linear-algebra

[frank gate]

x-y is at distance 3 from origin

x + y + z is at distance 0 from origin

Does the two planes intersect because the plane that is: x-y = 3 has a z = 0 which match the second plane?

[misty cairn]

The two planes intersect because you can find a solution that fits both equations

It's true that at z = 0, they intersect but you can find a more general solution by solving the simultaneous equation using both the plane equations, so you end up with the equation of a line

[jaunty fern]

Need help with finding the cost per km

[neat pond]

hey

is the standard matrix here just ```
[1 0 0 0]
[0 1 1 0]
[0 0 0 3]

please help

PLEASE

[astral junco]

@neat pond sry i can't help. you might want to post that in #help-1 as well so more people can see your question.

[rare barn]

how do you find a pagerank vector from the steady state vector?

[rare barn]

So from my alt transition matrix, i set up a system of equations and solved setting each row = to a,b,c,d resppectively and got a=1/3, b= 2/9 , c= 1/3, and d= 1/9

[sand oak]

Need some help

[rare barn]

idk where to go from there though, ik thats the steady state vector but idk

wait, is a pagerank vector the same as the steady state vector? im confused

[honest marlin]

In a video on the cross product by 3blue1brown, at this time: https://youtu.be/BaM7OCEm3G0?t=487, he solves for p1,p2,p3 but how does he know that they are specifically in that order? If I have x + y = 5 + 2, I do not know whether x or y is equal to 5 or 2.

[dusky epoch]

he can do this because $$p_1x + p_2y + p_3z = (v_2w_3-v_3w_2)x + (v_3w_1-v_1w_3)y + (v_1w_2-v_2w_1)z$$ is an equality meant to hold for \textbf{all} $x, y$ and $z$

[stoic python]

Ann:

[dusky epoch]

all he does is equate coefficients really

@honest marlin

[burnt oar]

I have a quick question about the distance function $\mathrm{dist}(A,SO(n))$ for matrices $A\in\mathbb R^n\otimes\mathbb R^n$

[stoic python]

Simplex:

[burnt oar]

Is it true that the taylor expansion near the identity is given by
$$\mathrm{dist}(\mathrm{Id}+A,SO(n))=\left| \frac{A+A^T}{2}\right| + O(|A|^2)$$

[stoic python]

Simplex:

[burnt oar]

or should i rather ask this in #❓how-to-get-help ?

[dusky epoch]

nah here is ok too i guess

[winter reef]

If f is an isometry f(a) =b and f(c) =d then is f(a x c) =b x d? (vector product)

And lets Say norm of a and c is 1

If that makes a difference

[worn crow]

@winter reef i think it's f(axc) = +- b x d

since an isometry doesn't have to preserve right-handedness

[winter reef]

yeah, thats what I thought

[worn crow]

but yeah, it preserves angles and lengths, so it's definitely +- bxd

[winter reef]

but does length need to be 1?

of these vectros?

[worn crow]

yeah, it can be arbitrary

f is linear anyway so the lengths don't matter

[winter reef]

makes sense, thanks a lot

t!cookie @worn crow

[winter reef]

@wintry steppe

[wintry steppe]

Okay, so I only see the channel when I am tagged

yep

I have muted this too

don't worry you're not the only one

[granite light]

Kinda lost, what im assuming I have to do is rewrite this into a bigger matrix and then make it into RREF and count how many pivot positions i have?

more lost on how exactly to make it into a bigger matrix, ive only done 2x2 but im assuming the first matrix is rewritten as
1,0,1,0,1,0

[half ice]

@granite light
Can you show this set is/isn't linearly independent?

There's a rather simple argument showing this isn't independent

Oh yeah, you could split these in half, then use rref

[granite light]

yea i got linear dependent

[half ice]

You can kill one of these before starting with a matrix

Make the matrix a bit smaller

So get rid of m5 = (0 0 0 1 1 1)

[granite light]

so gt rid of m5 and then compute it with m1,m2,m3,m4?

cuz this is how i originally approached problem, forming this:

and just RREF'd and saw wasnt LID

and found # of pivot positions

professor didnt cover this at all and the book is showing examples of dimension null space instead of regular

[half ice]

If you've already got the number of pivots, you already have it

Dimension = number of pivots

[granite light]

yea but

im not sure if i did the

matrix right

like did it riht

yakno

i cant check my answer anywhere

like idk if my process is right

[half ice]

Can do this
(1 1 1 0 0 0)
(1 0 0 1 0 0)
(0 1 0 0 1 0)
(0 0 1 0 0 1)
(0 0 0 1 1 1)

[granite light]

ohh !

ok

i think i get it

i think my problem is more of making these small matrixes into one big one

[half ice]

Make sense? They aren't the same elements, but still convey the same info

[granite light]

yea

got it

thanks !

[half ice]

Good luck, lmk if you need anything else

[granite light]

okieee

[half ice]

Oh, and remember you can kill one of the elements with a lin dep argument

[granite light]

correct

cuz span(v1, vk) = span (v1, vk-1) if vk is a linear combination of v1 to vk-1 correct

[sage mauve]

ye

[paper egret]

taking linear alg next semester, anything i should be expecting or be ready for?

[tranquil junco]

@paper egret death

[paper egret]

o awesome

like my last semester wasnt hard enough already with 3 math classes

[broken hawk]

@paper egret there’s two possibilities:
it’ll open your eyes to the true beauty of math… or it’ll just be really boring. depends on what the focus of the class is and whether you’ve got a good prof

[paper egret]

yea im reading up a bit on my professor atm

so far i see a full page of him flaming AP calculus in 2001

[tranquil junco]

lmfaooo

confirmed top tier

[paper egret]

or scoring rubric for 2001

wrote a shit ton of books 👀

wtf is the american mathematical society, is that huge?

[tranquil junco]

satan

[limber sierra]

the AMS is... just that

its an organization that acts as a public ambassador for mathematics

organizes meetings and events

has its own journal

writes articles on important happenings

etc

AMS fellowship is prestigious but just being an AMS member is fairly standard

getting published in the JAMS is a huge deal though

[tranquil junco]

they're different from the MAA right

[limber sierra]

yes

[tranquil junco]

these fucking acronyms god damn

[limber sierra]

but similar

[tranquil junco]

i hate MAA

for making AMC and AIME problems which end up on my tests

[paper egret]

wtf is new york doing, amc and aime problems on exams jesus christ

that's B O L D

[tranquil junco]

is not new york

my teachers are trying to 1 up each other for harder exam while not violating school policy

mine is winning

[paper egret]

wtf is the point, do they get a raise or something?

the people who dont like math suffer and end up hating it

[tranquil junco]

there's no one who doesn't like math in the honors classes at my school

you have to take a test filled with math to get in

and do well on math before to get in too

and you have to apply for it

[paper egret]

some people do honors, because it's "honors", regardless of the subject

at this level in high school

trying to be impressive in your classes is the mindset people have in preparation for college

honors this, honors that

[tranquil junco]

yah but its a stem school with a heavy focus on math

ik

i do it too lol

i've honestly never met someone who dislikes math in my geometry classes

[paper egret]

i feel horrible for kids that feel the need to take honors everything just to have a chance at better education

[tranquil junco]

maybe ppl who dislike the class cuz its hard but not math in general

yah that's sucky

tbh i just take all honors cuz non honors is boring

i'd rather hard and annoying than boring

[paper egret]

your time could be well spent on other things imo

[tranquil junco]

they aren't hard tbh

eh the only subject i don't really care about is english anyway

and there's no honors for that till ap eng lang in 11th

[paper egret]

ye i c where ur coming from

[tranquil junco]

idk its just like

if a class is hard at least i can bitch about it in earnest lmfao

when classes are too easy i actually sometimes end up doing worse cuz i can't bring myself to engage

[paper egret]

looking back i kinda regret taking some APs

complete waste of time

i could be doing other stuff

ima move this to #chill

[tranquil junco]

yus chill migration

[silver ore]

got a question

if I have a function, lets call it a that can be represented by a matrix A

If A^N=0

how can you proof how 0 is the only eigenvalue?

