linear-algebra

[sonic osprey]

You just need to see how to logically fit things together

You don't see why they're equivalent?

[wintry steppe]

no, because lets assume 1 norm is 3

then i have 3a <= 6 <= 3b

i can pick a and b that sattify this

[sonic osprey]

.

[wintry steppe]

and if 1 norm is x, i can do the same

[sonic osprey]

I'm just going to start quoting messages I've said already

because I'd rather stop repeating things for the fifth time

[sharp idol]

Yea. a and b must be constants

[wintry steppe]

okey, then what a and b will u chose?

u said they are equivalents, so there must be some constant values that sattisfy the inequality for all the polynomials

[sharp idol]

I recommend you go to #precalculus or one of the question channels. Regardless if this is linear algebra or not, this channel is already pretty involved in something right now.

[sonic osprey]

.

[wintry steppe]

tell me the a and b and i will try to find the polynomial

[sonic osprey]

Oh, are you talking about my example now

[wintry steppe]

yes

[sonic osprey]

To show they're equivalent, you want to find numbers a,b such that a||x||_1 <= 2||x||_1 <= b||x||_1 holds for all vectors x

Do you agree with this?

[wintry steppe]

yes

[sonic osprey]

So you just take a = 1 and b = 2 for example

[wintry steppe]

dfljhvasdjvhasdphfas

u just complained about this !!!

[sharp idol]

Hold on, shouldn't a and b be multiplying the same norm?

[sonic osprey]

For the sixth time, I complained about this because you are trying to show that two norms are not equivalent

[sharp idol]

Seems ya fixed it

[sonic osprey]

I just complained because I didn't understand you were talking about my example

I thought you were talking about your original problem

[wintry steppe]

ah okey, i see it

so a=1 and b=2 holds the inequality for any P(t)

right? right

[sonic osprey]

yes

[wintry steppe]

okey so, on my problem, they arent cuz i cant find a,b that holds for any P(t) cuz i can always increase the value of n, right?

please tell me yes

[sonic osprey]

yes

[wintry steppe]

hdxj

😄

||_but i can always increase n cuz i have infinite polynomials _||

[sonic osprey]

I agree

[wintry steppe]

okey xd

[sonic osprey]

But just saying that your vector space is infinite dimensional is not enough

Like i showed

[wintry steppe]

ye ye, okey, i got it

[sonic osprey]

So saying that you have infinitely many polynomials is not enough of a justification

[wintry steppe]

fair

ty

😦

///<

sorry if i missexplained but wasnt my intention to say inf-dimensional = justification. I was trying to say that i could do it cuz it was inf_dimensional

i dont use to talk about maths on english so sometimes is hard for me

[blissful vault]

thhanks to closed captions and it's sam i understand now

i have another question

this is the notes i took

i'm not sure what gamma is doing

it looks like the teacher just calculated T(1,0) and T(0,1) and then transposed them

[sharp idol]

That is normally what you want to do, because T(1,0) and T(0,1) transform the basis and therefore your vector space

Plus, they tend to be easier to evaluate

[wary lily]

for b, either two equations must coincide and the third is not parallel to them, or all equations are distinct and intersect at one point.

I'm not sure how to mathematically express this.

Please check:

For case one $\abs{\frac{a}{b}} = \abs{\frac{c}{d}}, k = l, ex + fy = ax +by$, this also works for any other pair to coincide and the third not parallel to them.

[stoic python]

az

[wary lily]

For case two $\abs{\frac{a}{b}} \neq \abs{\frac{c}{d}} \neq \abs{\frac{e}{f}}$ and there exists $(x, y) \in \mathbb{R}^2 | ax + by = cx + dy = ex + fy$

[stoic python]

az

[blissful vault]

then for this question: do i just evaluate T using beta and ignore alpha?

[spring pasture]

Alpha is the space in which u r doing transformation

But yeah u don't have to worry about it while doing transformation

[blissful vault]

alright thanks!

[dusky epoch]

i dont think youre supposed to do this in terms of the variables themselves tho

they ask for a desc of the relative positions of these lines

[wary lily]

true, I was trying to express it mathematically to learn more

thanks

[blissful vault]

HAHA my teacher wrote this https://media.discordapp.net/attachments/814219153669619731/817833486936309830/unknown.png

[nocturne jewel]

Yeah

[stable kindle]

depression

[raven wagon]

can someone help me with this problem ? I don't know where to start !!

[sharp idol]

I'm sorry, I do not know french well enough to be useful here.

[raven wagon]

can I try to translate it for you ?

[sharp idol]

I'd definitely appreciate that.

Can you also tell me what the upside down v means for this? I have never seen it used for vectors

[wary lily]

my wild guess is that 18 is the triple product so that would be the cross product, but then we would have a problem at trouvez l'angle entre...

[raven wagon]

So, in this tetraedron, we have the coordinates of W= (2,2,1). We also know that the face (triangle) formed by U and V=6 units.

The product of UV by W= 18

[raven wagon]

I thought about it but, nothing specifies that it's regular. I'm absolutely lost here !

[wary lily]

is that $(\ora{v} \times \ora{u}) \cdot \ora{w} = 18$?

[stoic python]

az

[raven wagon]

yes !

[wary lily]

this would make sense because the triple product is also the volume of the parallelpiped that is formed by the three vectors v, u, and w

if you find a relation between the parallelpiped and the tetrahedron that may help

[raven wagon]

can we say that all the vectors (u,v and w) have the same length ?

[sharp idol]

I think that is a bit of a jump

[raven wagon]

then, I know about W the most right ? ... what could I do with it ?

could I maybe divide 18 by W to find what the product of v and u should be ?

[sharp idol]

Sadly no. Dividing by vectors is a no no

Also, keep in mind a dot product is not "normal" multiplication

[raven wagon]

yeah right ! sorry haha

can I multiply a surface by a vector ?

[nocturne jewel]

you mean the area of a surface?

[raven wagon]

no sorry I mean the triangle made by U and V = 6 surface units

[sharp idol]

Wait, that's good!
The triangle made by $\vec{u}, \vec{v}$ is $\frac{1}{2}||\vec{u}\times\vec{v}||$

[stoic python]

dackid

[raven wagon]

should I isolate one of the 2 vectors ? or is it too complicated after?

I know what I did is wrong but it might take us somewhere ...

