linear-algebra

[rose parrot]

Thanks

[proud compass]

Are there any ways to reduce the paper I'm using from solving matrices?

Or any free apps for working with a matrix?

[gray dust]

got an rref calculator if you need: https://www.emathhelp.net/calculators/linear-algebra/reduced-row-echelon-form-rref-caclulator

symbolab eigenvector calculator: https://www.symbolab.com/solver/matrix-eigenvectors-calculator

[proud compass]

Thank you, these should help a bit

[paper egret]

ummmm wouldn't this just be like the opposite of identity or something?

instead of identity, all 1's become 0's and all 0's become 1 or something?

[pallid swallow]

What's $J_A$?

[stoic python]

Element118:

[paper egret]

oh T_A is linear transformation

our professor writes it a tad bit whack

[pallid swallow]

Oh, $T_A$ is the linear transformation defined by the matrix $A$

[stoic python]

Element118:

[paper egret]

ye

[pallid swallow]

So, I suppose you use the standard basis here

[paper egret]

question, when it says Z(v) = 0, does it mean Z(v) = (0, 0, 0, ... 0)?

[pallid swallow]

yes

[paper egret]

the resulting vector is all 0's

oh ok

[pallid swallow]

So, basically what this is asking is, find $A$ such that $Av=0$

[stoic python]

Element118:

[paper egret]

so it's telling me it turns any vector v, into a 0 vector

this A matrix

gotcha

oh so A

is just

some m * n matrix, with all entries equal to 0

wait idk if that's right

it sounds too easy to be rigt

[pallid swallow]

it is

how would they define the linear transformation if the matrix was something else?

[paper egret]

true

thanks for the help btw

[hardy blaze]

if a system has free variables, the solution set contains inf. many solutions

this is false right bc it can be inconsistent

im p sur eim right im just making sure ..

[proud compass]

yeah it can have 0 or more than 1 with free variables

[hardy blaze]

http://prntscr.com/p0hmdk could i use this as an example to prove that statement wrong

my professor is a headass

cuz thats inconsistent but x3 is free

[proud compass]

that might work, yeah

[sonic osprey]

Are you sure this system is inconsistent

[proud compass]

oh right,

[hardy blaze]

.

is it not? ;0

cuz r3 is 0 = 15

[proud compass]

add 2*R1 to R2 and x3 will just go away

[hardy blaze]

I-

it says its inconsistent in the book

[proud compass]

wait maybe im dumb

[slow scroll]

looks inconsistent to me

[hardy blaze]

http://prntscr.com/p0hn9f

[proud compass]

yeah it's inconsistent

[hardy blaze]

ok

ty all

& np^^

❤

this class is fun

idk y

[proud compass]

yeah linear algebra is a lot more fun than stats and logic

I'm guessing your class just started too?

[hardy blaze]

yes

[proud compass]

good luck man, we'll both probably need this channel a lot

[slow scroll]

i like linear algebra too. cant wait to take it next semester

[hardy blaze]

@proud compass u 2, & for sure 🤠

[cloud cedar]

really dumb question but i cant seem to make sense of it: i have a linear map between vector spaces (both 3D), and im supposed to "show that the map T: V \to W is an isomorphism onto a subspace of W." i got the matrix representation of the map, but it has determinant zero and cant be an isomorphism onto W. could the question mean that it's just an isomorphism of the eigenspaces with nonzero eigenvalue? that seems like a weird thing to prove

the fact that it says "onto a subspace of W" makes me think that it might be what's wanted, but that seems too trivial

[dusky epoch]

uh

show exact question statement?

[cloud cedar]

sure

my solution so far:

\Gamma(R^3) is the set of smooth vector fields on R^3

[dusky epoch]

dim Gamma(R^3) isn't 3

[cloud cedar]

isnt it spanned by the partial derivatives?

[dusky epoch]

pretty sure there's many MANY more smooth vector fields on R^3 than just the partial derivative fields

[cloud cedar]

i guess the partial derivatives only span the tangent space; do i need to do something fancier to talk about a basis for Gamma?