[broken hawk]

assume it has an eigenvalue other than 0

see what happens to its eigenvector

[silver ore]

yeah

I did that

but it was an exam

[broken hawk]

the harder part would be showing that there is a 0 eigenvector, imo

[silver ore]

I felt like I gave an incorrect answer

So im just here to confirm

when I think back, the logic is embarrasing

[broken hawk]

if a matrix is the 0 matrix, then multiplying it with any vector will give 0. but if v is an eigenvector of A of value λ, then it’ll be an eigenvector of value λ^n of A^n

[silver ore]

so I would rather not put it down

[broken hawk]

and in particular, not in the kernel

this is proven by an easy inductive argument

the existence of a vector not in the kernel then contradicts A^n = 0

note that this does not yet prove that A has any eigenvalues at all, however

but that’s also straightforward enough: if A is nilpotent, then A cannot have full rank. if A has not full rank, then its kernel is nontrivial. the kernel is the eigenspace of eigenvalue 0. so 0 is an eigenvalue

[silver ore]

ah

I see

I can't believe I didn't see this

@broken hawk thanks so much

I think the whole topic of eigenvalues just clicked for me suddenly

thanks so much

its a bit too late since my exam is done

but thanks anyways

[winter reef]

Let V be a linear euclidean space of dimension n. For what k, composition of k orthogonal symmetries to subspaces W_1,W_2,..... W_k of dimension n-1 changes orientation of V? Is it for k uneven?

[dense holly]

how do you find the change of coordinates matrix

[slow scroll]

[v1 v2 v3] is the matrix that takes vectors in the basis of {v1, v2, v3} and sends them to vectors in the standard basis for example @dense holly

[clear spoke]

Hey, what do I need to do to get the angle of a linear function?

for example

$y = \sqrt{3}x$

[stoic python]

Autistic Hoodie:

[clear spoke]

The angle of that is $\pi/3$

[stoic python]

Autistic Hoodie:

[half ice]

tan(angle) = slope

So in your case,
θ = tan⁻¹(√3)

[dense holly]

im reviewing for a final tm an i dont understand this problem and it seems really easy..

how do you solve these

[native lodge]

For part a, what do you know about x? You can see that it's composed of two parts

in fact, one of the parts comes from solving Ax=0

do you know which part of x that would be?

for part b, now you only have a single vector, x, that solves Ax=b
what does that mean about A? once you figure that out, what does that imply about Ax=0?

[idle echo]

Would it be accurate to think of matrices as multi dimensional numbers?

[slow scroll]

Ehh i don't know about that, but it would be accurate to think of numbers as one dimensional vectors, and linear transformations from R to R as 1x1 matrices

[dusky epoch]

"multi dimensional numbers" what's that even supposed to mean

[fringe cave]

@idle echo There was one thing I read that treated matrices as multidimensional lists, but honestly you're probably better off not doing multidimensional numbers as your analogy

Instead of matrices as multidim. numbers, you could instead think of them as linear transformations or (if you're a cs major) multi-(usually 2-)dimensional arrays

[slow scroll]

How might i try to prove that a an orthonormal basis of N vectors can be constructed from the eigenvectors of a NxN Hermitian operator?

[dusky epoch]

maybe try proving eigenvectors corresponding to different eigenvalues are orthogonal

[slow scroll]

i am more hung up on the existence of N orthonormal eigenvectors, not so much proving they are orthogonal if they do exist

[dusky epoch]

hmmm

[wintry steppe]

@slow scroll This is just the spectral theorem. See https://en.wikipedia.org/wiki/Spectral_theorem#Hermitian_maps_and_Hermitian_matrices

[wintry steppe]

someone can help me to find a good demonstration of the Perron Frobenius theorem ?

[slow scroll]

@wintry steppe I know what spectral theory is. I mean, I don't think I'm very good at it so I may have missed something. What is it about a Hermitian operator in particular that ensures its always diagonalizable?

[wintry steppe]

The proof is given on the page; essentially, it comes from the fundamental theorem of algebra and the fact that, for Hermitian operators, if V is an invariant subspace under A, then V^perp is also.

[slow scroll]

yea i just looked at it. makes sense

thanks

[inner isle]

Someone teach me this

[tranquil junco]

fam this isn't linear algebra what

[frosty vapor]

lol

[wintry steppe]

How do you prove the determinant of a matrix without using a calculator? Do you just solve it by hand?

[native lodge]

2x2 is quite trivial

3x3 is mentally doable as well

[wintry steppe]

5x5?

[native lodge]

That’s pretty hard to do in your head unless you have zeros all over the place, you would have to work it out by hand

[wintry steppe]

Got it, thanks

[native lodge]

Row reducing is an option available to you to make it a bit easier, no matter what you choose to do, it will take a bit of time

[broken hawk]

I now wonder… are QR-Decomposition or SVD ever easier to compute by hand than LU-Decomp

[winter reef]

Ahh shit, here we go eigen

[rotund cargo]

@clear spoke are the functions guaranteed to pass through the origin? if so, you can take arctan(y(1)) - can also replace 1 with any positive number

oops

for some reason my chat just updated after i sent that message

sorry!

[elfin schooner]

if <u,u> > 0, why do we have the modulus in Cauchy-Schwarz inequality?

Am I missing something?

nvm i figured it out

[winter reef]

what did you figure out?

im not sure why, since the modulus is before the square so why would that matter?

ohh, for complex?

[stoic python]

Display results online and refine query

Requested by dog#3227

[elfin schooner]

yuessss

thanks for the help though lmao

[rare barn]

is thjis true or not? and why... If a vector space V is spanned by {v1, . . . , vn}, then V is isomorphic to R^n

[swift plaza]

If V is a real vector space, then yes. In general, if V is an n-dimensional vector space over a field k then V is isomorphic to k^n, but the isomorphism is generally not canonical, i.e. it depends on the basis you choose for V and is therefore usually not particularly interesting.

The intuition is that you can just take the linear map that sends v1 to e1, v2 to e2, ..., vn to en where the e's are the standard basis, and then by linearity this defines a map from the entirety of V to R^n

[rare barn]

so then isn't it isomorphic all the time here?

[swift plaza]

actually, I just saw that you're original message said "spanned by {v1, . . . , vn}", in that case it isn't true, you also need that {v1, . . . , vn} is linealy independent, so it's a basis

[fringe cave]

Span =/= form a basis

[rare barn]

so it needs to be linearly independent as well to be isomorphic

[fringe cave]

i mean, obviously the linear independence is key, otherwise you'd be saying V (is spanned by {1, 1, 1}) isomorphic to R^3

[rare barn]

👌

[quaint heart]

The basis is the isomorphism

@swift plaza

[fringe cave]

o o f

[quaint heart]

That's what a basis is

Also what you said for finite dimensional vector spaces is true for infinite dimensional vector spaces as well

[rare barn]

srry ive got one more... The set of polynomials p ∈ P2 such that either p(0) = 0 or p(1) = 0 is a subspace of P2.

[broken hawk]

what happens if you take two polynomials which are 0 in different locations?