1/2 (uxv) =6

If we isolate u and say that u=3/v

Does it make some sort of sense ?!

[sharp idol]

Again, we cannot divide vectors

[stoic python]

dackid

[raven wagon]

So, 12 •w=18

Or not

[wary lily]

12 is the magnitude of h

[sharp idol]

No no. $\vec{u}\times\vec{v}\neq ||\vec{u}\times\vec{v}||$

[stoic python]

dackid

[sharp idol]

The left side is a vector and the right side is a scalar

[wary lily]

is that directed at me?

[sharp idol]

It is directed at this claim

[wary lily]

ah, having the magnitude of h may help here

[sharp idol]

Oooo, so here is something interesting!
$(\vec{u}\times\vec{v})\cdot \vec{w}=||\vec{u}\times\vec{v}||;||\vec{w}||\cos(\theta)$

This will actually get you really far!

[wary lily]

noice

[stoic python]

dackid

[sharp idol]

Yeah, that's what we need

@raven wagon you getting this?

[raven wagon]

Yup ! Got it

[sharp idol]

Okay. Use that identity and it should take you home free!

[raven wagon]

Thank you so much for the help ! It almost feels like I won the Olympics or something 😂🥳

[sharp idol]

It's a good feeling indeed :)

What was the problem anyways? You gave us a lot of givens, but not any goals xp

[raven wagon]

The final goal was finding the angle formed btw uv and w

I think I have everything I need with the last formula. Thanks again !!

[sharp idol]

Sweet! Glad we got there in the end

[nocturne jewel]

If I have 2 3x3 matrices M and N, is showing their determinants arent equal sufficient to showing they arent similar?

[stable kindle]

similar?

[wintry steppe]

yes

similar matrices have equal determinants

[stable kindle]

oh, right

[nocturne jewel]

Yeah wasnt 100% sure on if that was always the case

[wintry steppe]

equal up to change of basis

[stable kindle]

ok

[wintry steppe]

it's also one reason you can give a good definition of the determinant of a linear operator and not just a matrix

something to keep in mind

[nocturne jewel]

[stoic python]

dackid

[sharp idol]

Wait, you are talking about similar. What does it mean for two matrices to be similar?

Is it this?
A and B are similar if there exists a matrix P so that $A=P^{-1}BP$.

[stoic python]

dackid

[wintry steppe]

yes

that is the correct definition of "similar" for matrices

that is also the correct definition of "symmetric" for matrices

[sharp idol]

Okay thanks. I wasn't sure if my memory served correctly, so I just wanted to make sure.

[tame mural]

Hey TTera, you never had a YT channel, right?

actually it's too unlikely

[wintry steppe]

[sudden nacelle]

i need help finding the parametric representation of a hyperboloid

[ruby loom]

I'm having difficulty parsing the definition of the dual basis

And also really of getting an intuition for just what this is

[wintry steppe]

what part of the definition is confusing you

(share the definition you're working with)

[ruby loom]

Well, for one, it seems peculiar to me to define a basis by an indicator function

[wintry steppe]

indicator function?

[ruby loom]

Oh, I had thought 1 if k = j, 0 otherwise was a type of indicator function

perhaps mistaken

[wintry steppe]

you're defining a linear map by how it acts on a basis

this is perfectly fine, right?

since a linear map is completely determined by how it acts on a basis

you could write $\varphi_j(a_1v_1 + \cdots + a_nv_n) := a_j$ if you wanted

[stoic python]

(T*Terra, dqⁱ ∧ dpᵢ)

[wintry steppe]

it's basically a projection to the jth coordinate, in the basis (v_1, ..., v_n)

[ruby loom]

Hm

Okay, so I would like to relate this to my prior understanding of a basis

[wintry steppe]

sure

[ruby loom]

which was the notion of a basis of a vector space over a field, which was a set of linearly independent vectors

[wintry steppe]

what are you thinking

[ruby loom]

Well, are the elements of the dual basis also linearly independent? If they are, I'm not sure how to conceptualize what a linearly independent set of elements of linear functionals is

sorry if that is a bit disjointed, not sure I expressed that properly

[wintry steppe]

it's expressed perfectly and is a good question

they are indeed linearly independent

(it's called dual basis for a reason)

it's the same notion of linear independence, but the elements you're working with are linear functions V -> F

to say that something in V' is zero is to say that it equals 0 on every vector in V

so if you wanted to write out what the phrase "$\varphi_1, \dots, \varphi_n$ are linearly independent" means it'd be: if $a_1, \dots, a_n \in F$ are scalars such that for all $v \in V$, $$(a_1\varphi_1 + \cdots + a_n\varphi_n)(v) = 0,$$ then $a_1 = \cdots = a_n = 0$

[stoic python]

(T*Terra, dqⁱ ∧ dpᵢ)

[wintry steppe]

i.e. if the dual basis element $\sum a_i \varphi_i = 0$, then the scalars $a_i = 0$

[stoic python]

(T*Terra, dqⁱ ∧ dpᵢ)

[wintry steppe]

that's what that means

you should use this to verify that the dual basis elements are actually linearly independent, it's a good exercise

and then, since V and V' have the same dimension, it follows that the dual basis is actually a basis

[ruby loom]

Okay, that certainly makes sense, I'll try that exercise, as well

Ah, okay, the definition they provided was just a way of selecting each phi so that we have a linearly independent set of phi's much as we would with a basis for a vector space

[wintry steppe]

that's a nice way to think about it

it's basically defined to be a basis lol

[ruby loom]

okay! makes sense

thanks so much for helping me clarify that

[wintry steppe]

do make sure the construction makes sense to you

it's important

[ruby loom]

This particular one, or the general form?

[wintry steppe]

the dual basis construction

[ruby loom]

I'll make sure it makes sense, but I'm curious if that's because the dual basis construction pops up a lot or for some other reason?

[wintry steppe]

it's important in linear algebra in its own right

if you do anything that involves multilinear algebra you'll need it

e.g. differential geometry uses dual spaces a lot

[ruby loom]

Okay, yeah I can't say that I immediately see why this dual basis would be significant, but we've just touched on it

[wintry steppe]

it takes some working with it and seeing it in other places to see why it's important

imo

[ruby loom]

okay! noted

[fervent gulch]

<@&286206848099549185>

[dusky epoch]

15-minute rule!

you're not supposed to ping helpers right away @fervent gulch

[fervent gulch]

Ahhh. Thanks for letting me know

[dusky epoch]

so... what's giving you trouble here

[fervent gulch]

Ik that I have to do this: det([a - l, b]) = 0
[c, d - l] but I had a hard time doing this right

[dusky epoch]

this is badly formatted

but yes... you need to calculate $\det\bmqty{8-\lambda & -7 \ 6 & -5-\lambda}$ and set it equal to zero

[stoic python]

Ann

[dusky epoch]

do you have work to show as far as that goes?