[dusky epoch]

honestly

i think it'd be enough to say L is injective and X, Y and Z are LI

[cloud cedar]

ah

[hardy blaze]

getting the right answer when you have to explain something

is so fulfilling ..

idk why i like this class so much and its only been 1 week

[native lodge]

because linear algebra is great anyways

[dusky epoch]

those weren't linear algebra classes then

they were matrix-bashing

[hardy blaze]

our prof is only making us do hand calculations for the 1st 2 weeks

& then after that he said the calculator could do it but he just wants us to see wat is happening or smth

[half ice]

Multiply a 3×3 so you know how, then never do it again

[gray dust]

multiply a 4x4 even, then never again :>

[lone cove]

yeah, my linear class was taught by a cs guy so it was all just matrix algebra

it sucked ass

[dusky epoch]

homework: here are 10 pairs of matrices now multiply them together

[half ice]

Here are 10 matrices, give me the determinant of their product

[dusky epoch]

@half ice bonus: the matrices' dimensions are such that the product isn't square

[hardy blaze]

that sounds awful

i think we did multiply 3x3 once in cal 3 but he was showing us something from a higher lvl math

i cant imagine doing that that much ☠

[half ice]

A fun trick is getting the determinant of the inverse. I wonder how many kids would find the inverse first?

[jagged saffron]

when we say a function is periodic with period P do we define P as the smallest such number possible?

like f(x+2) = f(x) for all x

do we say it has period 2? or can it be period 4, 6,8 etc

idk where this goes

am working on this

I dont think it matters how we define the period for the first subset, because f periodic with period p implies f periodic with any multiple of p

yea

i was just wondering for the next ones if it would matter

i think if their period ratio is rational

it is closed under addition

ah ok

uh

let f be period a/b, g with period c/d, then f+g should be of period da

since a/b is rational multiple of p and d,b are integers it should be

rational multiple too

yea

i proved it already

thanks

last one is definitely not subspace if we just take two periods with irrational ratio

[stoic python]

Blitzkrieg:

Blitzkrieg:

[jagged saffron]

huh?

like make up a function like that?

I was thinking just contradiction

if F,g are have period of irrational multiples of P

assume (f+g) has some period, call it p_1

then (f+g)(p_1+x) = (f+g)(x) = f(x)+g(x) = f(p_1+x) +g(p_1+x)

for all x

wait

nvm

oof

oh wait i think i got it

i took f(0;1) with your example and f(0;sqrt(2)) and showed that if f(p;1) +f(p;sqrt(2) = f(0;1)+f(0;sqrt(2) then f(p;1) = f(0;1) and f(p;sqrt(2) = f(0;sqrt(2))

then p is a multiple of both 1 and sqrt 2

write p = x1 = ysqrt(2) we get sqrt(2) = x/y which is a contradiction

[coral sage]

does being closed under scalar multiplication imply that a set contains the zero element?

why not?

because doesn't multiplying by 0 necessarily give you back the zero element

well for 1 i realized I worded the q wrong

because i'm specifically asking for subsets of a field

so for some subset S over a field F where F contains a 0, does being closed over scalar multiplication by anything in F imply that S contains 0

i'd assume that it does, but that just leaves me confused as to why my linear algebra class listed those seperately when we were talking about subspaces

uh, that depends on what you mean by scalar multiplication

i mean scalar multiplication

yeah

I know that

i'm asking if there's any reason we have to state the fact that 0 is an element of a subspace seperately

since it's implied by the fact that subspaces have to be closed over scalar mul

[thorn robin]

maybe you're asking if you can replace the 0 \in S condition with S nonempty?

if you don't assume one of these, then your definition of subspace will include the empty set
which is not a vector space, since one of the vector space axioms specifically demands the existence of an element 0 \in V

[carmine terrace]

can someone help me understand what I did wrong

https://i.imgur.com/0RLrkzH.png

I had to miss the lecture on row echelon form and I'm extremely confused

how did they get the 1 on the second row, colum 2

yes

I'm trying to understand the row echelon, but all the examples i see online are nice ones where you can easily get leading zeroes

