linear-algebra

[undone garnet]

P is a invertible matrix

[edgy ridge]

Thanks! I'll take a look over it.

And your rank thing too.

You say the answer is two?

[undone garnet]

yeah

[gray dust]

D is a diagonal matrix whose entries consist of A's eigenvals. P is an invertible matrix whose columns consist of each eigenval's corresponding eigenvec

[edgy ridge]

Erm, does order matter with D? Is it, in this example, ((-2, 0, 0), (0, 1, 0), (0, 0, 3))?

Sorry - I don't feel like invoking the TeX bot.

[undone garnet]

no

it just affect the way

you construct P matrix

I mean, each permutation of diagonal of D

each P matrix

[edgy ridge]

So... it doesn't matter?

[undone garnet]

I mean

it doesn't matter what rank is

[gray dust]

the order of the eigenvals must be consistent with the order of eigenvecs

[edgy ridge]

I'll be honest, I'll have to take a look at diagonalization first. But thanks for the help!

[gray dust]

for example if you had eigenvals $\lambda_1,\lambda_2$ with respective eigenvecs $\xi_1,\xi_2$\\if you construct $D=$ diag$(\lambda_1,\lambda_2)$, then P's first column must be $\xi_1$ then $\xi_2$ is next

[edgy ridge]

Er, just to clarify some terminology, for some matrix A and vector v, if Av=cv, then c is the eigenvalue, and v is the eigenvector, yes?

Some reason, the teacher just started using terms without really defining them... it's fun.

But yeah, I think I might get what you mean.

Fair enough.

[gray dust]

yeah c is the eigenval and v is the eigenvec @edgy ridge

really doesn't matter what names you give em

i usually call eigenvals lambda and eigenvecs xi

[edgy ridge]

Anyways, I threw some numbers into a calculator regarding trace... Let's say a diagonal matrix with 1, 5, and 9 (3x3, 1-9). Its trace was 15. I then tried throwing in its square, so 1, 25, and 81... its trace was 107. However, squaring the matrix should result in 261 as its trace...?

[stoic python]

RokettoJanpu:

[edgy ridge]

Recall that I was given only the eigenvalues, and was told to find the trace of A^2

The values were -2, 1, 3.

3x3 matrix.

[gray dust]

,calc 1+25+81

[stoic python]

Result:

107

[gray dust]

tr(A)=sum of A's elements along main diagonal

[edgy ridge]

Yeah.

I guess the problem in this case is I don't know the matrix - only its eigenvalues.

So I know the trace of A, but what is A^2?

Erm, the trace IS the sum of the eigenvalues, right?

[gray dust]

why 261

[edgy ridge]

I just used a calculator... hold on a sec.

Anyways, Nguyen mentioned that if x is a eigenvalue of A, then x^2 is for A^2...

And since the trace is the sum of the diagonal, and is also the sum of its eigenvalues...

Can I say that if A has eigenvalues -2, 1, 3, then trace(A^2)=4+1+9=14?

[uncut forge]

Yeah, since it's diagonalizable(not sure if it's true in general)

Trace is sum of eigenvalues

And if A=$$PDP^{-1}$$ Then $$A^2 = PD^2P^{-1}$$

[stoic python]

AoiKunie:

[uncut forge]

So the squares of the eigenvalues to A are eigenvalues for $$A^2$$

[stoic python]

AoiKunie:

[dusky epoch]

bad tex

[undone garnet]

any idea?...

[pallid swallow]

A and B look almost similar

and M_2 is really whackable

only 2 cases essentially, invertible and rank 1

if A, B invertible we are done, because we have A=B

if A invertible we get AB=B^2...hmm

[undone garnet]

A^2 = tr(A)

hm...

A^2B = tr(A).AB

AB^2 = (A^2.B^2)/tr(A)

and we have

tr(A).AB = (A^2.B^2)/tr(A)

<=> tr(A)^2.AB = A^2.B^2

hm.../

[undone garnet]

problem is fixed

that is easier alot

😄

[wintry steppe]

Hello everyone. Would someone have any hint about the following:
https://math.stackexchange.com/questions/3445326/if-any-order-preserving-permutation-submatrix-of-a-otimes-b-belongs-to-a-b

[analog wasp]

<@&286206848099549185> how do we prove that an empty set is contained in a set that we will call X

[feral grove]

!15m

rule 4

[analog wasp]

?

did i break a rule?

[feral grove]

yeah

[analog wasp]

i dont think i did

[feral grove]

you need to wait 15 minutes after asking a question to ping helpers

[analog wasp]

oh

sorry!

[analog wasp]

maybe that rule has been changed or something because i couldnt recall it and i just looked again i dont see it am i blind?

[half ice]

@analog wasp
What's your question? How do we tell if the empty set is a subset of X? The empty set is a subset of every set.

[analog wasp]

apparently we need to demonstrate that

and i dont see a possible way to do so, maybe i miss understood what the teacher meant by that

if you understand any french then here you go, if not, the second question in Exercise 1 says : Demonstrate that Ø is included in X

[lone quail]

why are the eigenvalues associated to the symmetry of a line r passing through the origin always 1 or -1?

specifically f(r)=r (eigenvalue 1) and f(r2)=-r2

where r2 is the line perpendicular to r

what i dont understand is where f(r)=r and the other part come from

[clever cedar]

is it apporiate to say that a given linear transformation is not surjective because its rank does not equal m (the number of rows of the map)

[dusky epoch]

what's m

[clever cedar]

the number of rows

such that if T(x) = A(x) then m is the number of rows in the matrix A

[brittle juniper]

there are no "rows" to talk about when you have just a linear map and not a matrix

the rank of T is the dimension of its image

the image of T has a structure of vector space, it's a subspace of the output vector space
if the output space happens to be finite dimensional, then you do have T surjective iff dim Im T = the dimension of the output space

[hardy blaze]

http://prntscr.com/q1gj40

http://prntscr.com/q1gj6q

how does that make sense...

that last part :x

[gray dust]

looks like you only need to pick out an eigenvector corresponding to lambda_1 = 3

[stoic python]

RokettoJanpu:

RokettoJanpu:

[gray dust]

@hardy blaze ^

[hardy blaze]

ok

ty sm 🤓

this shit hard af

[gray dust]

no prob man

[wintry steppe]

How do you do 14a?

[grave plank]

multiply M and the vector (x,y) and see what happens

@wintry steppe

[wintry steppe]

Mx+My

Moved to #help-5

[vast torrent]

@sleek briar let's ask here. HP asked in the question chat and I dont know the answer

Questions gives two lines in space given by constant(x-x1)=constant(x-x2)=constant(x-x3) and similarly for the other line

question is to find the distance between the two lines

https://media.discordapp.net/attachments/490557019623915520/648235470211252245/IMG_20191125_002434742.jpg?width=298&height=398 his source gives this formula and I don't know its derivation

[wintry steppe]

How would I understand the concept for this?