[fervent gulch]

Not always

[dusky epoch]

what do you mean, "not always"

this is one particular problem

do you have any work that i could look through, or not?

[fervent gulch]

Like whenever my prof feels like it but not for this question

[dusky epoch]

okay then what exactly is giving you trouble with doing it for this question?

[fervent gulch]

The formatting really, like end result. That's all but you made it simple for me so thank you.

[dusky epoch]

you don't need to lie to me if what i've been saying has been utterly unhelpful to you all along.

[fervent gulch]

I really wasn't. Nono I don't mean anything like that.

[dusky epoch]

okay

i'll take your word for that

[fervent gulch]

Thank you and I'd like your help in 5 mins if you can wait.

[dusky epoch]

sure, ping me once you need

[fervent gulch]

I obtained this: 5l+lλ-8λ-82

[dusky epoch]

eh?

[fervent gulch]

and so I tried entering this but it won't accept I

[dusky epoch]

what's this business with l and lambda being separate things

[fervent gulch]

for the question earlier.

it isn't 1 but L

[dusky epoch]

i know it's a lowercase L

and i typed one in my message

what i'm asking is

why do you even have $l$ and $\lambda$ in there at once?

[stoic python]

Ann

[dusky epoch]

there was no way you could've taken $\det\bmqty{8-\lambda & -7 \ 6 & -5-\lambda}$ and ended up with something that has the letter $l$ in it

[stoic python]

Ann

[fervent gulch]

I simplified this: \left(8-l\right)\left(-5-λ\right)-\left(-7\right)\left(-6\right)

$\left(8-l\right)\left(-5-λ\right)-\left(-7\right)\left(-6\right)

[dusky epoch]

why? why do you have an L in there???

where's it coming from?

why do you write L in one spot and lambda in the next???

[fervent gulch]

oh my bad, typo. I meant to add lambda

[dusky epoch]

well this typo somehow slipped by you at all stages of the algebra

also, the lower-left entry of your matrix is 6, not -6.

[fervent gulch]

\quad λ=\frac{3+\sqrt{337}}{2},:λ=\frac{3-\sqrt{337}}{2}

[fervent gulch]

Eh thanks for that comment.

[dusky epoch]

redo the problem, but now be extra, extra careful about what youre writing.

you should start with: (8 - λ)(-5 - λ) - (-7) * 6

[fervent gulch]

yea so i did that and obtained lambda = 3+- sqrt 337/2

[dusky epoch]

you must have made an algebraic mistake somewhere.

i will need you to write out your work, in full, preferably on a piece of paper, and post it here so i can look over it and tell you exactly where you went wrong.

[fervent gulch]

Sure one sec.

[dusky epoch]

be extra careful, and triple-check every single move you make.

[fervent gulch]

So this matrix got me λ^2-3λ-82=0.

How do i texit this?

[dusky epoch]

i will need you to write out your work, in full, preferably on a piece of paper, and post it [your work] here

you keep giving me just the answer

i want your work

show your work

[fervent gulch]

okay imma send a picture gimme a min

[dusky epoch]

yikes

okay

so

youre skipping over ALL the steps i was interested in

and you keep writing -6 in the matrix when there was never any -6

the lower-left entry of your matrix is not -6

it's 6

positive 6

not -6

just 6

[fervent gulch]

hold on imma redo

so 2 and 1

@dusky epoch

[dusky epoch]

sigh

youre skipping over the step where you work out the determinant

which is what i wanted you to showcase

but yes now your determinant is correct, it's λ^2 - 3λ + 2

[fervent gulch]

ok I got another question

[dusky epoch]

yes?

[dusky epoch]

mkay, what's giving you trouble here?

[fervent gulch]

How do I do this.

[dusky epoch]

well they give you $P^{-1}AP = D$ for free

[stoic python]

Ann

[dusky epoch]

this could also be restated as $A = PDP^{-1}$

[stoic python]

Ann

[dusky epoch]

do i need to explain in more detail why those two are equivalent?

[fervent gulch]

I don't understand the K

[dusky epoch]

we'll get to that.

k here can be taken to be a nonnegative integer.

[fervent gulch]

Can you gimme 3 mins, my sibling is calling me

[dusky epoch]

okay

ping me once you're back

[wary lily]

I can row reduce augmented matrices, but I'm confusing Gaussian and Gauss-Jordan elimination. Is it important to know them from one another? And if so, can someone give me a rule of thumb that helps me remember them?

[dusky epoch]

it doesnt really matter

[wary lily]

OK

[reef sleet]

So we just started on determinants in class, and I was just checking my work for a problem using a matrix multiplication calculator when I saw tht you could calculate the determinant by finding the product of each element in the diagonal?

But only in RREF though?

[dusky epoch]

no, that trick works whenever your matrix is triangular

plus, RREF doesnt preserve det

[fervent gulch]

I'm alive

[reef sleet]

Oh yeah that makes more sense lol, thank you

Would you mind explaining why it works after you're done with Barbie?

Oh wait I see why it works

[fervent gulch]

I understand why it is equivalent.

[dusky epoch]

okay, great.

now, can you use the equation $A = PDP^{-1}$ to tell me what $A^2$ is, in terms of $P$ and $D$?

[stoic python]

Ann

[dusky epoch]

(note: the answer is \textbf{not} $P^2 D^2 P^{-2}$!)

[stoic python]

Ann

[fervent gulch]

Well A^2 is A times A. Which is

[dusky epoch]

which is...?

[fervent gulch]

25, 24, -16 and -15.

[dusky epoch]

uh

i couldn't give half a shit about what the actual entries of A^2 are

and you didn't answer my question

which i politely ask you to read in full

can you use the equation $A = PDP^{-1}$ to tell me what $A^2$ is, \textbf{in terms of $P$ and $D$}?