I can't find any with only two rows

https://i.imgur.com/b2WvpXb.png

so for this one, if i just switch the rows

that would be in row echelon?

https://i.imgur.com/AEmKl0L.png

dang, this is confusing me

I will, so in reduced row echelon form, we need to form a "staircase" of leading ones per row right?

the 0 need to be underneath the staircase

and the numbers above don't really matter what they are?

the part that confuses me

is that the staircase can go "two steps" ahead

so on this one for example, the 2nd row, the leading one could be on the 3rd column instead of colum 2?

so how would I know where to place the leading one?

does it simply come down to the algebra?

no rules?

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

because that one for example

switch row 1 with row 2

leading one in R1C1

then leading 1 in Row 2 Col 4?

hmm, so I could divide by 2 and have the leading 1 in row 2 col 2?

and since it's a system of equations, the place of the leading 1 doesn't really matter, because it should still give the same answer at the end of the day?

[gray dust]

row ops don't change the solution set

[carmine terrace]

ok, I'll try diving by 2 and see how it goes.

[carmine terrace]

i goofd and got the problem wrong, but I understand it now

thanks for the help

I managed to finally get it correct

[wintry steppe]

Can anyone help me

I

[lone cove]

quiz

[noble kettle]

@wintry steppe , first find f(g(x))

[broken hawk]

1. Requesting help during an exam a bannable offense.

this is awfully close to that

[wintry steppe]

Oh sorry

[earnest zinc]

hey, i was absent and missed a lecture for today, does anyone know of a good video that can teach me to solve these? thanks

or just list the approach i could use if you wanna be a real good dog owner

[gray dust]

i assume you're taking lin alg alongside DE?

[earnest zinc]

no

this is just some random ting in my lin alg class

he taught steps to solve this but i was gone

[gray dust]

this lin alg homework? lol

much of the work you have to do here is lin alg stuff... but solving this problem assumes you've had some experience with DEs

[earnest zinc]

:0

i think because we arent supposed to be rooted with DE knowledge

he just told us a set of concrete steps to do it

but i cant find those steps :c

[gray dust]

what the hell is your teacher doing lol

[earnest zinc]

😮

woof

[gray dust]

should've just given you a matrix and told you to find its eigenvectors and eigenvalues

[earnest zinc]

its meanie i dont even know where id start

i can do that

[gray dust]

end of

[earnest zinc]

but this is wacko tobacco

[gray dust]

but the second step is to plug those e-vecs & vals into solutions to the DE

[earnest zinc]

first step find the eigens of the 3x3 maytix?

[gray dust]

wait do you know how to find eigenvalues?

[earnest zinc]

ya

[gray dust]

like for any square matrix, 2x2 and 3x3

[earnest zinc]

yea yea so find eigens

[gray dust]

find eigenvals and eigenvecs first

[earnest zinc]

then plug into

[gray dust]

then lemme know when you're done and i'll show you the next steps

[earnest zinc]

oh im in high school class rn

no can do it

[gray dust]

wait hold on

[earnest zinc]

: )

[gray dust]

have you taken at least calc 1?

[earnest zinc]

ive taken up to multivariable

[gray dust]

k cool, so you've seen DEs before

[earnest zinc]

o

i guess yah

[gray dust]

it's just we're solving a system of DEs so we got derivatives of two quantities that depend on the quantities themselves

like dx/dt = 2x + 3y and dy/dt = -x + 5y

[earnest zinc]

class requires 0 calc knowledge

[gray dust]

yeah ik it's lin alg :>

[earnest zinc]

ok ok

[gray dust]

so ya, first get the eigenvals and vecs then lemme know

[earnest zinc]

actually ill see if youre on in an hour or so

i gtg but ty for giving me where to start

i might be able to derive steps

[gray dust]

alright np, i'll be around

[earnest zinc]

: D

woof

[proud compass]

you taking college classes in high school?

smart boi over here

[earnest zinc]

yeah just CS and math

oh ty

[proud compass]

im 3rd year uni just now in linear lol

[earnest zinc]

oh really?