[grave plank]

Well they are testing on whether you understand linear independence

in which case what you have to do is consider the linear dependence relation and create a system of equations to figure out w

[steady fiber]

similar matrices have the same eigenvalues with the same algebraic multiplicity (and also have the same determinant)

so just find those

and compare

[undone garnet]

rank(A) not equal rank(B)

obviously they are not similar

[edgy ridge]

Can anyone please help me with this question? Given a 3x3 matrix A with eigenvalues -2, 1, 3, find the rank(A+2I), where I is the identity matrix. A kind person was willing to help the other night, but I'm a bit lost still.

A has 3 distinct eigenvalues, so A is diagonalizable
A = PDP^(-1)
of course, I = PIP^(-1)
so
A+2I = PDP^(-1) + 2PIP^(-1) = P(D+2I)P^(-1)
rank(A+2I) = rank(D+2I)

D is diag(-2, 1, 3)
so
rank(D+2I) = rank(diag(0, 1, 3)) = 2

But where did rank(A+2I) = rank(D+2I) come from?

[undone garnet]

when you multiply a invertible matrix

it doesn't change rank

rank(P(D+2I)P^(-1)) = rank(D+2I)

[edgy ridge]

This applies when multiplying invertible matrices with that? Its inverse, or any other matrix?

[undone garnet]

just invertible matrix

I mean

rank(A.B) = rank(A)

if B is invertible

rank(A.B^(-1)) = rank(A) too

because B is invertible then B^(-1) is also invertible

so rank(PDP^(-1) = rank(DP^(-1)) = rank(D)

[edgy ridge]

Kinda confusing, but I'll take your word for it. 🙂

[undone garnet]

just thinking like this

rank(A.I) = rank(A)

right?

[edgy ridge]

What is the dot?

Multiply?

[undone garnet]

yeah

[edgy ridge]

Alright.

Makes sense so far.

[undone garnet]

rank(A.I^(-1)) = rank(A)

right?

so when you're dealing with rank

just imagine an invertible matrix is an identity

[edgy ridge]

Alright. I guess, so since in the case PDP^-1, it is obvious that P is invertible by the existence of P^-1 in the equation, it will not affect the rank, therefore rank PDP^-1 == rank D?

[undone garnet]

correct 😄

something like

rank(I.D.I^(-1)) = rank(D)

[edgy ridge]

Yep! Thanks. Now, next part is

D is diag(-2, 1, 3)
so
rank(D+2I) = rank(diag(0, 1, 3)) = 2

Is there a typo there? Is it not supposed to be

rank(D+2I)=rank(diag(0, 3, 5))=2?

[undone garnet]

D is diag(-2, 1, 3)

I is diag(1, 1, 1) then 2I is diag(2, 2, 2)

then

D + 2I = diag(-2, 1, 3) + diag(2, 2, 2) = diag(0, 1, 3)

obviously rank(diag(0, 1, 3)) = 2

just write it out

[edgy ridge]

Well, if we add up the diagonals, then for top left, -2+2=0, for middle, 1+2=3, for bottom right, 3+2=5, no?

[undone garnet]

ah yes

[edgy ridge]

Then should it not be (0,3,5)? Am I missing something that makes it (0,1,3)?

[undone garnet]

3+2 = 5

no no

(0, 3, 5)

it should be

[edgy ridge]

Ah alright. Thanks. It does not affect the rank, but it was making me puzzled.

[edgy ridge]

Can anyone please explain how I would go about this?

nvm

[wintry steppe]

Hello, I've been asked to prove that if a subspace is T-Invariant, then the following is true:

I'm unsure where to go from T(S) = S to this

[wintry steppe]

forget it, id missed an important condition

[wintry steppe]

Can i ask very simple question related to Laplace transform?

I have the answer but I think it's wrong

[dusky epoch]

don't ask to ask

[wintry steppe]

Transform u(t) to ..
Is 1/s
But teacher wrote number*1/s

"Level unit signal" (using google translate).

Sorry if it's not clear.. teacher isn't clear either..

@dusky epoch can you help?

[dusky epoch]

[wintry steppe]

???😟

[dusky epoch]

i really don't want to deal with this rn

[wintry steppe]

Then why bother writing "don't ask to ask"?

[dusky epoch]

someone else might have come and helped you

[wintry steppe]

It's not a long answer.

But I understand your frustration

I doubt someone will help

I'm stuck on it and ask my classmates

[vast torrent]

@wintry steppe if you're not sure which is correct, why not just evaluate the integral L{u}?

Or L{u(t-c)}

For more generality

[wintry steppe]

I don't know how

• teacher gives a table to do the transform

[vast torrent]

Well what's the definition of laplace transform, do you know?

[wintry steppe]

Yes it's transform from function time to function frequency

Next question!

[vast torrent]

Let's move to another chat, this isn't lin alg

[vast thicket]

if 2 vectors row reduce to the identity matrix does that mean they span all of R^2?

u = <6,3>, v = <-30, -12>

[gray dust]

if you cram those vectors into a matrix and can row reduce it to I, then yes

[vast thicket]

thanks

[winter siren]

If a set of vectors in R^4 are linearly independant. Are these vectors always a basis?

How do i determine if the vectors make a basis in R^4, otherwise?

Or in R^n in general.

[gray dust]

you need a set of n LI vectors from R^n to form a basis for R^n. Edit: thanks Ann

[dusky epoch]

of EXACTLY n LI vectors.

any more and the set will be guaranteed LD.

[winter siren]

Oh, it all makes sense now when looking at the question again. Thanks 😀

[vapid coyote]

the definition would require linearly independent and spanning

it's a theorem that in finite dimensions it's sufficient to have n linearly independent or n spanning vectors

[grave halo]

I'm tasked to the find the basis of the null space of the linear transformation (T_{A}:,\mathbb{R}^{3}\to\mathbb{R}^{3},, \mathbf{x}\mapsto A\mathbf{x}) where the matrix A is [\begin{bmatrix}
1 & 0 & 1\
0 & 2 & -1\
0 & 1 & -2
\end{bmatrix}]. Matrix A has full rank so the dimension of the null space is 0. Is the empty set a basis for a dimensionless null space? I'm not sure how to answer this. Thank you.

[stoic python]

Noether:

[grave halo]

Or is the answer, it doesn't have a basis?

[dusky epoch]

empty basis

[wintry steppe]

Anyone know about linear congruency

Dunno if this is the right place to ask about it

Solve this congruence relation to find the smallest possible value of x: 5x ≡ 1 (mod 21)

Tried running it into a calculator but it just made no sense to me

[grave halo]

How do I determine if these are vector spaces?