[rare barn]

would they not be a subspace

[half ice]

For example, p(x) = x and q(x) = x - 1 are both in that set, but (p + q)(x) = 2x - 1 is not

[rare barn]

yeah, so they aren't a subspace of P2

[placid oracle]

How can I show that this transformation T:V->P_1 defined: T(f)=f(1)+f'(0)t is linear

where V is thew vector space of differentiable fxns f: R->R

[quaint heart]

So clearly it takes the 0 function to 0

And then just use linearity of derivative

[placid oracle]

what do u mean

[brittle juniper]

tbh just use the definition

[fringe cave]

^

Anyhow, f` is the derivative, which you know is linear (as is multiplication by t if that t is what I think it is)

and f(1) is linear because (h+g)(1) = h(1)+g(1)

[placid oracle]

dont i have to use: T(u+v) = T(u) + T(v) and Tc(u) = T(cu) for the whole thing though

[dusky epoch]

that's the definition

and that is what @brittle juniper just suggested

but also... did you mean T(cu) = c T(u)

[fringe cave]

yes clearly

[placid oracle]

yes my b ann

ok but for when i use the definition, im confused as to what f im allowed to use, can it be any f that is differentiable?

[slow scroll]

f is a letter. g is also a letter. if you want those letters to mean differentiable functions then that's fine

[rare barn]

srry just reposting this if T : R3 → R4 has two pivots, then the kernel of T is a
line in R3. is this always true, sometimes true, or just false?

[slow scroll]

if T has two pivots in REF, then yea, then the nullity of the matrix is just 1.

[rare barn]

okay, but is there an exception to this that wouldn't always make the Kernel of T a line in R3?

[slow scroll]

hmm can't think of anything. This is a basic application of rank-nullity. We have a 4x3 matrix with two pivots in REF. That implies rank = 2, and rank + nullity = 3, therefore nullity = 1, a line in R3

[abstract bolt]

How do i get the equation for this?

[spice storm]

Why am I keep getting -7 when I try to do this?

It should be 0 in the position where 2 is

[slow scroll]

huh? is it row2 - 2*row1 youre asking about?

[spice storm]

I have to use the gauss-Jordon elimination

Yes

2x2

I have on my calculator -3-2*2

And keep getting -7

[slow scroll]

(2x - 3y) - (2x - 4y) = 0x + y

[spice storm]

I’m confused

I’ve been in the same problem since 9:30 and is really annoying me

PM

[slow scroll]

@spice storm its like elimination method of solving systems of equations. Now ur just doing it with matrices instead

[spice storm]

But why am I keep getting -7

When I literally just typed R2-2R1

R2 is -3 and R1 is 2

And yet I keep getting -7

[slow scroll]

(2x - 3y) - (2x - 4y) = 0x + y
2x - 3y -2x + 4y
combine like terms.

[spice storm]

Y=1

[slow scroll]

well, 0x + 1y, not -7y

[spice storm]

I’m still getting -7 😭😭😭😭 this is beyond annoying. Where did you even get 2x-4y

[slow scroll]

thats 2 times row 1

[sonic osprey]

R1 is -2 not 2

[spice storm]

So is R2+(-2)R1?

Because chegg has R2-2R1

So I don’t know which one is correct

[sonic osprey]

Those are the same

[slow scroll]

those are the same thing? I can't figure out what ur confused on exactly. Are you trying to use a calculator or something

[spice storm]

Yes I’m using a calculator

[slow scroll]

ok don't do that. aint necessary and you might be typing something in wrong

[spice storm]

I need a calculator because i have a disability

But here’s what I have on my@calculator

[slow scroll]

-3 -2(-2) is what you want

(-3) is the entry on the second row. (-2) is the entry on the first row, and you are subtracting 2 times that

[spice storm]

I got 1

[slow scroll]

alright cool thats right

[spice storm]

But why is R2+(-2)R1 wrong on my calculator but R2-R1(-2) is right?

[slow scroll]

R2 - R1(-2) is the same as
R2 + 2R1 because of the double negative

[spice storm]

Thank you!

[slow scroll]

npnp

[spice storm]

For the constant what should I do? Because I keep getting 19 and it should be 5. I’m doing -3-2(2)*-11

[slow scroll]

are you talking about the third column?

[spice storm]

Yes

[slow scroll]

(-11) -2(-8)

-11 comes from third column of R2. -8 comes from the third column of R1

[sour garden]

Prove (x-a)^k, k=0, 1, ..., n is a basis for polynomials of max degree n

[broken hawk]

as said, prove it by induction on n

what’re you having trouble with

[sour garden]

I wasn't trying induction

[broken hawk]

also, all you have to show is that they’re linearly independent (easy) and that there’s a way to represent a known basis with them

[sour garden]

I was expanding (x-a)^k with binomial theorem and trying to find the determinant of the transformation matrix

[broken hawk]

just try to represent x^n with the (x-a)^k

like, prove for all n (inductively) that x^n can be represented as a linear combination of the (x-a)^k for k≤n

and then also show that they’re linearly independent in some way

(that one’s easy)

I don’t see how or why you’d want to involve determinants here

[sour garden]

So x^(k+1) = x^k(x-a) + ax^k

So we have spanning by induction

[dusky epoch]

$x^n = ((x-a) + a)^n = \sum_{k=0}^n \binom{n}{k}a^{n-k} (x-a)^k$

[stoic python]

Ann:

[sour garden]

Ooh nice

[rapid robin]

Or you can use polynomial taylor formula

P(X) = sum P^(k)(a)/k!(X-a)^k

[stoic python]

kwak:

[dusky epoch]

weird. your reasoning looks fine to me

[sonic osprey]

What does R_3[x] mean? Up to degree 3?

[brittle juniper]

Usually yeah

[sonic osprey]

I'm not sure your scalars on u are correct, each of the x_i should be reals

But you might mean that already

[sonic osprey]

Oh, even none of the f(P_i) are 0, their linear combination can still be zero

[placid oracle]

Can someon ehelp me with this problem? In the year 2000, the population of two cities are both twenty million, but at the end of every year, 5% of citizens from city 1 move to city 2, and 15% of citizens from city 2 move to city 1. If m_x is the population of city 1 after x years after 2000, what is the limit of x as it approaches infinity of m_x. (lim_x->infinity of m_x)

In the year 2000, the population of two cities are both twenty million, but at the end of every year, 5% of citizens from city 1 move to city 2, and 15% of citizens from city 2 move to city 1. If m_x is the population of city 1 after x years after 2000, what is the limit of x as it approaches infinity of m_x. (lim_x->infinity of m_x)

So Ik that to find it after one year, you multiply the migration matrix by the population matrix

and to find it after another year, you multiply the migration matrix by the matrix you found in the previous step

and so on

but here, it doesn't seem to be converging anywhere and im also not supposed to use a calculator, so im pretty sure theres another way to do this without multiplying the matrices over and over

[wet finch]

first question for you: what are the dimensions of your migration matrix and population matrix?

(and can you write out your migration matrix and the starting populatino matrix?)

[placid oracle]

2x2 and 2x1

and yes one sec

[wet finch]

( so far so good!)

[placid oracle]

is this right?

[wintry steppe]

Can someone help me how to solve this problem?
Determine algebraically the zeroes of f(x) = 3x^3 + 21x^2 + 36x
I am used to doing it with 4 terms
but I don't know how to do it if 0 was the 4th term
I try and group it but I can't find 2 of the same expressions

[placid oracle]

i did the thing where we keep multiplying them but it doesnt seem to converge anywhere (i also stopped at four because it became hard to do by hand)

@wintry steppe try prealg-algebra

[wintry steppe]

ok

[wet finch]

@placid oracle that is correct

[placid oracle]

Okay, so what do i do next...

[wet finch]

the trick is to not multiply them out the way you did

instead of calculating Av, A(Av), A(A(Av)), etc

use the associativity of matrix multiplication

and look at Av, A^2 v, A^3 v, A^4 v, ...

i.e. calculate just the powers of A

and see what that converges to, and THEN multiply by v

(it's still going to be hard to do that, so there is another trick...)