im 3rd year too basically

just high school

[proud compass]

decided on math major last year

[earnest zinc]

nice, good decision

oh shoot brb

[proud compass]

yeah cs + math should get you very far. gl with school man

[earnest zinc]

@gray dust

[gray dust]

sup

[earnest zinc]

ooo

hi hi

[gray dust]

[earnest zinc]

ok so i did the eigenval vecs

and i got -1 as the eigenval and [1;1] as the eigen vector

[gray dust]

https://cdn.discordapp.com/attachments/540211747613704221/618540560856252426/image0.png

which problem are you doing?

[earnest zinc]

1

[gray dust]

ok lemme see, one sec

hold up

only 1 eigenvalue?

[earnest zinc]

ohm

well the thought the eigen vector was C just in general

then i multiplied A and C

then found what value would relate the product of A and C, back to C

which i found was -1

[gray dust]

C?

oh don't think about C yet

that's part of the diffeq part of the problem

[earnest zinc]

oh

brajrj how do i start

[gray dust]

the way to find eigenvalues and eigenvectors should yield 2 evals for this matrix

and 2 corresponding sets of evecs

[earnest zinc]

hmm

how

how do you find eigenvalues and vectors

if you give a bried summary it might come to me

ah

[gray dust]

do you know how to calculate determinant?

[earnest zinc]

yea

[gray dust]

do you know the identity matrix I?

[earnest zinc]

ohh

i know hold on let me recalculate

[gray dust]

i just want to be sure you know the way to do it

$\det(A- \lambda I) = 0$

[stoic python]

RokettoJanpu:

[earnest zinc]

okay eigenvalues are 1 and -1

to get eigenvectors, you then multiply A by the eigenvalues?

oh

wait nvm

[gray dust]

nah

for each eval you calculate A - (lambda)I

then let v be lambda's corresponding eigenvector

and solve this

$(A - \lambda I)v = 0$

[stoic python]

RokettoJanpu:

[earnest zinc]

i see

i got

0 -2 0
0 -2 0

while solving for the eigenvector

OH

[gray dust]

??

[earnest zinc]

okay i got it

at least i found eigen vec and vals

[gray dust]

the evals are 1, -1

what's your evec for 1?

[earnest zinc]

thats 1, 0

in vector vertical form thing

theen for -1 its 1,1

[gray dust]

you didn't use symbolab did you? 🙂

[earnest zinc]

nah

[gray dust]

ok cool

[earnest zinc]

i was just confused since i got 0 -2 0; 0 -2 0

[gray dust]

so let's keep those evals and evecs to the side somewhere

[earnest zinc]

so i thought that x1 was just completely gone, but its a free variable

and x2 is just equal to 0

[gray dust]

yikes lol

[earnest zinc]

ok

kept in the side

[gray dust]

let's name the evals and evecs though

[earnest zinc]

oh

cool

[gray dust]

$\lambda_1 = 1, \lambda_2 = -1$

[stoic python]

RokettoJanpu:

[earnest zinc]

: D

[gray dust]

their corresponding evecs will have the same subscripts

$v_1 = \langle 1, 0 \rangle, v_2 = \langle 1, 1 \rangle$

[stoic python]

RokettoJanpu:

[earnest zinc]

yes yes

[gray dust]

now we're at the differential equation part

[earnest zinc]

i know the equation of the solution or whatever

i just don't know what c1 or why e is in there

[gray dust]

had you have taken a DE course, you would know that if you see a DE of this form

$\frac{dy}{dt} = ky$

[stoic python]

RokettoJanpu:

[earnest zinc]

no

oh

[gray dust]

k is a constant

[earnest zinc]

yeah ive seen that

[gray dust]

this usually leads to the solution y to take the general form of an exponential

$y = ce^{rt}$

[stoic python]

RokettoJanpu:

[earnest zinc]

oh

[gray dust]

i think you might have done something like this at the end of calc 1, idk

[earnest zinc]

ya ya

[gray dust]

but you go into it a little deeper in a DE class

so when you deal with 1st order or 2nd order (meaning the highest derivative in the equation is the 1st or 2nd derivative) DE it's usually a good idea to assume the general solution(s) take the form

of y = ce^(rt)

[earnest zinc]

yes

[gray dust]

what we have here is a system of equations though, a bit different

we have two derivatives of different quantities, but those derivatives depend on both quantities themselves

$\frac{dx}{dt} = Ax + By \\ \frac{dy}{dt} = Cx + Dy$

[stoic python]

RokettoJanpu:

[gray dust]

it becomes a little trickier to solve because we have coupled rates of change

[earnest zinc]

yea

[gray dust]

so what we can do is still assume that, because this is a system of 1st order DEs, the solution(s) still take the general form y = ce^(rt)

but now we rewrite the system using matrices

[earnest zinc]

oohoh

[gray dust]

so if you imagine <x, y> as a vector in the xy plane

<x, y>' = A<x, y>

where A is a matrix containing all those constant coefficients A,B,C,D

then you apply lin alg skills to get A's evals and evecs

[broken hawk]

(that is horrible notation, <x,y> reads as a scalar product)

[gray dust]

and those things will be what we use to build our solutions to the system

[broken hawk]

(or a span)

[earnest zinc]

oh i see

does the t in e^rt alternate signs

like it does in the determinant

[gray dust]

not sure what that question means

[earnest zinc]

o

wai tnvm

[gray dust]

t is our independent variable

[earnest zinc]

mm

[gray dust]

so instead of just y = ce^(rt) we also have x = ce^(rt)

[earnest zinc]

mhm

[gray dust]

we have 2 evals and 2 evecs from matrix A, right?

[earnest zinc]

yeah

[gray dust]

that gives us 2 solutions to the system

and since you're in lin alg, i think i can tell you that we will be taking linear combinations of these solutions

the linear combo of two solutions to the system is also a solution 🙂 (to be explained later if you take DE)

[earnest zinc]

is this the answer

x(t) = 1[1 0]e^t + 1[1 1]e^-t

asuuming you transpose the row matrix things

[gray dust]

hold on a second

what i should actually do is not use the terms x and y

[earnest zinc]

i just used everything we named and plugged into the equation

oh

[gray dust]

but x_1 and x_2

[earnest zinc]

x1(t)

[gray dust]

and x will be a vector made up of components x_1 and x_2

[earnest zinc]

h m m

[gray dust]

so here's the solution, ready?

[earnest zinc]

: )

[gray dust]

$x = c_1 v_1 e^{\lambda_1 t} + c_2 v_2 e^{\lambda_2 t}$

[stoic python]

RokettoJanpu:

[earnest zinc]

thats the eq i used

[gray dust]

there you go

i just wanted to give you some idea of where this comes from

oh yeah, and c_1 and c_2 are arbitrary constants that we solve for when we apply initial conditions

[gray dust]

,rotate

[earnest zinc]

: ( or : )

[gray dust]

👍🏽

[earnest zinc]

OH

ARE YOU A TEACHER

BECAUSE HOT DOG YOU SHOULD BE

phd in benevolence

[gray dust]

i've considered it 🙂

but i am not

[earnest zinc]

why

do you have phd

[gray dust]

yes i do

[earnest zinc]

you spontaneously worked me through the problem

[gray dust]

in benevolence 🙂

[earnest zinc]

: D

ty for the big teacher help

[gray dust]

yup no problem!

[earnest zinc]

: ) : ) : )

[gray dust]

you did pretty well for a hw assignment like this

[earnest zinc]

: D : D

[gray dust]

to be fair, most of it was lin alg stuff you already know. it's just i don't know why the teacher had to turn it into a system of DEs

[earnest zinc]

such kind words from teacher man

yes current guy bad man

thanks again teacher man bye!!!