[sonic osprey]

What does it mean for something to be a vector space?

[grave halo]

Closed under addition and scalar multiplication, right?

[sonic osprey]

Are you sure?

[grave halo]

Uhh

A real vector space satisfies x,y\in L implies x+y\in L

And if x\in L then cx\in L if c\in R

[sonic osprey]

Are you reading this somewhere

[grave halo]

Yeah my notes lol

Is it wrong?

@sonic osprey Can you help me?

[sonic osprey]

Sorry

That is what you would call a subspace of a vector space

[grave halo]

What do I have to check?

[sonic osprey]

It depends what the problem is asking for

Are you trying to check that they're vector spaces? Or subspaces of other vector spaces

[grave halo]

That they're vector spaces

[sonic osprey]

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

[vast torrent]

What does it mean to increase the determinant

[half ice]

@grave halo
For the first one, you know that R⁴ is a vector space. So, you can prove A is also a vector space by proving it is a subspace

[vast torrent]

What does it mean to increase the determinant

[grave halo]

@half ice Thank you. I did just that 🙂

@wary shoal I don't know if you mean increase it as in you want it to be 16721692 bigger, ie 906340587+16721692 or 16721692*906340587 but if that's what you want you can always use the the property of determinants: (\text{det}(cA)=c^{n}\text{det}(A)) somehow?

[stoic python]

Noether:

[grave halo]

Well find the c^n that satisfies that! 🙂

n is the dimension of the matrix

Uhh

Those are a bit trickier rules

I'm not sure how to do it algebraically but you could brute force it numerically.

[vast torrent]

Have you tried randomizing it using a computer

[grave halo]

^ what I meant by numerically

[vast torrent]

Until you get one that works?

Ya

[grave halo]

Did you try every single combination?

[vast torrent]

You sure there's a solution?

[grave halo]

What kind of algorithm did you use in order to not brute force it?

[vast torrent]

So leave your pc running for an hour and go cook dinner

[grave halo]

Can you somehow express the determinant algebraically with your required conditions?

[vast torrent]

Why are you using doubles for integers

[grave halo]

We wish haha

[vast torrent]

He wants to permute the elements

Yes

Why are you using doubles for integers

[grave halo]

Yeah thank God

[sonic osprey]

The permutation definition of the determinant might help here

At least, it may reduce down your search space

I mean, with a 9x9 matrix

It's going to be hard to determine what's invertible and what's not

Without actually calculating the determinant

[vast torrent]

Why are you using doubles for integers

[grave halo]

Ayy

[vast torrent]

I wonder if a greedy algorithm will work

Put all the 9s in one row and expand on that row

Worth a try

Well alternating i mean

9 1 9 1 9 1 9 1 9

In the first row

Try that

And the second row will be like

x 8 2 8 2 8 2 8 x

Something like that, still thinking it through

x x 7 3 7 3 7 x x

x x x 6 5 6 x x x

Worth a try

[clever cedar]

what is happening in step 1

where did i = 1e^pi/2 come from

[half ice]

Do you disagree that i = e^(π/2)?

[clever cedar]

i have no idea what that means

:p

idk where it came from

or why it was introduced

wdym gary

[half ice]

Useful here is euler's formula:
e^(it) = cos(t) + isin(t)

Put in π/2, you get i = e^(iπ/2)

[clever cedar]

oh looks similar to de'moivres theorem

[half ice]

It's an extension of demov

[clever cedar]

oh

[half ice]

Is very useful

[clever cedar]

i did ctrl+f in my textbook for euler

didnt show anything 😠

but ill write it down

[half ice]

Whaaaa

He developed the field lol

[clever cedar]

oh lol

sorry, using that formula do i plug in pi/2

for t

[vast torrent]

Wtf kind of book are you using

[half ice]

In fact your book presents re^(iθ) as polar form, your book must have introduced euler already

[clever cedar]

written by the profs at my school

i can't seem to find euler, sorry kaynex

[half ice]

There's no understating how important this is, complex numbers exist because of it
e^(it) = cos(t) + isin(t)

So commit that one to memory lol

[vast torrent]

Post the book so we can roast it or prove your ctrl.f wrong, either way

[clever cedar]

ok one second

[vast torrent]

It's hard to believe this could be omitted

[half ice]

So yeah, let t = π/2, you've got e^(iπ/2) = i

[clever cedar]

https://moodle.yorku.ca/moodle/mod/resource/view.php?id=2156315

[half ice]

Or, think of a complex number with arg π/2 and magnitude 1

You've got i

[vast torrent]

Needs a password rip

[clever cedar]

how come u chose pi/2

[half ice]

You're right, I could have chosen 5π/2

It doesn't particularly matter

[clever cedar]

oh

[half ice]

In fact, π/2 + 2πk for any k works as the exponent

You may see they have that in the proof

They have θ = π/2 + 2πl

[clever cedar]

oh ok

[vast torrent]

Page337 and Euler isn't mentioned

[clever cedar]

:p

is he important

in lin alg

[half ice]

Euler did everything lol

He's math daddy

[clever cedar]

oh lolol

i thought he was just the guy who said 'i dont have enough paper so i cant write the proof'

[half ice]

Well, lin alg in its modern form is a 1900's thing

[clever cedar]

ah

[half ice]

No that was Fermat

[clever cedar]

oh ya

[half ice]

He also had some good stuff. Two very important theorems are named after him

[clever cedar]

oh damn

[half ice]

Well, maybe the last theorem isn't crazy important. Still interesting though

Fermat's little theorem is useful for modular arith

ANYWAY Still need help with it?

[clever cedar]

uh im watching a video

so i dont annoy u

if i still have q's ill ask

thank you !

[half ice]

Sure sure, good luck!

[edgy ridge]

What does it mean when a matrix is close to another? In this case, it's question B.

[vast torrent]

That's a weird phrase

I guess just calculate A¹⁰⁰ and compare it

[steady fiber]

if you make a small approximation at one point, you get exactly the same matrix that you are supposed to be "close" to

you just approximate an extremely small number to be 0

[edgy ridge]

Erm, I don't understand? I ran it through a calculator, and this happened...

[wintry steppe]

Someond know how to do laplace transform with sin?

[vast torrent]

You want to take the diagonilization

Before raising it to 100

@edgy ridge

[half ice]

Oof that's nuts lol

[edgy ridge]

So, what’s supposed to happen? Will the resulting matrix equal the one depicted in b.?