[stoic python]

Ann

[fervent gulch]

Hold on

[fervent gulch]

Yeah you can.

By doing:

@dusky epoch

[dire thunder]

she asked for A^2 in terms of P and D, not in terms of their values

[dusky epoch]

that is STILL not what i'm asking you to do.

[dire thunder]

just find A^2 for arbitrary P and D

[fervent gulch]

PDP^{-1} -> is the equation that can be used

[faint lintel]

Pretend you don't have values for A, P, D, or P^-1

[fervent gulch]

We add ^k to d.

[dusky epoch]

wow, way to jump the gun

also, we don't "add ^k", we "raise to the k'th power"

[shy atlas]

hi ann

[fervent gulch]

I'm trying but you have to stop with the sarcastic comments, everyone has there own pace.

*their

[dusky epoch]

i'm not being sarcastic

you went from 0 to 100

in a near instant

$A^k = PD^kP^{-1}$ is what i was trying to build up to

[stoic python]

Ann

[dusky epoch]

i expected you to say $A^2 = PDP^{-1}PDP^{-1}$, then cancel out the $P^{-1}P$ in the middle to get $A^2 = PD^2P^{-1}$

[stoic python]

Ann

[fervent gulch]

Okay so we have to add K in the matrix next to 3 and 1 in matrix D.

[dusky epoch]

no you don't "add k"!!!

nothings getting added!! the k is an exponent, not an addend!!!

[fervent gulch]

I know what you mean I am great at visuals then explaining I guess.

[dusky epoch]

it is important that you learn how to explain yourself properly

[fervent gulch]

^

[zinc copper]

is the channel free or does barbie still need help?

[fervent gulch]

still

[dusky epoch]

also, i don't really know how to feel about you saying "i know what you mean" when you spent the last 10 or so minutes not knowing what i meant at all

[zinc copper]

alright

[dusky epoch]

anyway, ok, fine, you jumped ahead, and now we know $A^k = PD^k P^{-1}$

[stoic python]

Ann

[dusky epoch]

it is now that we remember we know what P and D are, and that we know how to raise D to the k'th power

[fervent gulch]

mhm so I was doing the math and I obtained the anser

*answer

Gimme a min

Row 1 is( 3^k+2 ) -4 and 2 times (3^k+1) -4

and row 2 is -2 times (3^1+k) - 4 and -4 times (3^k) +4

[dusky epoch]

$\bmqty{3^{k+2} - 4 & 2 \cdot 3^{k+1} - 4 \ -2 \cdot 3^{k+1} - 4 & -4 \cdot 3^k + 4}$

[stoic python]

Ann

[dusky epoch]

is this your answer?

[fervent gulch]

I'll show my math it u want

yea it is

[dusky epoch]

this is incorrect.

[fervent gulch]

I'll show my math

[dusky epoch]

yes, please do.

i mean, idk about you, but the fact that this doesn't give the identity when plugging in k=0 or A itself when plugging in k=1 is a very big red flag that something went horribly wrong somewhere.

[fervent gulch]

Ok I'm uploading

[dusky epoch]

okay so

$A^k = PD^k P^{\color{red} -1}$, not $PD^kP$ as you did.

[stoic python]

Ann

[fervent gulch]

gotcha so i gotta do

so if we look into my previous answer, for row 1, I remove the ^2 to 1 and -4 to 2, then add in (3^k+1) -3

[dusky epoch]

..

what.

[lavish jewel]

you have to do it all again

[dusky epoch]

^

[fervent gulch]

I did

[dusky epoch]

then show your new work.

[fervent gulch]

and I am modifying it

[dusky epoch]

no

do not modify your old work.

go directly back to the start, do not pass go, do not collect $200.

[fervent gulch]

alr imma show u

[wary lily]

Quick question, when a homogeneous linear system has the same number of equations as unknowns, is it guaranteed that it has always only the trivial solution?

[lavish jewel]

depends on the rank

[dire thunder]

no

assuming system is linearly independent - yes

but consider
x+y = 0
2x+2y = 0

[wary lily]

O, sorry, linear homogeneous system in row reduced form.

[dire thunder]

x = 0
0y = 0

[lavish jewel]

that it is in row reduced form changes nothing

1 1 0; 0 0 0 is row reduced

[wary lily]

true, true

if it has the same number of pivot columns as rows/equations

[dire thunder]

x+0y = 1
x+0y = 1

oh wait columns

[wary lily]

nah, that's not row reduced

[lavish jewel]

yes, same number of pivot columns as equations -> full rank matrix

i.e. the columns are lin indep

(for a square matrix)

[dire thunder]

anyway, it is all contained in more simple statement

"rows of matrix/columns of matrix are linearly independent"

[wary lily]

I'm still not there formally, but I get it

thanks

[zinc copper]

that's the same as last time??

[dusky epoch]

this is all the exact same

[fervent gulch]

wrong photo

[wary lily]

can't help but notice the similarity

[fervent gulch]

like i said wrong photo

[wary lily]

dw

jk

[dire thunder]

hi az

[wary lily]

hey

I'm starting some linear alg

was told multi var calc without lin.alg isn't that useful/productive

[fervent gulch]

@dusky epoch

[dusky epoch]

okay... let me check this now

okay now you inverted P

but u

uh

something weird happened in your multiplication

[zinc copper]

still using the old matrix it seems

not the inverted one

to multiply

[fervent gulch]

wdym do u not see the -1

[zinc copper]

the 1,1 element uses the -3 and 2 elements from P in its dot product

[fervent gulch]

wdym

[zinc copper]

i mean look yuor result is the same

so that should be a clear sign youve gone wrong

and where youve gone wrong is youve inverted P

but still used P as the right multiplicator

[fervent gulch]

this is the last result that when wrong.

[zinc copper]

and not P^-1

[dusky epoch]

theres also the issue of $-3^{k+1}$ morphing into $-3^k + 1$ as if those aren't two very different expressions

[stoic python]

Ann

[zinc copper]

oh yeah didnt even catch that

you're gonna have to do it again and carefully. Maybe writing more neatly can help, and on paper that hopefully hasnt been attacked by kindergartners

[fervent gulch]

my now result is not the same as

this one

[zinc copper]

that's not what you wrote lol

[dusky epoch]

i transcribed your earlier answer here. this is not mine. it's yours.