[gray dust]

yep no prob, if you ever run into trouble with the rest of your hw feel free to ask here 🙂

[carmine terrace]

Hey guys, what are the rules for row echelon on solutions?

https://i.imgur.com/0zBD3vt.png

i set it in the form. but from what I remember, when you get a row full of zeroes that = to no sols

if it's 0 0 0 1 does that mean it's just one solution? or no solution?

[grave plank]

is linear algebra done wrong good to study while taking a linear algebra class?

[gray dust]

@carmine terrace if you correctly applied row operations but end up with a row of zeros with a non zero number at the end, that's implying that 0x_1 + 0x_2 + 0x_3 = 1, which implies 0 = 1... which is a load of horse crap. we call that an inconsistent matrix, and it has no solutions

[carmine terrace]

^ thank you, that's what I was trying to remember. An inconsistent matrix

one more question, when does it have an infinite number of solutions?

[gray dust]

when you have at least one free variable

[carmine terrace]

alright sounds good, thanks again. I feel far more comfortable about my quiz tomorrow.

It's still the easy portion, so I better be ready.

[gray dust]

yup no prob man. good luck on your quiz!

[carmine terrace]

thanks!

[charred stirrup]

hi i want to do a CS degree and i have a question about linear algebra

this one is lin alg 2 (which i will take)

but i want to know if linear algebra 3 will be useful in CS at all

I don't know if the topics will be really high level academia that won't apply much

[sonic osprey]

Really depends on what field of cs you do

Obviously

But the first half of stuff is more theoretical stuff that might be useful if you do a lot of algorithm type stuff

The second half will be very, very essential for a lot of different cs fields

[mint surge]

I have just started linear algebra but I had taken a break off my math classes for a semester and I guess I forgot everything? I don't know if this belongs here... I'd like to know how the coefficient matrix and the constant vectors correlate

My initial thought was to take the rref of v1, v2,v3, v4, v5

But then I realized that does nothing with M

by the way its referring to rref the command in Matlab.

My second thought was to attach V_x to M_x to make a 6 by 5 then rref that, but that seemed more programmy than mathy

[wintry steppe]

i don't get the remark lol

i mean

what is $\phi_T: V \cross V^* \to \bR$

[stoic python]

matqew:

[wintry steppe]

and now does T((.), v) come about?

@barren plank decided to get a real education

basis was confusing me lmao

<- was the person with the crank la a few weeks ago

[carmine terrace]

I'm blanking on this matrix
https://i.imgur.com/kD4WiKf.png
So I break it down to reduced echelon form
1 0 3 -2
0 1 -1 -2

It's tripping me up, because I can't seem to solve the values for it.

[dusky epoch]

x_3 can be made free

[carmine terrace]

how so? IDK how to really proceed from here

[dusky epoch]

wait. it sounds like you fucked up the arithmetic already

maybe not, lemme check rq

[carmine terrace]

I'll recheck as well

[gray dust]

i got top row 1, 1, 2, -4

[dusky epoch]

no, everything is in order

false alarm

[gray dust]

did you add -r2 to r1?

[dusky epoch]

you have the equations
x_1 + 3x_3 = -2
x_2 - x_3 = -2

[carmine terrace]

I divided first row by 5

[dusky epoch]

you can make x_3 a free variable and express everything else in terms of it

your row reduction is fine.

[proud compass]

x_2=x_3-2
x_1=2-3x_3?

[carmine terrace]

yea that's what's confusing me ^

The 1 on row 2 is tripping me up

[proud compass]

once you hit a wall like that where it looks unsolvable, I think you can usually put it back into variable form

[gray dust]

$x_1 = -3x_3 -2$

[stoic python]

RokettoJanpu:

[proud compass]

what you had so far works I think

I think you're done messing with it in matrix form

[carmine terrace]

yea, cause I understand that I have to allow x_3 = t (I need in terms of r, s, t

but IDK what to do with the -1 from row 2

[proud compass]

oh

multiply it by an integer

wait no, what -1?

the -x_3?