[placid oracle]

yeah isnt that also tough to do by hand

[wet finch]

yes but it's the key insight

that you should look at powers of A first

so the strategy now is to diagonalize A

write A = P D P^(-1) for some diagonal matrix D

and now the great part

$A^2 = A A = P D P^{-1} P D P^{-1} = P D^2 P^{-1}$

[stoic python]

Buncho Bananas:

[wet finch]

and in general, $A^n = P D^n P^{-1}$

[stoic python]

Buncho Bananas:

[placid oracle]

omg i completely forgot abt this

[wet finch]

how does that help? because calculating powers of a diagonal matrix is SUPER EASY

[placid oracle]

yep totally, im gonna try it on my own gimme 2mins

[wet finch]

so then to find the lim as n -> infinity of A^n, you just need to do it for D^n and then conjugate by P

[placid oracle]

so what do i plug into the power though?

[wet finch]

what do you mean

you're looking for the limit as n goes to infinity

[placid oracle]

Ok so ive diagonalized it:

so basically 1^infinity is 1 and 4/5)^infity is 0

so then i resolve the matrix

[wet finch]

so the limit as n goes to infinity of A^n

is P [1 0; 0 0] P^{-1}

[placid oracle]

yes so it convereges to 30mil and 10mil

[wet finch]

great

:)

[placid oracle]

so the answer is 30mil, cool!

@wet finch i got 1 more question for u if u got time

[wet finch]

ok

[placid oracle]

Let transformation T: V-> P_1 be defined by T(f) = f(1) + f'(0)t and V be the vector space of differentiable function f: R->R.
I first had to show that it was linear, so I said that we know f' is linear and so is the mulptlication by t. And that we know that f(1) is linear because (h+g)(1)=h(1)+g(1) in any possible function in this example
is that enouigh to show that its linear?
and second, I have to prove that it is an onto function, but i am lost as to how to do so

[wet finch]

one minute

you need to show that T(f+g) = T(f) + T(g) and T(af) = aT(f)

it sounds like the things you said will certainly help you show those two facts

but at the end of the day that's what you need to show

[placid oracle]

yes but im not sure how

[wet finch]

as for the surjectivity, you need to show that "for any real numbers a and b, there is a f such that T(f) = at + b"

i.e. f(1) = a and f'(0) = b

[placid oracle]

again, im not sure how

[wet finch]

I don't believe that :)

write out T(f+g)

[placid oracle]

T(u+v) = (u+v)(1) + (u+v)'(0)t

but idk how to shjow it equalsT(u)+T(v)

[wet finch]

well, what is (u+v)(1)

how do you evaluate the sum of two functions

[placid oracle]

u

u(1)+v(1)

but is that all the justificaiton i need?

and (u+v)'(0)t = u'(0)t+v'(0)t

[wet finch]

what do you mean "all the justification"

you are correct that that is u(1) + v(1)

now write out T(u) + T(v)

so literally: write down T(u) and write down T(v) and add them

[placid oracle]

u(1) + v(1) + u'(0)t+v'(0)t

[wet finch]

is that the same as what you got when you wrote T(u+v)

[placid oracle]

so yes, they are equal

[wet finch]

please notice that it wasn't actually hard to do this!

you are just psyching yourself out

[placid oracle]

i had already done this

[wet finch]

to show that T(u+v) = T(u) + T(v), you just do it. write out both sides and see that they are the same

[placid oracle]

but i though that this wasn't justificationm

okay

[wet finch]

why wouldn't it be?

that's literally the definition of linear (along with T(av) = aT(v), which you still need to show)

[placid oracle]

im not sure, ur right i was psyching myself out because i thought it was too simple

[wet finch]

math doesn't have to be hard

just because it's simple doesnt' mean you're wrong

sometimes it is just simple

[placid oracle]

so T(cu)=cT(u)

(cu)(1)+(cu)'(0)t = ((cu)(1))+(c(u)'(0)t

which works

but now what abt to show that its onto

[wet finch]

onto just means that for any a + bt, there is some f with T(f) = a + bt

i.e. there is some f with f(1) = a and f'(0) = b

[placid oracle]

okay, but again i dont see how i can show that

pretty sure im missing smth dumb again

[wet finch]

once again, just do it

write down a linear function with f(1) = a and f'(0) = b

for example, f(x) = bx + (a-b)

[placid oracle]

ok but arent there linear fnxs where this doesnt work?

[wet finch]

maybe my use of the word "linear" in my last sentence was misleading

i meant "linear" in the sense of "degree-1 polynomial" not in the sense of "linear transformation"

was that your question?

[placid oracle]

yes but also, im confused as to why we're allowed to pick any linear function, there are clearly ones that dont work for the condition as well?

[wet finch]

what do you mean? the problem is asking you to show "for every a + bt in P_1, there is some differentiable function f such that T(f) = a+bt"

but T(f) = f(1) + f'(0)t

so if I give you something that looks like a + bt

you need to find some f with f(1) = a and f'(0) = b

so that T(f) = f(1) + f'(0)t = a + bt"

so now YOU get to find f(x)!

and you can do whatever you want to find it

[placid oracle]

oh so i just need to show that there exists a function that satisfied that condition and that works?

i was confusing it with all fxns need to satisfyu the condition which is obv not true and it wasnt making sense

[wet finch]

i mean it's obviously not true that every function f satisfies T(f) = 0

but to show surjectivity you just need to find AT LEAST ONE function f with T(f) = 0

and AT LEAST ONE with T(f) = 1 + 3t

and AT LEAST ONE with T(f) = pi + 42t

that's what surjectivity means

[placid oracle]

ok the at least one idea makes a lot more sense, i rlly need to brush up on this

[wet finch]

yeah if you have trouble with surjective and injective and stuff like that, you need to review it before you do more linear algebra

that kind of stuff is the foundation of linear algebra (and beyond) and you need a strong foundation before you move on!

[placid oracle]

for sure, thanks for all the help tnite!

[wet finch]

yep, gl

[sonic osprey]

@wintry steppe see my comments above

[sonic osprey]

Think about f(-14P_1 + P_2 + P_3)

If I did my calculations correctly

[sonic osprey]

@wintry steppe

What do you mean by distribute f?

If you use the linearity of f, you get -14f(P_1) + f(P_2) + f(P_3) which is 14 - 3 - 11 = 0

@wintry steppe

[plain fjord]

If I have an upper triangle matrix, and need to find a basis for it, doesn't identity matrix suffice? I can construct any triangle matrix with that.

I mean, if that is so, it's quite easy exercise...

[sonic osprey]

This makes no sense. You don't find basis for matrices, you find basis for linear spaces

[plain fjord]

Whoops, you are right. Let me rephrase

I had an exercise where I had to prove that upper triangle matrices form a vector space. I did that, 8 things to check. Then I need to find a basis for that vector space, and construct a matrix which represents that basis

But I was thinking, isn't identity matrix one of such matrix?

[broken hawk]

ignore me for I am talking BS

or at least not thought through stuff

[plain fjord]

I know that with identity matrix, I can form matrices that are not upper triangle matrices, so it contains "extra" stuff, but I was thinking, does it matter?