[fervent gulch]

yea I know

I'm saying that the previous one is not the same as the one I gave in rn. Narwhal said this: #linear-algebra message

[dusky epoch]

this is the correct answer. it looks like this is what symbolab gave you.

[zinc copper]

both results are like this one is what im saying

[dusky epoch]

and it is also what i got.

[fervent gulch]

it's the only think that let's me format correctly cause I am bad at texit

I'm not sure where to find matrices on google docs

[zinc copper]

Alt + = for equation

then use equation inserter

or if you know microsoft language for math it's something like \matrix[a&b@c&d] or something

oh nvm google docs

that's word

idk then

[fervent gulch]

Thank you Narwhal. :p

[zinc copper]

i dont think you can

i think ima just ask my q in a question channel and come back here when it's free

[fervent gulch]

thank you

[fervent gulch]

@zinc copper i think you could use this, free rn

[zinc copper]

Since ive posted in a question channel ill just make a reference: #help-3

It's a problem im having with a step by step proof of bezout's theorem in "rational points on elliptic curves" by silverman

[lavish jewel]

shouldn't be 0, the rows are lin indep

use the properties of row operations

what happens if you scale a row, swap rows around, add rows together

[tame mural]

Kind of a silly question, but is there a name for joining two columns or rows together into a fatter matrix?

join([1 0 0], [0 0 0]) = [1 0 0; 0 0 0]

[fervent gulch]

i can modify, the default is 0

[dusky epoch]

@tame mural vertical concatenation?

[hollow finch]

Identify the row operations used to get to the second matrix and then modify the value of the determinant based on how those row operations affect it

[fervent gulch]

how do i obtain this?

[spring pasture]

Get the row echelon form

Do the row operations

[fervent gulch]

row 1: -125, 25, -5, 1
row 2: 0,72/5,48/25.152/125
row 3: 0,0,-12/5,12/25
row 4: 0,0,-16/15, 208/225

I did that

[spring pasture]

Ok

So what about right side

[fervent gulch]

row 1: -118
row 2: 0
row 3: 0
row 4: 0

[spring pasture]

So by row 3 you have -12/5 c +12/25 d =0

Bring the equations back

And try to solve them

But I think you can make it simple than this

[fervent gulch]

wait i'm confused

[spring pasture]

Just like this

Write all the equations back from the matrix

Like row 1

-125a+25b-5c+d=-118

[fervent gulch]

0a+72/5b+48/35c+152/125d

[spring pasture]

=0

[fervent gulch]

0a+0b-12/5c+12/25d

[spring pasture]

=0

Look at right side too

[fervent gulch]

0a+0b-16/15c+ 208/225d=0

[spring pasture]

Yes

But I think that u can bring it in more simple form

The calculations are too big

@fervent gulch I'll tell u some row operations you follow them

[fervent gulch]

ok

[spring pasture]

First do R2 -- R2+R1

Then R1 -- R1-R2/2

[fervent gulch]

0, 18,0 2

[spring pasture]

27 0 3 0 will be row 1

[fervent gulch]

ok wb the next

[spring pasture]

Divide row 1 by 3

Row 2 by 2

[fervent gulch]

0,0, 0.48 and 12/25 then 0, 7.2 and 0.608. DEf feel way off

[spring pasture]

Why that

Ok I'll just try it and see

@fervent gulch

Write this

[fervent gulch]

yes

[spring pasture]

Left side I got

9010

0901

1,16,9,0

8,-8,0,0

And right side

12

6

44

0

Ok

So the equations are

a+c=12

b+d=6

a+16b+9c=44

8a-8b=0

Now by 4th one I get 8a=8b

So a=b

[fervent gulch]

this is the question:

[spring pasture]

Oh

So p is like ax³+bx²+cx+d

So yeah first row

27 9 3 1

Yes all rows are right

And after row operations see the equation we got

[spring pasture]

From here

So I get a=b

And c=12-a

Now substitute this in 3rd equation

We get a+16a+9(12-a)=44

Get a by this

16a+108=44

a=-4

So b=-4

c=12+4=16

[fervent gulch]

i was gonna type a for ya

[spring pasture]

d=6+4=10

[spring pasture]

I think u can do that so np

We got all the values

So write the polynomial

[fervent gulch]

-4+16-4+9(12--4)

=152

[spring pasture]

Wat is this

Why this

[fervent gulch]

[spring pasture]

We got the values put them in p(x)

p(x)=-4x³-4x²+16x+10

[fervent gulch]

[spring pasture]

Do u get it?

[fervent gulch]

yes i was just being a bit dramatic

[spring pasture]

Ok do the next part then

[fervent gulch]

i was gonna say something but i doubt myself

[spring pasture]

What is it?

[wary lily]

Is it a bad habit if I row reduce my systems with a CAS? it's boring and I make silly mistakes. I don't see much learning in doing them by hand.

[floral thistle]

If you still make silly mistakes, then I would practice more

Until the task becomes so trivial that needs to be automated

What kind of mistakes you make?

[wary lily]

like forgetting a minus sign, or miscalculating a product and addition when doing part of it mentally, etc.

not conceptual, I dare say

[floral thistle]

Oh

Then go ahead and automate it

if it's not required for a test or something

The conceptual part is the important thing

[wary lily]

yeah, no, I guess it's partly bc I'm not motivated enough anymore to do them

because they started to become boring

thanks

I would first check to see which of the two points lies on the line, A or B

suppose it's A

then by definition, there is a line that goes through B and intersects the line at 90degs

you mean how to check this, or why doing this?

OK, you have two points of a triangle, and looking for the third

you also know that the third lies on a given line

so, by definition, this given line is one of the three sides of the triangle

right?

when you find which point lies on this line, you also find out which point does not lie on it

bc there is C on this line

you find the other one

so the remaining is one point that does not lie on this line

right?

nice

[wintry steppe]

When the null space of a certain matrix is
x_2 u + x_4 v + x_5 w

Where u, v & w are vectors

Is the basis for such a null space just

Span{u, v, w}?

[limber sierra]

not unless u, v, w are all linearly independent

[wintry sphinx]

and the span isn't the basis

[limber sierra]

oh yeah

that as well

[wintry steppe]

I presumed that they were linearly independent, I guess I misinterpreted what the book is trying to say

In fact, the construction of u, v, and w automatically makes them linearly independent, because equation (1) shows that 0 = x_2 u + x_4 v + x_5 w only if the weights x_2, x_4, and x_5 are all zero.