[vast torrent]

Some of the 0s will instead be (number less than one in abs value)^100

@edgy ridge

[steady fiber]

you did no approximations like I said you should do

so you get that thing

which is also pretty close to what it needs to be

just looks disgusting

@wintry steppe don't think this is the channel you want for laplace transforms, probably #real-complex-analysis or #multivariable-calculus

[vast torrent]

I thought that was the definition of orth. proj. What's the defn you have for ortho. proj?

[earnest onyx]

If it's a projection and it equals it's own tranpose

[vast torrent]

Ah okay

That's actually probably the definition

I just took the formula for granted

So

Call the matrix A(A'A)-¹A' =P

First prove P = P²

Then prove P = P'

Ah that's not enough, is it

[earnest onyx]

still need to prove it's the range of A don't I?

orthogonal projection ONTO the range of a

[vast torrent]

Yes that's why it's not enough

Thinking

[earnest onyx]

On these questions, you have to remember to show that the orthogonal projection P is actually the projection onto the indicated subspace, that's some clarification he gave us

[vast torrent]

Yes yes that's what i meant by not enough

[ruby niche]

Question 3.12

Whats the area of f(triangle PQR)

When triangle PQR has a area of 6

Isnt it just 6 times (2x2+3x8)?

[dusky epoch]

no

[vast torrent]

Hope this helps but i dont have time to think it through rn

[earnest onyx]

aaaaaaa

[ruby niche]

ok solved it, just a misstake, instead of substitution i used addition

thx anyways @vast torrent

[earnest onyx]

just so happened to have my solution as well

lol

[opal plaza]

could use some help getting started with this problem 🙂

[vast torrent]

@earnest onyx post your work, i wanna see

[opal plaza]

I thought isomorphic meant it had to be invertible

[vast torrent]

S will be invertible, yes

A map is one to one and onto iff it is invertible

[half ice]

In linear algebra, you can quickly show two spaces are isomorphic if there's an invertible matrix that takes the basis of one to the basis of the other

Wait that doesn't work here does it?

Still, think in terms of bases

[vast torrent]

Trying tothink if there's a quick.way to do this rather than needing to.solve a system

My gut tells me it will be easier to find S-¹: V to R²

And then invert it

[opal plaza]

I'll have to look at it a bit longer

ty for the info

[vast torrent]

Anyway the idea is

To pick a basis of R2 {e1,e2}

and simutaneously solve

S[e1] = [1 1 1]^t

S[e2] = [2 0 -1]^t

Because to define a map on vector spaces you only need to consider how the map acts on a basis

If you choose {(0,1),(1,0)}

Shouldn't be too painful

[opal plaza]

oh that's interesting

I'll put some thought into it 🙂

[vast torrent]

Feel free to come back later :)

[uneven peak]

Hey the way my prof teaches the course makes it hard for me to actually learn the material and I tend to learn better in a lecture setting. What YouTube channels or website do you recommend to use as review and help me understand the material?

[timber minnow]

@uneven peak khan academy and 3blue1brown series are great resources

[uneven peak]

Thank you so much, gonna binge watch those over the next few weeks

[vast torrent]

Ted Shifrin has good videos if your course is heavy on Calculus @uneven peak

Depending on what the curriculum is

[uneven peak]

Ok will check it out too thanks

[north sierra]

what is this symbol called?

im studying eigen vectors/values and they dont say what it's called

[vast torrent]

Greek letter lambda

[north sierra]

oh okay

thank you

[vast torrent]

👍

[lone quail]

@opal plaza i think theres a better way to do it

for a function to be linear and one to one it means that:

1)its ker(f)={0}

2)its space of arrival must be R2

what does it mean to be onto btw?

[vast torrent]

Surjective

[lone quail]

ok cool thanks

i think you can just stick in the values made from the bases of arrival?

Just like this no?

It is linear and one to one

And onto

[vast torrent]

Thats what you would get presumably solving the system of equations

You just did it by inspection , right?

[lone quail]

yeah

i mean you can sort of see it straight away

as if its rank is max so 2

it means kerf={0}

(by considering a matrix A made from the two bases of V)

which means that only the 0 vector goes into 0 (injective);

you also know that dim(kerf)+dim(imf)=2, so that makes dim(imf)=2, which means that its also surjective

(you can also check it by the rank of the map)

@vast torrent you got it?

[vast torrent]

Well, careful

It's not enough to show the image is 2d

You need the image to be V

[lone quail]

it is

[vast torrent]

It is, but rank nullity doesn't tell you that

[lone quail]

you can use it to tell if a function is surjective

[vast torrent]

I think you're making it too complicated

[lone quail]

if dim(imf) is equal to the dimension of the space of departure

[vast torrent]

If the functions maps a basis to a basis

[lone quail]

it has to be surjective

[vast torrent]

That's all you need

[lone quail]

if the rank of the matrix of arrival was 3 (which is not this case)

it would not be surjective

by definition

each vector has to depart and land in the same dimension

[vast torrent]

If you prove S maps any basis of R2 to a basis of V

You don't have to check anything else

[lone quail]

S?

[lone quail]

ah yeah

[vast torrent]

Youre not saying wrong things but youre overthinking it

[lone quail]

yeah but you dont have to do any calculations this way

[vast torrent]

We can use your function

And note that it maps the standard basis of R2 to the given basis of V

That's it

[lone quail]

yes but you have to make a function first

you cant make one by guessing

[vast torrent]

You did that for us

[lone quail]

yeah but that function was made using that reasoning

[vast torrent]

And you can make a mapping by inspection, sure

Trial and error

[lone quail]

yeah but which takes more time?

[vast torrent]

I don't know, which?

[lone quail]

i just think its an elegant approach to the problem

[vast torrent]

You don't need to check the kernel though

[lone quail]

you can see it straight away anyways

[vast torrent]

Do whatever you think is easier for you

[lone quail]

i got a bit carried by my enthusiasm xd

sorry if i was a bit overbearing

[vast torrent]

Not a problem at all

It wasn't my question btw

[lone quail]

yeah

[vast torrent]

Enthusiasm is good

👍

[north sierra]

could someone explain to me why the basis is what they showed?

the very bottom

[vast torrent]

They picked first x2=2 and x3=0

Then x2=0 x3=1

it says "a" basis, they just picked ones to avoid fractions and because setting each variable to 0 except for one variable is an easy way to pick the vectors

[north sierra]

having trouble with this

[sonic osprey]

What have you tried?

[north sierra]

nothing

i dont understand it

dont know where to start

[vast torrent]

What does it mean for x to be an eigenvector with lambda an eigenvalue?

You might need to go back to the definition, nothing wrong with that

[north sierra]

true

yeah i couldn't tell you that lol

so for x to be an eigen vector with lambda, the eigen vector * lambda equals Ax?