[broken hawk]

also btw, if you’ve already shown that the space of all nxn-matrices is a vector space then you did too much work

[sonic osprey]

The identity matrix is in that linear space yes

[broken hawk]

you could’ve just shown that the triangular ones are a subspace

which is only three things to check

(nonempty, closed under addition, closed under scalar multiplication)

[plain fjord]

I mean, if the diagonal has even a single zero, that would mean the matrix is not invertible and can't form a basis

Yeah actually I thought about that, and did it at first, but then I read about more properties that vector space has, and checked them all. was tedious though

[sonic osprey]

I think you're confusing yourself here

The basis you want to construct is for the vector space of upper triangular matrices

Not for R^n or something

[broken hawk]

there’s a theorem that:

1. a subspace of a vector space is a vector space over the same field
2. a subset of a vector space is a subspace if it
   a) is nonempty / contains 0
   b) is closed under addition
   c) is closed under scalar multiplication

it then inherits all other axioms from the larger space

this means in most cases you don’t have to prove it satisfies all 8-9 axioms

but only those three things

that theorem gives you everything for free

[plain fjord]

Okay. Good to know.

Zopherus, hmm, yes that's true, it is for whole R^n, so identity matrix is a "larger" basis than what was asked, but I'm not sure is that a bad thing.

[broken hawk]

R^n isn’t even in consideration here

the space is TriUpp(n) or however you wanna denote it

which is isomorphic to $\mathbb{R}^{\frac{n^2 + n}{2}} = \mathbb{R}^{\frac{n(n+1)}{2}}$

[stoic python]

Sascha Baer:

[broken hawk]

(by simply ignoring all the zero entries of the matrix and considering it a weirdly arranged vector)

[plain fjord]

Hmm, oh yeah, if identity matrix would represent the basis, then it wouldn't be closed under addition, i mean, i could get out of the Upp(n) space with standard basis

So probably not a good idea after all

[sonic osprey]

Yeah you could use the identity matrix as one of your basis matrices, but probably not the nicest way to do it

[broken hawk]

I’m not quite sure what they ask to do with that matrix representing the basis tbh

are you sure you’re not supposed to find matrices which form a basis

[hot ivy]

Please help me answer this question!! It’s multiple choice! Thanks!

If a function is not one-to-one on an interval I, it cannot have an inverse function on I.
a) True. If a function is not one-to-one then at least one of the range values of the function has more than one image when the inverse is defined.
b) False. The function can have an inverse as it is not one-to-one on the interval I.
c) True. If a function is not one-to-one then at least one of the domain values of the function has more than one image when the inverse is defined.

[plain fjord]

Actually, now that you mention that, I'm supposed to find a basis for the Upp(x) space, not necessarily a matrix representing it...

[broken hawk]

good. that’s much easier cause I actually know what that means :P

consider the matrices as weirdly arranged vectors

[plain fjord]

And count the dimension to it, but that must be n i suppose

[broken hawk]

it’s definitely not n

[plain fjord]

I know that the matrices are vectors in disguise... heh.

[broken hawk]

(I actually spoiled it earlier on)

[plain fjord]

I mean, the columns of a matrix are the basis vectors

[broken hawk]

not in your case

in your case, vectors are matrices

[plain fjord]

hmm, you mean like (1,0) is the first thing in a matrix, i mean 1 in the diagonal

and then expand to n coordinates, that example above would be case if n is 2

[broken hawk]

I have no idea what you just said

[plain fjord]

hehe okay, doesnt matter

Back to the original question, the basis. So it doesn't necessarily have to be the standard basis, since that is too large, right?

[broken hawk]

I think you’re still triyng to find a basis for the wrong space

this is what a vector in your space looks like (for $n=3$):
$$\begin{pmatrix}
a & b & c \
0 & d & e \
0 & 0 & f
\end{pmatrix}
$$

[stoic python]

Sascha Baer:

[plain fjord]

Yes, true

[broken hawk]

this is a vector any nothing else

the columns of this thing are meaningless

in this exercise

[plain fjord]

V = {(a,0,0),(b,d,0),(c,e,f)} in other form, right?

[broken hawk]

wait what

that’s a set of three vectors

in ℝ³

[plain fjord]

Look what i wrote, the first thing becomes a column in that matrix you wrote

[broken hawk]

are you just using really really bad notation?

pls don’t interrupt

also this is not linear algebra

[hot ivy]

sorry wrong discord

[plain fjord]

It's possible I have my terms mixed up

[slow scroll]

catnose, i think you are getting confused between finding a basis for the column space of a matrix and finding a basis for the vector space of triangular matrices

[broken hawk]

what you wrote there is “the vector space V is a set with three elements: the vectors (a,0,0), (b,d,0) and (c,e,f)”

this is false

[plain fjord]

Actually, what i tried to write was another way of saying the same thing you wrote, and you wrote it in matrix form.

[broken hawk]

then you’re using ridiculously confusing notation

[plain fjord]

kxrider, hmm, it's possible

[broken hawk]

I hesitate to call any notation bad but I think you got pretty close to it

{ } has a very established meaning

[plain fjord]

Hey come on, I'm just learning the stuff, that's why I am asking 😃

So I'm happy if you point out that my notation is messed up

[placid zephyr]

Anyone wanna get in vc and talk about some linear ?

[plain fjord]

Hmm, perhaps { should be <

[slow scroll]

if you said "V = span{(a,0,0),(b,d,0),(c,e,f)}"
that would be the column space of that upper triangular matrix, not the space of triangular matrices

a, b, c, d, e, f can be any element of the field, so they each need their own degree of freedom

[frank gate]

Hello everyone! If i want to find a normal in a plane given to vectors v1 and v2
that would be that the normal n = v1 x v2
but! does it mather the order i cross them in? if i do v2 x v1 instead i get different values, but is that equivalent?
v1 x v2 <=> v2 x v1 ?

[plain fjord]

But... hmm

[slow scroll]

thats why the space of nxn upper triangular matrices would have n(n+1)/2 vectors in its basis

[broken hawk]

my god can you people stop interrupting

[plain fjord]

Wait, so for n = 3, that would lead to (3^2+3)/2 = 6

But, isn't that impossible, a basis can have at most n vectors...?

[broken hawk]

you're still confused as to what the vector space is

[slow scroll]

a nxn matrix has n^2 entries lul

[plain fjord]

Yes, but...

[slow scroll]

for a 3x3
1 degree of freedom in first column
2 degrees of freedom in second
3 in third....
= 6 total

[placid zephyr]

Degrees of freedom sound like the dimension of the linear vector space

Or are they not related

[broken hawk]

it's literally what this is about

[placid zephyr]

O

Well wouldn’t the dimension of an m x n matrices just be MN

[slow scroll]

these are triangular matrices

[broken hawk]

@plain fjord you're stuck thinking that "vectors are columns and matrices are a didferent thing"

[plain fjord]

Actually in this case, we are limited to nxn spaces

[broken hawk]

here, the vectors are matrices

[sonic osprey]

the elements of your vector space are matrices

[plain fjord]

Oh yeah... now I think I'm starting to get it...

Hmm...

It's starting to get a little clearer, but I gotta think this a little 😃

so, in two dimensional space, if i had a matrix [a,b; 0,c] that would be a*[1,0;0,0] + b*[0,1;0,0] + c*[0,0; 0,1] - (in this ; means change to next row)

$$\begin{pmatrix}
a & b
0 & c
\end{pmatrix}
$$

[stoic python]

Catnose:

[plain fjord]

Ah damn, misspelled it

0 and c are meant to be in the next row

Anyway, you get the idea what I wrote? I don't know the syntax to that bot yet

[swift plaza]

[1,0;0,0] , [0,1;0,0] and [0,0; 0,1] would indeed be a basis for the space of 2 by 2 upper triangular matrices

but be careful, you said in two dimensional space, the matrices are 2x2 but this space is 3 dimensional

[plain fjord]

Actually, in the exercise, it's n-dimensional, but we are getting there

[broken hawk]

no, it's not

the matrices are nxn, this does not mean the space is n-dimensional

[plain fjord]

Uh, sorry, i meant, that the matrices are square matrices, and nxn

[placid zephyr]

Wouldn’t n x n be square ?