I don't really understand why this is the case

When can I, and when can I not conclude that they are linearly independent?

[limber sierra]

you can verify linear independence by checking the definition of linear independence

in this case, because you row reduced the matrix, the rows are automatically linearly independent

(as you ended up with no 0 rows)

[wintry steppe]

I have, it's just difficult for me to follow to be honest

[limber sierra]

i mean the definition of linear independence is that

this equation

is 0 only when x_2, x_4, x_5 are 0

but that must clearly be the case here

[wintry steppe]

What do you mean with 0 rows? Since the latter is all zeros. Or do you mean zero on the left side? It would always be fully 0 if the left side is 0, no?

[limber sierra]

sorry, let me rephrase

you ended up with rows where only one entry is nonzero

in order for the sum of these to be 0

since the circled entries are the only nonzero entries in their row

they must all multiply with their scalars to 0

which means x_2 must be 0, x_4 must be 0, and x_5 must be 0

hence these vectors are linearly independent

[wintry steppe]

That makes total sense, thank you

[wary lily]

asterisk can be any real number

I need to decide existence and uniqueness

I thought this is inconclusive

[wintry sphinx]

what do you mean? existence of solutions to Ax = b?

[wary lily]

that's an augmented matrix, sorry

system of equations

[tame mural]

Another definition for independence is that every vector contributes to the span.

[wary lily]

bc of $a_{14}$

[stoic python]

az

[wary lily]

if it's anything other than 1, this becomes inconsistent

[wintry sphinx]

That's not true

that's only if the other asterisks are 0

[wary lily]

yes, I assume that mentally

but it's inconclusive, right?

[wintry sphinx]

existence depends on the values in the asterisks

but uniqueness doesn't

[wary lily]

if it is consistent, there are infinitely many solutions

but we can't say if consistent or not

[wintry sphinx]

yep

[wintry steppe]

If they were not linearly independent, would that mean the basis would be the zero vector?

[limber sierra]

what

no

the zero vector is never in any basis

[wintry steppe]

Then what would be the basis if that was not linearly independent?

[limber sierra]

find a set of vectors that is linearly dependent

remove one in such a way that makes the set linearly independent

repeat until all your vectors are linearly independent

that's a basis

[wintry steppe]

That is what one does for the basis for a null space? I didn't come across an example like that yet

[limber sierra]

i feel like youre getting wires crossed a bit here

[wintry steppe]

But row reduction would make this unnecessary, right?

[limber sierra]

im talking about bases in general

in the context of working with null spaces and row reduction you typically dont run into these issues

[wintry steppe]

Yeah, I am. Sorry

Thanks for the help

[wintry sphinx]

okay I'm sure you've heard of a vector space

this is just a set of vectors

a subspace is a vector space that is a subset of another vector space

a basis of a vector space is a set of linearly independent vectors such that the vector space is the span of these vectors

because the null space of a linear transformation is a subspace, it has a basis

[wintry steppe]

I feel bad for wasting your effort but my lecturer skipped over the chapter covering vector spaces :(

I find it admirable that you two can discuss these kinds of topics with individuals like myself, who are obviously misinterpreting things and reasoning incorrectly. My understanding of mathematics is so shaky that if I was in your shoes, such a conversation would confuse me to the point where I'd doubt what I had learned

[wintry sphinx]

so you don't know exactly what a vector space is?

[wintry steppe]

I don't

[wintry sphinx]

you can look up the definition on Google or Wikipedia, but I'll try to give you some intuition for it

do you know what a set is?

[wintry steppe]

I do, but I'm worried that being introduced to this would confuse me to the point where I couldn't really return to the material I have to know for the course lol

[wary lily]

please, keep it on for the sake of the rest of us

damn

[wintry sphinx]

okay

So a vector space is just a set of things that are called "vectors"

that can be added, subtracted, and scaled by scalars, which are just numbers

forgetting about the field for a second

all of the axioms that you find on Wikipedia are basically just guaranteeing that addition / subtraction / multiplication work like how we're used to them seeing

[jaunty sage]

can anybody just tell me what is actually asking to find?

im confused

[wintry sphinx]

Projection is a linear transformation

so you can find a matrix that represents this transformation

[jaunty sage]

do i stack the columns together and do rref?

i know the formula for projection but not sure how to apply it

do i just use the vector a and then use this

[wintry sphinx]

first, do you understand the question?

[jaunty sage]

not sure

if i do

would be nice if you could help me understand it

[wintry sphinx]

what part don't you understand about the question?

[jaunty sage]

lets start from the beginning

a

isnt y 2 times a ?

[wintry sphinx]

so do you know what a projection is?

[jaunty sage]

ye i mean i did go over it once

[wintry sphinx]

going over it is different from knowing what it is

do you know what an (orthogonal) projection is?

[jaunty sage]

yes

[wintry sphinx]

So if we say that we're projecting a vector v onto span(a), we expect to get another vector out, right?

[jaunty sage]

yes

[wintry sphinx]

It turns out that this function that does that is a linear transformation

by that I mean if you project x + y, you get the sum of the projections of x and y

I'm sure you've also heard that every linear transformation can be represented by a matrix

They're asking you to find that matrix

[jaunty sage]

yes

[jaunty sage]

is this not the formula tho?

[wintry sphinx]

It is the formula for it

but you should be able to understand why

and not just blindly plug in a formula

[wary lily]

there is a matrix that if multiplied by v does the same as this formula?

[wintry sphinx]

yeah

[wary lily]

multplied by u

[wintry sphinx]

it's actually straight from that formula

if you write

[jaunty sage]

yeee alright that makes sense

[wintry sphinx]

$\mathbf z = \frac{\mathbf u \cdot \mathbf v}{||\mathbf v||^2} \mathbf v = \mathbf v \frac{\mathbf v^\top \mathbf u}{\mathbf v^\top \mathbf v}$

[stoic python]

Saccharine

[wintry sphinx]

just moving the v to the other side

Now if you say $\mathbf z = A\mathbf u$, it's pretty clear what A is equal to

[stoic python]

Saccharine

[wary lily]

nice, I still have to learn some notation to understand this but sounds cool. I understand the vector form of the projection formula because it's just the magnitude of u time the cos of the angle times the unite vector that we are projection onto

this makes intuitive sense

really excited to learn this

[wintry sphinx]

$\mathbf v^\top \mathbf u = \mathbf v \cdot \mathbf u$

[stoic python]

Saccharine

[wary lily]

O

[wintry sphinx]

$A^\top$ is just the transpose

[stoic python]

Saccharine

[wary lily]

transpose is when you convert rows to columns?