[gray dust]

aye, $Ax=\lambda x$

[stoic python]

RokettoJanpu:

[distant chasm]

I'm trying to convert a complex number to polar form

z = 3 + 3i

how do i find theta if the point is (3,3) on the unit circle?

how do i find the angle of (3,3) on unit circle?

[vast torrent]

the point (3,3) is not on the unit circle

[half ice]

@distant chasm
Draw a line to (3,3). What's the angle between it and the x-axis?

Note (3,3) is not in the unit circle. That is, the modulus is greater than 1

[sinful heron]

To polar form you need to find “r”

Take the coeff of the real and imaginary parts, take the sum of the squares and square root that value (like Pythagorean)

sqrt(3^2+3^2)

=sqrt(18)

To find the angle, the equation works out graphically to be cosx=coeff real / r and sin=coeff imaginary / r

It’s the angle shared by those 2 values

For cosx=3/sqrt(18)

Bad question though no exact values

For angles

The formula for polar form is

re^i(x)

If I recall correctly

(3,3) has nothing to do with what’s on the unit circle

[north sierra]

for this question i get the zero vector

[steady fiber]

if you multiply the vector by the matrix, you get a zero vector

is that what you mean?

[north sierra]

yeh

yeah

so what does 0 vector tell me?

[vast torrent]

Is 0 a scalar multiple of (4,-3,1)?

[steady fiber]

what is the scaling factor

[north sierra]

i was gonna say that its not an eigen vector but i checked the answer and it said it was idk im confused

[steady fiber]

if there is one

is there a constant you can multiply (4, -3, 1) by

to get (0, 0, 0)

[north sierra]

0

[steady fiber]

then that's an eigenvalue

isn't it?

by definition 0 would be an eigenvalue

the vector is being scaled by a constant when multiplied by a matrix

[north sierra]

true

[vast torrent]

An eigenvector can't be 0, but 0 can be an eigenvalue

[north sierra]

i see

thanks

why can't the eigenvector be 0?

[half ice]

We define that the zero vector doesn't count as a basis eigen

Note that the zero vector is in every eigenvector space

[sinful heron]

I’m confused I thought eigenvector was defined as (lambda*I-A)x=0

[half ice]

Nuu. Eigenvectors and values are defined as v and λ that satisfy
Av = λv

[sinful heron]

Oh okay

I thought we just use that to check your work

[half ice]

That is, v is the vectors that don't change direction through your transformation

And λ is the amount they are scaled by

You can use that to show that
det(Α - λI) = 0
But that's a result and not a definition.

[sinful heron]

I see okay

[half ice]

It's useful to find them with

The only way 0 is an eigenvalue is if there's a non-trivial way to send a vector to 0.

Ergo, non-invertible matrices have 0 as an eigenvalue

[sinful heron]

Sorry for the blurry pic but this is mostly what you meant right

[half ice]

That looks right, yeah

You can keep going to get det(λI - A) = 0

[sinful heron]

The notes makes sense now thanks

[edgy ridge]

Hey, so I ran a matrix through a calculator, and it says that the eigenvectors are (4, 2, 0) and (-3, 0, 2). But I got, from my own calculations, (2, 1, 0) and (-3/2, 0, 1), which are simply multiples of the the calculator's result. Am I correct, is the calculator, or is it equivalent?

[dusky epoch]

equivalent

if v is an eigenvector then so's αv for any nonzero α

[edgy ridge]

Alright. Thanks!

🙂

[pliant thistle]

does anyone know where to get started? I know that (3,3) must be 0 , apart from that I'm stuck.

[subtle wharf]

Could someone explain to me why R_y(theta) or a rotation about the XZ plane is different from the other two?

In other words, why is theta negative here?

[dusky epoch]

it's... not?

[subtle wharf]

It is, theta is negative in that rotation matrix compared to the others

cos(-x) = cos(x) and sin(-x) = -sin(x), it's just simplified

[quartz compass]

I don't really get your thing but maybe it's something to do with these matrices being formulated in a different handed coordinate system than the one you're using?

[subtle wharf]

I'm mainly asking because I want to develop matrices for rotations in 4D and I'm curious if the XW, YW, and ZW planes will have similar exceptions like that

[quartz compass]

it's not an "exception" it's a choice

flip the sign if you want

there's no god-given coordinate system or choice of where to rotate in any given plane

[subtle wharf]

Got it

[sturdy abyss]

how can I prove the first part

[empty copper]

Start writing it out

[winter moat]

What would it mean by e is eigenvector of G? Can you write it in a mathematical equation?

[empty copper]

$G\mathbf{e}=\lambda\mathbf{e}$ for some $\lambda\in\bR$, assuming you're working with real numbers

[stoic python]

Rijinaru:

[sturdy abyss]

yeah i got to that part I just can't seem to show it

since we are adding identity matrix

it seems confusing

[dusky epoch]

what is $(G + kI)\mathbf e$?

[stoic python]

Ann:

[dusky epoch]

@sturdy abyss

welp they're offline

[winter siren]

Checking if a matrix A is orthagonal by taking the matrix itself multiplied by its transpose A^T equals the identity matrix.

If I have (1 / 3) in front of it and get a diagonal matrix with 9s? Is it wrong to simplify it to the identity matrix?

[dusky epoch]

1/3 in front of what

[winter siren]

The matrix calulated by:
A * A^T

So

(1 / 3) (A * A^T)

[lone quail]

is A simmetric?

[dusky epoch]

...

[winter siren]

A is a diagonal matrix with 0s and 9 in the diagonal.

[dusky epoch]

...

.....

[lone quail]

so it is simmetric

[dusky epoch]

what the fuck are you on about.

you're giving the same name to what appears to be 3 different things

who cares if it's symmetric

who cares if A is symmetric

[lone quail]

was just trying to help but ok

[dusky epoch]

if AA^T = 1/3 * 9I = 3I, then NO, A is NOT orthogonal.

[winter siren]

Okay, thanks.

[lone quail]

if f(v)= λv, then why [f(v)]^2= v*λ^2 ?

whouldnt v also be squared?

[dusky epoch]

uhh

what are v, V and λ

and f

[gentle knoll]

maybe you mean f^2(v)? because that would be f(f(v)) = f(lamda v) = lambda f(v) = lambda^2 v, assuming f is linear

@lone quail

[lone quail]

@gentle knoll ok thanks perfect

I got it

[distant chasm]

could anyone give me any tips on how to start this problem

can't use quadratic formula since too many terms

should i convert it to polar form?