[plain fjord]

Thanks for clarifying that :). Yeah you are right, I gotta be more careful with what I type

Yes we are dealing with square matrices

[placid zephyr]

Ok

[plain fjord]

So I just gotta generalize what I have now.

Thanks for your help, even though I haven't yet completed it, but I learned a lot

[oblique crag]

Need help with 27. Completely clueless on how to even do part a

[oblique crag]

<@&286206848099549185>

[dreamy quest]

Ahhhhh this

Linear transformations with respect to 2 bases

It took me more than 5 hours to understand this

The most important thing I learnt in those 5 hours was that a linear transformation has something absolute about it, and that, given a basis, its representation in that basis (or with respect to two bases) is simply one of infinitely many ways to represent that linear transformation.

[stoic python]

Tony Wang:

[dreamy quest]

^

Though for the first question you don't need this.

But the idea of a base or basis remains the same: you can represent every number or vector in your domain of interest uniquely using that basis. For numbers, base (n) dictates that the (k)-th digit to the left, (a_k), must be between 0 and (n-1), and that it represents a value of (a_k k^{n-1}). But note that \emph{every single number has exactly 1 unique representation in a base} -- in particular, two numbers are the same iff each digit is the same. Similarly, given a basis, every vector in your vector space has exactly one unique representation. For the first part of 27.\ it's just a matter of finding that unique representation.

[stoic python]

Tony Wang:

[dreamy quest]

Here, I think you will find that the correct "digits" of the vector are (89 , -15, -6)

Because 89(1) + (-15)(5-x) + (-6)(2+3x-x^2) = 6x^2 - 3x + 8

A nice parallel that might help you understand the idea of basis more is if you consider working in base (x) from 0 up to (x^n), where (x) is an unknown positive integer variable. Then the coefficient of each (x^k) is really just (a_k) as I talked about earlier. So for example, here, (6x^2 - 3x + 8) would be represented as ((6, -3, 8)). In fact, you may notice that working in this base is equivalent to working with the basis (B = {x^{n-1}, x^{n-2}, \dots, x, 1}). (This is a basis for polynomials of degree less than (n).)

[stoic python]

Tony Wang:

[dreamy quest]

@oblique crag

[modest jasper]

Does anyone know how to do a question like this?

[slow scroll]

just want to point out that you can also just create a change of basis matrix that maps every vector in the standard basis to a linear combination of vectors in B. @oblique crag

@modest jasper have you computed the eigenvalues of A?

[modest jasper]

Yeah @slow scroll 😃

[slow scroll]

and the eigenvectors i presume?

[modest jasper]

Yes.

[slow scroll]

ok so you need the matrices that do the right stuff (im tired lol). Have you done this before? Typically I see A = PDP^-1, but you can invert matrices to get D = P^-1AP

[modest jasper]

Wait, I got it.

Thanks 😄

[slow scroll]

nice np c:

[modest jasper]

Btw am I insane, or should this answer be working???

[oblique crag]

@dreamy quest it says <95,-18,-6> but im@still confused on exactly how u came up with those numbers

[dreamy quest]

Do you understand why it is those numbers though @oblique crag

[oblique crag]

No

Whats the computation process

[slow scroll]

@modest jasper ur answer is a little off according to wolfram alpha lol

[oblique crag]

I see why they make sense but my question is more of how did u get them without guessing random numbers @dreamy quest

[slow scroll]

pahy, would you know how to construct the change of basis matrix from the standard basis to B?

[oblique crag]

Honestly ive come to realize my class uses some rly weird terms compared to friends taking classes at other school

So i have no idea

[modest jasper]

@slow scroll Oh, sorry, I should have posted this.

[oblique crag]

Imma say no

[slow scroll]

thats right according to WA @modest jasper 🤷

[modest jasper]

I know!!

I don't know what the problem is.

[slow scroll]

okay pahy, there is a linear transformation that takes any vector in B to the standard basis. First, do you know how the standard basis of polynomials works?
(1, 0, 0) = 1
(0, 1, 0) = x
(0, 0, 1) = x^2

[oblique crag]

Yep i understand that

[dreamy quest]

Oh right so for that one it's quite easy because the only way you can get an x^2 is if you use the third vector

so you need a -6 there

[oblique crag]

Ohhhh

That makes a lot of sense

[dreamy quest]

Yeah and you work downwards, so the next term you look at is the x term

[oblique crag]

Then u get -18x from that

[dreamy quest]

yeah XD

[oblique crag]

And have to make it -3x with 2nd term

[dreamy quest]

Though for other bases which arne't so easy you set up a system of linear equations and solve

[oblique crag]

I understand how to do b,c,d

A just got me rly stuck

[dreamy quest]

essentially you have a(1) + b(5-x) + c(2 + 3x - x^2) = 6x^2 - 3x + 8 and you solve for a b and c by comparing coefficients of 1, x, and x^2

[oblique crag]

Still interested in learnin bout converting to standard basis though if ur keen kxrider

[dreamy quest]

and in this case it's very straightforward

[slow scroll]

$B = {1, 5-x, 2+3x -x^2}$, $$\begin{pmatrix} 1&5&2 \ 0&-1&3 \ 0&0&-1 \end{pmatrix}$$ is the matrix that takes vectors in $B$ as an input and outputs vectors in the standard basis. The inverse of this matrix takes in vectors from the standard basis and outputs vectors in $B$, the transformation you are looking for.

[stoic python]

kxrider:

[oblique crag]

Oh ok so u do that, then u multiply it by the 6x^2-3x+8?

Or somethin else

[slow scroll]

right, but not the matrix in that picture, but the inverse of that matrix

[oblique crag]

Inverse?

[slow scroll]

the one in the picture says that if put in (0, 1, 0) for example, I get out (5, -1, 0). The vector I plugged in was from the basis of B, and the transformation converted it to the standard basis. The inverse of this matrix allows me to input vectors from the standard basis, and get that vector in the basis of B

[oblique crag]

Right, how do u get the inverse though

[slow scroll]

have you inverted matrices before?

[oblique crag]

Oh right yes i have

Need to look at my notes to recall though

Wait i think i remember

Can add the columns with leading 1s

Then do rref

[slow scroll]

yea so rref the matrix augmented with the identity matrix

[oblique crag]

Yep thanks

[slow scroll]

np

[woeful laurel]

Anyone know some tricks you can pull on matrixes

Like this one, multiplying with that column matrix gives you the sum of the rows

I know it's not really trickery but still

[broken hawk]

not really sure what you’re looking for tbh

[rigid cypress]

what do you mean by when you say trick?

[plain fjord]

Probably some handy things to know, to simplify calculations?

But they are just curiosities and fun to know things, usually

[quaint heart]

I guess you should know the relation between matrix multiplication and the dot product

[plain fjord]

Of course. But I guess what he was trying to say, are there other nice simplifications like that, or other interesting things to know.

[earnest vessel]

But a trick should be something that simplifies calculations

What they posted IS the calculation

So I'm not sure what they are referring to

[plain fjord]

Well, perhaps something, that is interesting to know?

[rigid cypress]

identity matrix, maybe?

elementary row operations?

[plain fjord]

Well, not maybe related to his thing, but i got an exercise, to prove that a row operation on a matrix, is the same as multiplying on an elementary matrix from the left side.

Which is kinda easy if you try it with finite matrix, like 3x3 matrix, and multiply it with a matrix that has two rows exchanged, it exchanges the rows on the original matrix too

But the proof should be about any matrix, not just 3x3

[earnest vessel]

Yes, that is at least theoretically useful

[plain fjord]

Yeah, my exercise has nothing to do with what he said, but I just mentioned it, since the topic came up.

