak matrix]

can you prove condition ii) now?

this one, for any scalar k and any vector u in C

[fossil pagoda]

yes

k is the scalar and u is the vector (0,t)

lets say that u=(0,a) which can be any real number and in this case is 450

(0,450) * K which can be any real number which in this case lets say is 1.5

[peak matrix]

Don't say the 450 part, you need your proof to be general. a can be any real number, not just 450

[fossil pagoda]

ok

[peak matrix]

do (0, a) * k instead

[fossil pagoda]

(0,a) * K will always equal (0,some real number)

because any real number * 0 is 0

[peak matrix]

yes, to be exact, the "some real number" will be aK

[fossil pagoda]

and a is any real number multiplied by a real number

correct

[peak matrix]

k(0, a) = (0, ak)

ak is real

[fossil pagoda]

exactly

[peak matrix]

so (0, ak) will be in C

and that's it

[fossil pagoda]

yep

[peak matrix]

now this

try proving condition i)

[fossil pagoda]

it does go through 0,0

[peak matrix]

really?

what t makes it go through (0, 0)?

[fossil pagoda]

t can be any real number

[peak matrix]

can you name few vectors that are in D?

[fossil pagoda]

(1,430) (1,532.32) (1,23)

[peak matrix]

neither of these is actually in D

D contains vectors of form (0, t) + (1, 1)

[fossil pagoda]

right the x needs at least a 1

woops

[peak matrix]

so it contains e.g.
(0, 0) + (1, 1)
or
(0, 5) + (1, 1)
or even
(0, 3.14159) + (1, 1)

[fossil pagoda]

do the vectors have to be written with the +(1,1)?

or can the 1,1 just be added

for examples

[peak matrix]

you can just add it

[fossil pagoda]

(1,2) instead of (0,1)+(1,1)

oh ok

[peak matrix]

it contains
(1, 1)
(1, 6)
(1, 614)

etc

in general, it contains any (1, t)

does it contain (0, 0) though?

[fossil pagoda]

no

so it is not a subspace

[peak matrix]

yes, it doesnt

[peak matrix]

mhm

can you draw it?

what would it look like?

[fossil pagoda]

it would be a plane with an empty space at 0,0 going in the positive direction?

[peak matrix]

not quite

[peak matrix]

this plane would also contains vectors like (5, 713)

does D contain that?

[fossil pagoda]

no

x has to be 1

so its a line

[peak matrix]

yes, x has to be 1

yes

x = 1

[fossil pagoda]

on x=1 in the y direction