[wintry sphinx]

yes

[wary lily]

O, we have to transpose for matrice multiplication to make sense

it becomes the dot product

[jaunty sage]

how do i approach this question

[proper sand]

Can anyone give me a hint for starting this?

[digital bough]

It is only a transpose matrix multiplikation like that if the vectors are given in an orthonormal basis. Otherwise you cant do that!

After reading some LA questions here i can conclude american LA notation is complete garbage

[wary lily]

what language do you learn it and what books do you propose?

[digital bough]

I have nothing to propose

LA notation seems to vary a lot but i mean it is not that bad

It is just bad notation for beginners imo

[wary lily]

that's true

I'm reading this book and they propose different notation already for matrices on introduction

[jaunty sage]

i solved part a and it basically is 0 because A^T b was 0

so what would the comment be?

[fervent gulch]

i was looking over and i thought maybe p(x) is x^3 + x^2 + 3x - 3.

okay so 5^3 + 5^2 + 3x5 - 3 is p(5)

[sharp idol]

Do you need help with that problem?

[stoic python]

dackid

[fervent gulch]

yeah

[sharp idol]

And solve for $(A-3I)\vec{x}=\vec{0}$

[stoic python]

dackid

[sharp idol]

The solution to that should give you your eigenvector.

That should give you a good starting place

[fervent gulch]

could u show me, i have a hard time visualizing this

[sharp idol]

Have you solved the equation
$A\vec{x}=0$ before?

[stoic python]

dackid

[fervent gulch]

I think so but I tend to forget

[sharp idol]

Okay, use gaussian elimination as best you can, and then let's see what we can get from there.

[fervent gulch]

row 1: 1 0 0

row 2: 0 1 0

row 3: 0 0 1

[sharp idol]

That is definitely not what will happen

[fervent gulch]

before that is -20 15 -15

and 0 -3/2 -15/2

lastly 0 0 3

@sharp idol

[sharp idol]

Sorry Barbie. I fell asleep. Do you still need help?

[nocturne oracle]

what is this even asking me to do, isnt this like trivially true from the definition of a basis

[sonic osprey]

yes

[nocturne oracle]

thanks zopherus! :) @sonic osprey

🙂

[sonic osprey]

fucking cringe

[nocturne oracle]

🥺

i am reporting you to the moderation

[dire thunder]

prolly point of this ex is just for you to remember defn

[dire thunder]

i am reporting you to the gulag

[nocturne oracle]

i decline your offer

[dire thunder]

i decline your decline

[grand imp]

jeez zoph lol

[burnt parrot]

https://www.reddit.com/r/cheatatmathhomework/comments/6s0nuy/linear_algebra_find_two_different_ways_to_express/

for this problem, does it have to be row reduced into that specific form to get the answer?

[dire thunder]

your system will have free variables

which you can vary

[burnt parrot]

when i reduced it to row 1 [ 1 2 0 -2] row 2 [ 0 0 1 -1] i didn't get the right answer

is there a rule where you always have to row reduce in a way that leaves the free variable as the last vector?

[wintry steppe]

Why is Spider-Man so good at comebacks?
Because with great power comes great response ability.

[dire thunder]

i mean

[wintry steppe]

I was wondering why the ball was getting bigger. Then it hit me

[stoic python]

Commander Vimes

$$\begin{pmatrix} 1 & 2 &  -3 \\ -3 & -8 & 7 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$

[dire thunder]

@burnt parrot this is matrix equation

agree?

[burnt parrot]

yeah, agreed

[dire thunder]

so if i am not wrong in RREF it would be

[stoic python]

Commander Vimes

$$\begin{pmatrix} 1 & 0 &  -5 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ -2\end{pmatrix}$$

[dire thunder]

we can check it

,w RREF {{1,2,-3}, {-3, -8, 7}}

[stoic python]

Results provided by WolframAlpha

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

Requested by Commander Vimes#1488

[dire thunder]

yep so here we go

@burnt parrot

[wintry steppe]

A police officer just knocked on my door and told me my dogs are chasing people on bikes. That’s ridiculous. My dogs don’t even own bikes

[dire thunder]

holy fuck

i forgot add z

imagine there is also z in unknown vectro

[stoic python]

Commander Vimes

[dire thunder]

now it is better

so it is just the same as saying
x-5z=5
y+z=-2

[burnt parrot]

ohhh, so that specifically is the correct row reduced form. I guess I have a fundamental misunderstanding as to why only that form is correct, because i do see how and why it's correct, just not why it's the only correct form

[dire thunder]

yor moving z which is free variable to the RHS we arrive at
x=5+5z
y=2+z

and now you have vector of coefficients in your linear combination as (x(z), y(z), z)

[dire thunder]

well it is becuase of the way how we define RREF

[wintry steppe]

Two fish are in a tank, one says to the other "how do you drive this thing?"

[dire thunder]

shadow please move to #chill

[wintry steppe]

😄

[dusky epoch]

@wintry steppe jokes are fine but don't spam over ongoing convos with them

[wintry steppe]

just tryna brighten up ur days

[wintry steppe]

alright

[dire thunder]

but like permuations of rows of the same system can give diff rrefs

(afaik)

[dusky epoch]

no

rref is invariant under elementary row ops

[dire thunder]

oh nice then

[burnt parrot]

i do remember learning something about reduced row echelon form being unique; probably should have remembered that. not sure how I convinced myself the other one was a valid rref

thank you for the help!

[dire thunder]

yw

[nocturne oracle]

is b ... asking me to prove that or what

i would assume so

[wary lily]

I looked up a solution

after row reducing, we get the simplified system of:

sin(alpha) = 0
cos(betha) = 0
tan(gamma) = 0

now, there are 3 solutions for eq. one, 2 solutions for eq. 2, and 3 solutions for eq. 3.

I can't make the mental connections as to how this leads to having 18 solutions.

This is maybe connected to the fact that I'm used to linear systems and this isn't linear.