[oblique crag]

Anyone help with part c

[oblique crag]

<@&286206848099549185>

[sonic osprey]

What are you confused about?

[oblique crag]

part c. i dont understand how i am supposed to compute it. completely lost on even the first step @sonic osprey

like i think im supposed to multiply [t1]bb' by the 2x1 matrix of <-7,5> but thats not giving me the right answer

[sonic osprey]

It would be (-7, 5) if the bases being used were the standard bases

But they're not the standard bases

[oblique crag]

Right... so how do i approach this

Hmm could it be -7(1)+5(1-x)

So -5x-2

<-2,-5> multiply by [t1]bb’?

<11,-22,-35>

-11(1)-22(1+x)-35(3x-x^2)

Okay got it, tysm zopherus

[oblique crag]

Alright so why is part g process different from part c. Or is it the same and im doing my math wrong

[inland anvil]

In general, if you have a matrix that's mostly empty and repetive

actually, not just one, but a series

for each size or so, it's like a patern

what are good ways to find the determinant in terms of some sort of recursive formula

asking because I have such a series and I have a conjecture for a formula the determinants follow

but I want to be able to prove it

[copper fossil]

You mean like a sparse matrix?

[inland anvil]

I don't know the definition of that

lemme see

oh, it's pretty informal actually

yeah

exactly what I'm thinking

[copper fossil]

The minor expansion formula is a recursive definition of a determinant

http://mathworld.wolfram.com/DeterminantExpansionbyMinors.html

[inland anvil]

that's the one I know

[copper fossil]

Oh

I don't know of any others

[inland anvil]

eh, maybe I should just focus on using that more

I'll read http://mathworld.wolfram.com/Determinant.html as well

silly me, not researching my problems well before asking online

[woeful laurel]

@broken hawk yeah it's kinda hard to explain

something kinda like the example that I gave

it's not a trick per-say but it's something nifty

yeah that still makes zero sense

[hasty rune]

I've gotten a bit rusty with my definitions - given a Matrix, is the Dimension the number of independent rows and the Rank the number of independent columns?

[swift plaza]

Rank is the dimension of the column space, which is indeed equal to the number of independent columns. 'Dimension' on it's own really has no meaning, but the dimension of the row space is equal to the number of independent rows, which in turn is equal to the rank.

[placid oracle]

is this always true? why? In an inner product space, if <x, y>= 0, then either x = 0 or y = 0.

[dusky epoch]

[placid oracle]

?

[dusky epoch]

well, what do you think

is this always true?

[placid oracle]

no?

[winter reef]

no

[placid oracle]

what

[winter reef]

its not always true

[dusky epoch]

@placid oracle well, is it not always true?

how do you prove that a statement that begins with "for all ..." is false?

[placid oracle]

u just find one examplee where it isnt

[dusky epoch]

well

if you claim that it is NOT the case that

In an inner product space, if <x, y>= 0, then either x = 0 or y = 0,

then find me a counterexample!

[broken hawk]

and do it fast cause I have a question

[winter reef]

[dusky epoch]

also, i might not have made this clear but

don't overthink it

[placid oracle]

well doesnt it work if they are both = 1

[dusky epoch]

who are "they"

[placid oracle]

x and y

[dusky epoch]

what does the < , > symbol denote?

[placid oracle]

wdu mean

[dusky epoch]

i mean exactly what i said

there's this symbol which you've used in your statement

composed of two angle brackets and a comma

what does it denote

[placid oracle]

for inner product doesnt it mean they are functions

[dusky epoch]

answer my question

what does < , > denote?

wouldn't be a good idea to use the symbol if you don't know what it means, would it?

[placid oracle]

i just said it, what inside is a function

[dusky epoch]

no, you did not answer my question

< , > denotes THE INNER PRODUCT.

and what does the inner product take as its inputs?

[placid oracle]

functions

[dusky epoch]

wrong

vectors

elements of your inner product space

[broken hawk]

well, functions are vectors

of some spaces

(it may be that there's information we were not told)

[placid oracle]

"The inner product of tweo functions f(x), g(x) on the interval [a,b] ios a number denoted <f,g> given by <f,g> = integral from a to b of f(x)g(x) dx"

[dusky epoch]

ok but that's just looking at a particular kind of vector space

with that particular product

and in that vector space, the vectors are functions

[broken hawk]

and you didn't indicate the question was about that particular space

[dusky epoch]

but you are working in the more general context of an inner product space, whose elements may or may not underlyingly be functions

[broken hawk]

so we were assuming full generality

[dusky epoch]

should i give the most straightforward counterexample to "If <x,y> = 0 then x = 0 or y = 0" or nah

[brittle juniper]

||>two functions f(x), g(x)
🤢 ||

[placid oracle]

no then ur right

[dusky epoch]

who

and right about what

[placid oracle]

to assuime full generality

and i literally pasted ^ from my textbook

[dusky epoch]

okay but

you're trying to find a counterexample to this claim, yes?

[placid oracle]

yes

[dusky epoch]

so you need to produce a vector space, and two vectors therein, such that neither of the vectors is the zero vector, but their inner product is equal to zero.

what vector spaces do you know of?

give me the simplest vector space you can think of

[placid oracle]

R^n

[dusky epoch]

alright

so can you find two such vectors in R^n

[placid oracle]

but doesnt the vector not matter if you can just choose an interval where the inner product will equal zerop

[dusky epoch]

what interval

look

all i'm asking you to do now

is find two vectors in R^n

neither of them the zero vector

such that their inner product is zero

can you do this

[broken hawk]

(the standard inner product, unless you wanna make things complicated)

[dusky epoch]

hell

you can even pick a particular value for n

if you want

[broken hawk]

(don't pick 1)

[placid oracle]

(1,1) (-1,1)

[dusky epoch]

there we go

was that so hard

@broken hawk you had a question of your own?

[broken hawk]

aye, sec

Let $x \in \mathbb{C}$ and $A \in \mathbb{C}^{n \times n}$. Define $K = \langle x, Ax, A^2 x, \dots \rangle \subseteq \mathbb{C}^n$.

Is $\exp(A)x \in K$? If no in general, under which condition is it?

[stoic python]

Sascha Baer:

[dusky epoch]

did you mean $x \in \bbC^n$

[stoic python]

Ann:

[broken hawk]

yes, sry

[dusky epoch]

ok

exp(A) can be expressed as a linear combination of A^k for 0 ≤ k ≤ n-1

because by cayley-hamilton, chi_A(A) = 0

so that gives you an expression for A^n in terms of those

and once you have A^n you can express all the powers above it as well

and nothing should diverge

[broken hawk]

still an infinite sum tho, so I suppose I’d have to somehow reason it’s a cauchy sequence, right?

which, tbf, it probably is

the same reasoning should also apply to exp(iAt)x, right? (where i is i and t is a positive real number)

still in the same K, but the exponential plays well with scalars

[placid oracle]

can someone explain to me what im doign wrong here: im supposed to show the least squares solution to the system of equations written below by both solving the normal equations and through the Ax=projColA b method

but im getting two different answers for both methods

<@&286206848099549185>

[placid oracle]

anyone?

[slow scroll]

yea idk personally 🤷

[placid oracle]

i have no clue what im doing wrong and ive been trying to redo this problem for th epast hour

still getting the same stuff

[paper vector]

could someone tell me what the difference is between linear and abstract algebra

like what some of the main focuses are in each kind of algebra, all i know is that abstract algebra has groups and vectors and rings but not much about linear algebra

[hasty rune]

Linear is about vector spaces that obey a particular set of properties

IIRC, vectors must be scaleable and addable

[slow scroll]

@paper vector Linear algebra is just focused on vector spaces and linear mappings between them while abstract algebra is the study of groups, rings and fields

[limber sierra]

well, abstract algebra is much broader than that

but thats the focus at a general level, yes

[paper egret]

how come abstract algebra isn't early university?