[vast torrent]

use the quadratic formula bruh

$x^2 + 2x + (1+i)$

[stoic python]

gfauxpas:

[distant chasm]

oh wow

tyvm

ill give it a shot

[stoic elm]

Can someone tell me what I'm doing wrong lol

not sure if i'm even supposed to be doing that for these types of questions

[distant chasm]

u found the nullspace

you want to find which vectors

form a basis such that you can form other vectors for them

since you found from RREF that the first two vectors in the matrix are linearly independenat( pivot columns)

then the vectors that span the basis are

the first two vectors

i.e the standard basis of [1,0], and [0,1]

sorry since its R^4 it should be

[1,0,0,0], [0,1,0,0]

[stoic elm]

:/ that didn't work

[wintry steppe]

does this prove that tr on tensors is well defined?

or can anyone give me a hint

[wintry steppe]

<@&286206848099549185>

[north sierra]

so if you have a lower triangular matrix or upper triangular matrix, the eigen values are the diagonal entries of the matrix?

[wintry steppe]

hey guys

what one does Linear Algebra fall under

[sharp totem]

@north sierra yes

[wintry steppe]

pz

im tired and i need to submit college app

[dusky epoch]

probably "advanced mathematics" since that's the one that isn't obviously not fitting

[wintry steppe]

thankd

[silver ore]

quick question

Lets take <v1, v2> as an example

I know that for inner product spaces, there is conjugate symmetry

but for the dot product, if I have computed <v2, v1> first, I would get the same answer if I computed <v1,v2> first

why is that?

[dusky epoch]

$\left< (x_1, y_1, z_1), (x_2, y_2, z_2) \right> = x_1\overline{x_2} + y_1\overline{y_2} + z_1 \overline{z_2}$

[stoic python]

Ann:

[dusky epoch]

this is the complex dot product.

[silver ore]

oh right

thanks

[steady fiber]

those looks like high school subjects

so linear algebra probably falls under none of them

[north sandal]

I'm confused by this question in my linear algebra book, any help is appreciated. Trying to understand it.
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 11.
,tex $ A = \begin{bmatrix}
1 &-4 & -2 & 0 & 3 & -5 \
0 & 0 & 1 & 0 & 0 & -1 \
0 & 0 & 0 & 0 & 1 & -4 \
0 & 0 & 0 & 0 & 0 & 0 \
\end{bmatrix}
\sim \begin{bmatrix}
1 & -4 & 0 & 0 & 0 & -5 \
0 & 0 & 1 & 0 & 0 & -1 \
0 & 0 & 0 & 0 & 1 & -4 \
0 & 0 & 0 & 0 & 0 & 0 \
\end{bmatrix}$
I reduced to echelon form and can see that columns 1, 3, and 5 have pivots, thus leaving columns 2 and 4 as free columns. Writing it out in linear form gives:
,tex $
x_1 = 5 + 4x_2 \
x_3 = -1 \
x_5 = -4 \
0 = 0 \
$
I think the final parametric form would look something like:
,tex $
{ x_2 \begin{bmatrix} v_2 \end{bmatrix} + x_4 \begin{bmatrix} v_4 \end{bmatrix} : x_2, x_4 \in \mathbb{R} }
$
But I'm not sure what v_2 and v_4 would be... or if I'm correct with my approach. Thanks in advance.

[stoic python]

czloc:
Compile Error! Click the reaction for details. (You may edit your message)

[half ice]

@north sandal
That's correct. What are v2 and v4?

The final parametric form is
x1 = 5 + 4t
x2 = t
x3 = -1
x4 = s
x5 = -4

[north sandal]

oh, i think i overcomplicated it then

thanks

[cobalt dagger]

is there any easy way to solve this?

sqrt((5^12-5^10)/6)

[dusky epoch]

$5^{12} - 5^{10} = 5^{10}(5^2 - 1)$

[stoic python]

Ann:

[cobalt dagger]

i know but like

im studying for this psychometry test

and i have to find a simplified form of this equation

[gray dust]

Equation??

[cobalt dagger]

the answer is 2*5^5

but I don't know how it got there

[gray dust]

Use what Ann said to rewrite the expression. And...

$\sqrt{ab}=\sqrt{a}\sqrt{b}$

[stoic python]

RokettoJanpu:

[cobalt dagger]

ok so the top part of the fraction is sqrt(5^12)-sqrt(5^10)

what after that?

[gray dust]

Cannot split a difference under a sqrt like that

[cobalt dagger]

you can cause the whole thing is under sqrt

oh wait

there's a minus there

oof

[wintry steppe]

Good books to study linear algebra?

[gray dust]

you may want to get comfortable with prealg-algebra before tackling linalg

[wintry steppe]

I'm doing it already lol

Guess I'll finish the basics and get on prealg

[lone quail]

if i know that det(M)=0 where M is any given square matrix

then does det(M-lamda*I)=0, whatever lambda (the eigenvalue) value is?

[vast torrent]

what do you think?

[lone quail]

probably

no actually of course it is

because im setting the equation to 0 lol

sorry im a bit tired, i prob did 10hr study today

[vast torrent]

@lone quail im not sure i understand the q actually

[gleaming topaz]

If det(M) = 0 then there is no identity matrix or am I wrong?

[lone quail]

yeah but if det(M) is 0, it means whatever the eigenvalues are, then det(M-eigenvalue *I) will always be 0

[vast torrent]

There's a theorem, not sure what you're looking for

[lone quail]

yeah i found what im looking for dw

it was pretty obvious

[vast torrent]

detA=0 iff 0 is an eigenvalue

[lone quail]

yeah

[vast torrent]

Not the only eigenvalue

Doesnt have to be

[lone quail]

yeah, what i was asking was that if the other eigenvalues subtracted to A would give determinat 0

which of course must be

[vast torrent]

Well you're solving for those lambda that make it true

[lone quail]

yes

thats exactly why its obvious

xd

[vast torrent]

As long as your scalars are algebraically closed, there's at least one

[lone quail]

yeah

[vast torrent]

If the scalars are R,not necessarily

is there a soln

[lone quail]

what do you mean by scalars are R?

[vast torrent]

If you insist on your scalars being real

[lone quail]

yeah ok i get what u mean

[vast torrent]

det(A-lambda I)=0 might not have a solution lambda

[lone quail]

yeah thanks

[vast torrent]

Np

[lone quail]

i think i did my record study time today

im fused

[young pasture]

Ifk how to do q8

I know the answer is (7^n) * -2

Idk how to find n

[young pasture]

<@&286206848099549185>

[feral grove]

det linearity

[young pasture]

What?

[feral grove]

like

when you have a determinant, and you multiply say 1 row by a scalar, it multiplies the determinant by that

there's a fairly simply proof for that

[young pasture]

Yeah i know that

How does that help me find the answer?