[torpid portal]

this amounts to counting all the possible ways you could choose 3 solutions, 1 from equation 1, 1 from equation 2, and 1 from equation 3. By the multiplication principle, that's 3*2*3=18

[wary lily]

O, true, that's all the permutations of (alpha, betha, gamma) tuple

[torpid portal]

ye

[wary lily]

thanks

[torpid portal]

no worries

[wary lily]

the difference is that when we dealing with linear systems

there is either a unique solution

or infinitely many

but not something in between

that's maybe what confused me

[torpid portal]

I mean there could be no solutions

[wary lily]

true

I was referring to consistent systems

[torpid portal]

It might help to think of it as being linear in sin(a), cos(b) and tan(c)

but after getting a unique solution for sin(a)=... cos(b)=...

etc

you've only solved for sin(a), cos(b) and tan(c)

not a, b, c

if that makes any sense

[wary lily]

somehow but not fully

can you explain further?

[torpid portal]

You can uniquely determine sin(a), cos(b) and tan(c) since the system is linear in those variables, however determining those values does not uniquely determine a, b, and c.

That probably doesn't illuminate it much further but I'm not sure how to explain it further sorry

Partly I don't quite understand what your gripe with it is

like sin(a)=0.5 doesnt uniquely determine a ofc but you understood that already

[wary lily]

what does it mean the system is linear in those variable? Are we referring to the output of sin(a), etc?

[wary lily]

helped me further understand

I think I'm starting to see

[torpid portal]

Oh sure, so in the same sense you might say e^(2x)+e^x+2 is a quadratic

but its not a quadratic in x

its a quadratic in e^x

you can say this system is linear in sin(a), cos(b) and tan(c)

[wary lily]

it's like we are packaging the non linearity away and look at them from another level

where they look linear

[torpid portal]

Oh it would have been a much better idea to give sin^2(x)+sinx+2 as the example of a quadratic lol

then that analogy extends all the way through subbing say u=sinx to solve for sinx=some value being the same idea as subbing sin(a)=x, cos(b)=y, tan(c)=z

and yeah I think you got the idea

[wary lily]

good explanation, thanks

I'm starting to better understand the idea of substitution

[sweet vine]

can someone tell which book it is

pdf viewer im using lets me invert

thats all i have
and i hv found it
it is a lecture note from

https://www.math.lsu.edu/~lawson/

[waxen flume]

https://i.imgur.com/7C8B2O7.png

For a, I got always

C, I got never

I'm not sure about B

for B

isn't it never because they're in R3

and a plane is in 2 space?

so

Always, never, never?

[lavish jewel]

your answer for c is wrong

and so is your reasoning for b

[waxen flume]

why

since null A has only the 0-vector in it

I thought there's 1 solution

so it cant be any choice of b

[lavish jewel]

there is only one solution to Ax = 0

but that's not what they ask about there

do you know what they mean by "consistent"?

[waxen flume]

ye

at least one solution

oh

its always

[lavish jewel]

there ya go

what about part b

[waxen flume]

the span of v1 and v2 is just a 3 x 2 matrix

[lavish jewel]

no

the span of v1 and v2 is all vectors w such that w = av1 + bv2, for any a and b

so taking what you said, the span is all vectors that come out from multiplying that 3x2 matrix with any vector [a;b]

[hoary osprey]

might wanna double check ur answer for a too

[lavish jewel]

how come? if the null space has only the 0 vector in it, the columns are linearly independent

should be rank 3

care to explain?

[sweet vine]

oo there is an extra condition
ur right then

[waxen flume]

im confused

is my a right

[lavish jewel]

it is

a and c are both "always" given the conditions in the problem

b should also be always

[waxen flume]

ty

https://i.imgur.com/Yl0WDhH.png

for my 3x3

I have [0, 1, -2] for my first column

how do I figure out the other two columns such that the matrix A has only 1 solution

[lavish jewel]

you can't

[waxen flume]

because for their to be only one vector [0 1 -2] in the solution set

it means the RREF matrix has only 1 solution

[lavish jewel]

that's wrong

[waxen flume]

the other two columns will have free variables

[lavish jewel]

this will have infinitely many solutions

[waxen flume]

ah

im confused then

[lavish jewel]

it's telling you the column space of A is spanned by a single vector

so it's just a line, instead of all of R3

any b such that Ax = b will be multiples of this vector you were given

[waxen flume]

its asking for a 3x3 matrix not a multiple of the [0 1 -2] vector?

[lavish jewel]

it's asking for a 3x3 matrix such that, no matter what x you multiply, the result is a multiple of that vector

[waxen flume]

how do i come up with the matrix then

[lavish jewel]

start from the rref form and work backwards, i guess

[waxen flume]

like that?

[lavish jewel]

do you know what an eigenvalue decomposition is?

[waxen flume]

bro what

we havent even learned that stuff lol

im just gonna ask my teacher later today np thanks anyway

[lavish jewel]

anyway. the easiest way would be to have a matrix A where all columns are identical

[waxen flume]

like [0, -1, 2; 0,-1,-2; 0,-1,-2]?

[lavish jewel]

mhm

[waxen flume]

https://i.imgur.com/SZ4bdaP.png

this is the same

instead of Col A

oh wait

nvm

this is Null A

so my 3x3 matrix * [1 0 1] = 0

right?

[lavish jewel]

mhm

[waxen flume]

maybe

[-1,1,1; -1,1,1; -1,1,1]

because when i multiply the -1 and 1 by the given vector

they become -1 + 1 = 0

and the middle 1 is irrelevant because its multiplied by 0 anyway

[lavish jewel]

this is wrong because the basis would have 2 vectors

you also don't get the 0 vector

,w {{-1,1,1},{-1,1,1},{-1,1,1}}*{{1},{0},{1}}

[stoic python]

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Requested by Edd#5182

[waxen flume]

see

i get the 0 vector

LOL

[lavish jewel]

you do get the zero vec, nvm, i was reading those as columns

[waxen flume]

eyyy

[lavish jewel]

but anyway this is wrong

[waxen flume]

noo

LOL

[lavish jewel]

this is rank 1

the basis of the null space has 2 vectors, not just 1

[waxen flume]

what would i modify in my matrix?

[lavish jewel]

you need to have 2 linearly independent columns

[waxen flume]

column 1 and 2 are linearly independent

not multiples of each other

[lavish jewel]

...

col 2 = col 1 * (-3/4)

you also ruined the null space condition