[limber sierra]

its worth noting that linear algebra is a subset of abstract algebra, but they're usually treated pretty differently

@paper egret because its research-based pure math, all of which fits better under "Advanced Mathematics"

and in some systems (particularly american), math majors might not even encounter it till, like, the 3rd year of ug

[paper egret]

ah ic ic

im assuming early = freshmen/sophomore, advanced = junilr/senior

[limber sierra]

the distinction is mostly meant as a divide between research-adjacent fields and more "computational" or "general" fields

so like, everything up to intro to proofs fits under "early university"

then anything actually rigorous and proof-based is "advanced"

but of course, this isnt a 100% hard-and-fast rule

and its largely contextual

still, its usually clear enough (except for that poor kid asking about exponential functions in #groups-rings-fields)

[placid oracle]

how do i show that in an inner product space if ||u+v||^2 = ||u||^2+||v||^2, ui and v are orthogonal?

[winter reef]

it jsut follows kinda from definiton, (a+b)^2 = (a)^2 + (b)^2 + 2<a,b>

if u and v are orthogonal then it means <u,v> = 0

[half ice]

@placid oracle
What's u²? <u, u>?

[placid oracle]

yes

i think i get dogs explanation

[clear spoke]

How would I go about solving the system of equations

$c_1 \cdot \sin(\pi - c_2) = 1 \ c_1 \cdot \cos(\pi - c_2) = 0$

[stoic python]

Autistic Hoodie:

[clear spoke]

Hmm, maybe I should actually use my brain instead of mathematical procedure

c_1 could be 1 and c_2 could be pi/2

Yup 😃

But the answer is -cos(x)

[wintry steppe]

Hi y’all can I have a hint on q9 tyty

[placid oracle]

how do i do this?

[native lodge]

inner product

physics language detected

[hasty rune]

Isn't inner product also on usual math LA courses at the end when generalizing LA concepts?

[native lodge]

it is lol, but I hear that word used more often with physics

[hasty rune]

True

[quaint heart]

um no

@hasty rune inner products are very useful in math

oh shit

[hasty rune]

They're akin to Dot Products, right?

[quaint heart]

I meant to tag other guy

yeah they are

[hasty rune]

Oh lol

[placid oracle]

@hasty rune yes same thing

so how do i start

[placid oracle]

<@&286206848099549185>

[wintry steppe]

Sorry, can anyone give me a hint on q9?

I posted above <@&286206848099549185> sorry n.n

[solar rock]

15 minute rule

[native lodge]

!15m

doesn't work here lol

[wintry steppe]

I posted my first one a few hours ago

same question but someone else posted

after

[gray glen]

remember how you would normally compute the dot product

also that $\mathbf a \cdot \mathbf b =|\mathbf a||\mathbf b|\cos\theta$ where $\theta$ is the angle between them

[stoic python]

CaptainLightning:

[wide rain]

how would that help

isnt that what you have to show

[gray glen]

well it says to show they're perpendicular

[wide rain]

ok?

[wintry steppe]

Ree ty for the tips will try

[wide rain]

so you want to show that cos 90 = 0 ????

[wintry steppe]

Ye

That the dot prod =0

[gilded junco]

Yo guys question about left inverse and right inverse. Something in my textbook is throwing my mind in loops

Left inverse is intuitive 'Is there a function,g, that you can apply to function f to return back x?"

Right inverse is like "Is there are function,g, where you can apply before hand to f, and you'd get x?"

So g circle f = idx (left inverse)

and f circle g = idx (right inverse)

BUT MY TEXT BOOK SAYS "f circle g = idy"

but f by itself = idy. So I just don't get it

Thanks i'll leave now, hopefully some kind person knows what i'm missing :*(

Oh, I was using idx in the sense of "What's the first thing you put into the function"; and in fact, if g: Y -> X, then the first thing you put into the function is idY. Ok I get it now, thanks

[sonic osprey]

Yw

[warm stirrup]

Glad I could help

[gilded junco]

Can I ravage someone about inverses

[native lodge]

matrix inverse?

[gilded junco]

functions inverse

[native lodge]

linear algebra isn't most suited for that topic lol, doesn't really fit

[gilded junco]

excuse me, i'm shook

maybe it's cause this class is so early on, that they felt to include it even though it isn't part of linear algebra

[slow scroll]

well if the functions are linear transformations it probably fits 🤔

[native lodge]

you need to show problem then

[gilded junco]

the problem is my understanding

Forgive my laziness, but I would love if I explain things how I understand it, and you correct me

So let f: X->Y. The left inverse will take where you mapped to with f, and give you where you started

The right inverse..... ?

[native lodge]

so this was matrix stuff

left and right inverse

[gilded junco]

It's pretty much synonymous dude

Matrices in some way come from functions; excuse my butchering of the concept

[native lodge]

function inverse I typically think of f^(-1)(x)

maybe because I do more calc than LA

[slow scroll]

A right inverse to f:X -> Y is a function g such that f is a left inverse to g @gilded junco

[gilded junco]

Yeah, I got to that realization

This surely shows where my iq limit is, or atleast that's what i've been telling myself

[sonic osprey]

What are you asking?

[gilded junco]

to uNdErStand

I see the right inverse as nothing more than a left inverse from a different view point

(as kxrider pointed out)

I'm on a mission to understand the other viewpoint - why would you use a right inverse

[slow scroll]

"I see the right inverse as nothing more than a left inverse"
careful, the right inverse is not a left inverse, but if
gf = id, then g is a left inverse to f, and f is a right inverse to g. thats all i was saying

[sonic osprey]

Something that might help is noting that a function has a left inverse if and only if it is injective. And a function has a right inverse if and only if it is surjective

[gilded junco]

Zopherus ^ I know that factoid too, if you think it'd help in explaining the difference, please elaborate

So @slow scroll , let's start with
"Left inverse helps you get back the domain when a function has been applied to it"

"Right inverse..... "

[sonic osprey]

If you're looking for differences between the two, there aren't really any differences between right and left inverses. The ideas are dual to each other

[gilded junco]

When would you use one over the other? In particular the RIGHT inverse?

The lecture slides I posted above seem to have just that, I just can't comprehend

I gotta have dinner now. Sorry. afk for 15

[slow scroll]

they don't always exist. A function has a left and right inverse iff its bijective

[gilded junco]

I did it boys

[sonic osprey]

When he says you can make the argument of why do

That makes 0 sense

idy is a function from Y to Y, but f is a function from X to Y, so having f(x) be idy makes no sense

[gilded junco]

oh

soz

thanks for pointing that out!

Would you say:
You can make the argument of why do f ( g(x) ) to get an element in the codomain when you could've just done f ( x ) to get the same element, in half the steps

[slow scroll]

if g(x) = y, then f(g(x)) = f(y)
not f(g(x)) = f(x)

[gilded junco]

ok that point is a whole mind bobble, that i'm trying to get my head around; I think this is where I became messed up in the first place.

Reading your thing now kxrider

[sonic osprey]

Not really, still doesn't make much sense

Half the steps doesn't really matter

[gilded junco]

I'm content. Thanks guys

[jaunty fern]

<@&286206848099549185>

Need help with 13.b

I made y the subject

for the first equation

[dusky epoch]

and what did you get

[jaunty fern]

dont know what to do from here

y= 7/2 - b/2x

[dusky epoch]

-b/2 = 4

[jaunty fern]

?