Cant see how that helps find n

[feral grove]

imagine multiplying every column by 7

row and column is interchangeable in this

how would that change your determinant

[young pasture]

That would multiply the dterminant by 7

Or rather 7^n where n is the number of columns

[feral grove]

yes

[young pasture]

Ok

[feral grove]

precisely

[young pasture]

How does that help me find n?

[feral grove]

n is the dimension of the matrix

[young pasture]

Ok whats n in this question?

Whats the value of n

[feral grove]

have you tried putting n in?

it depends on the dimension of your matrix

[young pasture]

As an answer?

[feral grove]

7^n *-2

[young pasture]

Oh good idea dude let me try that

You are a legend it worked

I overthought it

I thought you had to find n but n is actually just a part of the answer

Thank you bullton

[gray dust]

@cursive junco express x,y,z in terms of two new variables t_1,t_2 as shown above then substitute into the vector (x,y,z)

[charred stirrup]

why wrong?

nvm you just have to factor it

[gray dust]

What’d you try

[gray dust]

Show work?

Neither point lies in the intersection

Review your notes on finding plane intersections (you need a direction vector and a single point)

[gray dust]

Definitely google this, attempt the q, come back if stuck

[wintry steppe]

if anyone has any questions ping me

;question

[sonic osprey]

@wintry steppe does it have to be linear algebra

[wintry steppe]

yeah

or AP calc bc

which is calc 1 and calc 2

im taking calc 3 next semester

i have a 99 in linear algebra rn

so id say im fairly good at it

@sonic osprey

[lone quail]

@wintry steppe

Why is it that A is orthogonally similar to B if and only if h=0?

(the two matrices in the middle of the page)

the definition of orthogonally similar tells me that there exists an orthogonal matrix P such that P^-1AP=B

the answer in the textbook however, tells me that for A to be orthogonally similar to B it must be symmetric, why is that?

[wintry steppe]

Well assume A is orthogonally similar to B

Then A = Q^T * B *Q

For some orthogonal matrix Q

Then we get A^T = Q^T * B^T * (Q^T)^T = Q^T * B *Q = A

so since B is symmetric, A is also symmetric

[lone quail]

ohhhh ok that makes sense

thanks a lot

[wintry steppe]

No problem

[young pasture]

https://media.cheggcdn.com/media/ba8/ba80d618-5fe1-474a-a2da-d6ed60377a41/php84WShf.png

How would I solve this question?

I tried using cross product but did not work because you can't use cross product with matrices

[native lodge]

We can answer this quite easily

Really think about what $E\vec{v}=\vec{0}$ means

[stoic python]

⚡Amphy⚡:

[native lodge]

the important thing is that v is a nonzero vector

[young pasture]

I am not sure I know where to go from there

[native lodge]

can you recall that matrix multiplication can be thought of as a combination of the columns?

[young pasture]

Yup

Oh wait

detE = 0

[native lodge]

you see it now?

[young pasture]

yup

[north sierra]

Cool question

[native lodge]

we have a nonzero vector in the nullspace of E, so then we are working with a singular matrix

[young pasture]

Yup, I got it. Thanks dude. E's rank is at most 2

[native lodge]

[austere creek]

If $\mathbf{v}_R$ and $\mathbf{v}_I$ and the real and imaginary parts of an eigenvector, what does the notation $P=(\mathbf{v}_R,\mathbf{v}_I)$ mean? Is $P$ just a coordinate?

[dusky epoch]

can i have some context

where'd you encounter that

[stoic python]

Gamedolf:
Compile Error! Click the reaction for details. (You may edit your message)

[austere creek]

"An introduction to Eigenvalues" section in this book

it says predict what shape you will gert when you multiply some points in a previous step by P^-1

i guess P can't be a vector because no inverse

[dusky epoch]

no like can you give me a page

or sth

i want to see what context this notation is used in

[austere creek]

sec

[dusky epoch]

is B a two by two matrix or

what

[austere creek]

yea

[austere creek]

I think P is just a point and P^-1 is just a reflection in y=x

[dusky epoch]

no

P is a two by two matrix

[austere creek]

hmm

[austere creek]

oh is it the $\begin{pmatrix}a&-b\b&a\end{pmatrix}$ thing

[sturdy abyss]

how can I determine the values of k.

In the mark scheme, it says something about taking the vector product

please help

[stoic python]

Gamedolf:
Compile Error! Click the reaction for details. (You may edit your message)

[tidal kettle]

@sturdy abyss any value of k that makes anything in parentheses to the n-th power less than 1

because then as n goes to infinity the terms will goto 0

specifically work the 7k to find it because making sure the largest coefficient term is less than 1 for some k in general makes all the smaller terms less than 1 for that k aswell

@austere creek what book are you using

[austere creek]

Applied Linear Algebra: The Decoupling Principle

By Lorenzo Adlai Sadun

[strong kelp]

help me

there was a method to solve linear equations by matrix method named after a guy i forgot the name

[dusky epoch]

Gaussian elimination? Gauss-Jordan elimination?

[strong kelp]

umm not gaussian

cramer's rule got it

[eager kestrel]

Hello, I am not sure my math is correct in this mechanics problem, I seem to got something wrong here, would appreciate all help I can get, tried to solve it for several days and this is my best attempt (most of the problem is just pure mathematics, linear algebra and system of equations, if you are not comfortable with the physics, just ignore it and focus on the math, thanks)

my try: https://imgur.com/gallery/UfF55tq
I tried to find the spring constant ”k”
The unstreched spring has a length of ”d”, and the table mass ”m”, express the spring constant ”k” in terms of, ”m,g, a” and ”d”

[clear otter]

@eager kestrel the second term in your cross product is incorrect. It should be a positive 48

Where is F_mg acting? Is it acting at C or is it at the center?

Never mind. I see.

[eager kestrel]

@clear otter yes it should be 48 i saw that after posting

But the answer is still wrong

Also mg should be negative

But still wrong

I would guess its acting at c

Have you done mechanics?

Its just a basic course, would mean the world if you tried to solve it

[gleaming topaz]

You should have positive 96 as well but unsure if that affects the answer

[clear otter]

@eager kestrel you're taking the cross product because you're looking at moments about point O?

[eager kestrel]

Exactly

Thanks for pointing it out

@clear otter is not harder than that

And since i know its suppose to be equalibrium

Just set each component to zero

And solve system of eq for k

[clear otter]

Correct.

Sum all of your forces and set X Y Z to zero. Then sum all of your moments about some point and set them to zero.

[eager kestrel]

Outside atm, home in 15 will check if pos 96 nails it

[clear otter]

Personally, I'd recommend choosing a different point than O for finding moments because it'll let you ignore the contributions from a different force.

[eager kestrel]

Yes i did or tried do that in notes as u prob.can see

But it does aleeaddy

[winter siren]

I have a short question. If you have some vectors in the room R^3.
How can you determine the point / line / plane for these vectors?

What do I need to learn in Linear Algebra to accomplish this (It is a Matlab assignment)?

[clear otter]

You did not. You're taking your cross product with respect to O.

[eager kestrel]

But it does aleeaddy

[clear otter]

Oh I see, there's more than one page 🙃

[eager kestrel]

Since at O it gets rid of upward force at O

From the cone

[clear otter]

Ah right

@eager kestrel I'll try to quickly solve this and let you know what I get for k.

[lone quail]

if A and B are both similar and diagonalizable, so there exists matrixes P and Q so that:
(P^-1)AP=D and (Q^-1)BQ=D.
by transitivity, the textbook says, that the basis by which f is represented by B is formed by the columns of P(Q^-1), why is that?

For reference A is actually the representative matrix of f while B is the botton matrix

[eager kestrel]

okay great, means alot to me, when i changed it to 96 I got the answer:

(21mg)/(64*(5*a-d))

[lone quail]

Oh sorry thought you finished

[clear otter]

@eager kestrel is that the correct answer?

[eager kestrel]

nope

https://gyazo.com/7588dfcf6f50e7f780b2c761c28676aa

" ... is not correct."

@clear otter

[clear otter]

@eager kestrel this system of equations is a pain.

[eager kestrel]

Just solve for A like I did

Then put A in both of the eq equal to each other

From x and y

And then put inside sys of eq for z

You dont have to simplify, i can plugg in big answers!

@clear otter

[clear otter]

Lol,I know how to solve

It's just longer than I'd like. You just need k right?

@eager kestrel k = 21mg/(24*(5a - d))

[eager kestrel]

https://gyazo.com/47a41c9791399d4f2b3b0c92d180659c

Somewhere you missed something aswell

thanks for trying, although you seem to missed something, unfortunately

@clear otter

[clear otter]

HMMMMMM

Is that green underline where it's saying the problem is?

oh shoot. I see where my problem is.

[eager kestrel]

no its just where I put the cursor when I type the answer in

ah great that you found it! really getting me excited

[clear otter]

@eager kestrel try 3mg/(4*(5a-d))

Ahhhh

Reduce that 😅

Fixed

[eager kestrel]

https://gyazo.com/7de2c199bf06eab342c467aa8120eac9 feel bad for you now :/ welcome to my world!

[clear otter]

@eager kestrel I'm not sure then. I've double checked my moment equations and they all look fine. That should be the correct answer.

When doing your cross product for Fmg, the second component should be negative. That should be a negative 3a.

Your 6a should also be positive in that cross product. Look at your origin.

[grave halo]

Let's say I want to project (\mathbf{x}) onto the subspace (V) where (V=\text{span}\left\lbrace\mathbf{b}{1},\mathbf{b}{2},\cdots\mathbf{b}{n}\right\rbrace) then I know that (\text{Proj}{V}\left(\mathbf{x}\right)=A\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}\mathbf{x}). However I also know that if my basis vectors for (V) are orthogonal then I can also compute the projection with simple dot products. Can someone explain to me how the general matrix product simplifies when the basis vectors are orthogonal? Thanks!

[stoic python]

Noether:

[grave halo]

I forgot to add that A has the basis vectors as its columns.

[eager kestrel]

@clear otter https://imgur.com/gallery/1qwe6UJ

new solution, i think that a problem might be thr gravitations force in the ceneter and the force at A

since if you look at the second to last page

you see that the magnitude of vector |Ts| is equal to a vector, which is impossible

but if I dont equate the A vectors, I dont get the problem

how to deal with this? not sure..

[opal plaza]

If someone has a sec could you please help me check work. I found a vector of 6x_1 = 9x_2 so I think the basis would be [2,3]? Also trying to conceptualize what exactly this vector means in relation to the original matrix? ty ❤️

[vast torrent]

is 6(2) = 9(3)?

[opal plaza]

haha

where do I go to learn multiplication?

[vast torrent]

I make that mistake too sometimes

you meant [3,2]

how many eigenvalues and eigenvectors do you have?

[opal plaza]

just one, 4

[vast torrent]

with dimension 2?

or dimension 1

dimension of the eigenspace I mean

[opal plaza]

I believe it's 1?

[vast torrent]

I dunno, I didn't solve Av = 4v, I'm assuming you did

I don't know if the solution space has dimension one or 2

[opal plaza]

one

[gray dust]

gonna quickly plug into symbolab eigen finder to confirm

[vast torrent]

lol I just did for wolfram alpha

then you only need one vector for the basis

if the vector satisfies Av=4v, you're done

[gray dust]

d'oh, too fast for me

[stoic python]

Noether:

[grave halo]

Solved that one btw

Idk how to delete

[vast torrent]

you dont need to delete, let other people see it

[clever basalt]

Guys I need a math pro to be my guru 😭

[lone quail]

if A and B are both similar and diagonalizable, so there exists matrixes P and Q so that:
(P^-1)AP=D and (Q^-1)BQ=D.
by transitivity, the textbook says, that the basis by which f is represented by B is formed by the columns of P(Q^-1), why is that?
For reference A is actually the representative matrix of f while B is the botton matrix

https://cdn.discordapp.com/attachments/540211747613704221/650392600208998430/image0.jpg

[timber minnow]

@finite sleet you can do this a better way

responding to your question earlier in #help-2

use this:

https://en.wikipedia.org/wiki/Sherman–Morrison_formula

and decompose your matrix into A = (a-1)I + 1*t(1)

then you have the inverse to be:

[timber minnow]

$\frac{1}{a-1}
\left( I_3
-\frac{1}{(a-1)+\mathbf{1}^\mathsf{T}_3\mathbf{1}_3}
\mathbf{1}_3\mathbf{1}^\mathsf{T}_3 \right)$

[stoic python]

davidL:

[timber minnow]

expanding:

,w (1/(a - 1)) ({{1, 0, 0}, {0, 1, 0}, {0, 0, 1}} - (1/((a - 1) + 3)) {{1, 1, 1}, {1, 1, 1}, {1, 1, 1}})

[stoic python]

Display results online and refine query

Requested by davidL#1578

[timber minnow]

solving:

,w Simplify[((1/(a - 1)) ({{1, 0, 0}, {0, 1, 0}, {0, 0, 1}} - (1/((a - 1) + 3)) {{1, 1, 1}, {1, 1, 1}, {1, 1, 1}})) . {1, 2, 3}]

[stoic python]

Display results online and refine query

Requested by davidL#1578

[timber minnow]

as a specific example, when a = 0:

,w {(a - 4)/((a - 1) (a + 2)), 2/(a + 2), (3 a)/((a - 1) (a + 2))} /. a -> 0