[carmine terrace]

my bad, I mean row two completely

cause row 2 is 0 1 -1 -2

[proud compass]

row 2 can be expressed as $x_2=x_3-2$

[stoic python]

Ping Warrior:

[proud compass]

I think

so you just move it to the other side

[gray dust]

$\begin{bmatrix} 1 & 0 & 3 & -2 \ 0 & 1 & -1 & -2 \end{bmatrix}$

[stoic python]

RokettoJanpu:

[proud compass]

am I understanding that right? I'm kind of at the same level in my class so I could be wrong actually

[gray dust]

@carmine terrace this is the matrix you have at the end right?

[carmine terrace]

yes that's the matrix i got

[gray dust]

so this is your system now

[carmine terrace]

yea I'm not sure how to proceed ping. I'm stuck on this

yes that's my current system

[gray dust]

$x_1 + 3x_3 = -2 \\ x_2 - x_3 = -2$

[stoic python]

RokettoJanpu:

[proud compass]

this is equivalent to your matrix ^

[gray dust]

so all we gotta do is just isolate x_1 and x_2

[carmine terrace]

Ohhhhhh

[gray dust]

$x_1 = -3x_3 - 2 \\ x_2 = x_3 - 2$

[stoic python]

RokettoJanpu:

[proud compass]

and then you replace x_3 with t and get:
$[-3t-2, t-2, t]$

[stoic python]

Ping Warrior:

and then you replace x_3 with t and get:
$[-3t-2, t-2, t]$
```Compile error! Output:

! Missing $ inserted.
<inserted text>
$
l.11 and then you replace x_
3 with t and get:
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

LaTeX Font Info: Try loading font information for U+msa on input line 11.
(/usr/local/texlive/2018/texmf-dist/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2013/01/14 v3.01 AMS symbols A

[carmine terrace]

moment of truth, punching in the answer now

[proud compass]

oh god idk how to use the bot but that should be your answer lol

[carmine terrace]

got it

thanks guys.

[proud compass]

good shit dude, you had it the whole time just didn't know

best way to be stuck imo

[gray dust]

$x_3 = t, -\infty < t < \infty, \text{ your solutions are }(-3t-2, t-2, t)$

[carmine terrace]

it honestly is.

[proud compass]

whats your major?

[stoic python]

RokettoJanpu:

[carmine terrace]

so whichever variable is free from row 1, that's the value I solve for?

comp sci

I missed class so I fell a little behind

[gray dust]

if you can express the other variables in terms of a single other variable then let that variable be free, set equal to t or whatever, done

[proud compass]

whenever you can (and must) express a variable in terms of another, the one you're letting "vary" is free

[carmine terrace]

alright sweet, thanks.

[proud compass]

like y=2x
x is free and that eq can probably be expressed as the augmented matrix [1 -2 | 0]

[gray dust]

if x_1 = 2x_2

[proud compass]

does that work, roketto?

[gray dust]

x_1 - 2x_2 = 0

[1 -2 | 0] i guess lol

[proud compass]

now I can use that to mess with my non-math major friends

[carmine terrace]

it makes more sense guys thanks.

[gray dust]

no prob

[carmine terrace]

Just two more problems to go

[proud compass]

GL man. I spent like 6 hours on linear alg homework the other day, takes way too much practice and paper

[carmine terrace]

Thanks, I have a quiz tomorrow, but I'm pretty sure it'll be easier

[proud compass]

GL fam

[undone garnet]

$rank(M)=1$\
Prove that $trace(M)^k = trace(M^k)$

[stoic python]

Nguyễn Thành Trung:

[winter reef]

if dim M = 1obvious, if dim M =n and rankM = 1, then it's possible to write it in a form that only first row is non zero, right?

[undone garnet]

sorry

dimM = 1 obvious?

I don't understand a lot

[noble swallow]

Didn't you already prove that if rank(M)=1, then M^2=trace(M)M in an exercise you asked about previously?

[undone garnet]

woops

I just remember that

i prove

M^2 = trace(M).M

ahhh

well

then we can trace both side

so we get

trace(M^2)=trace(M).trace(M)=trace(M)^2

[noble swallow]

Yes you can also use M^2=trace(M)M to prove by induction the statement above for all k

[undone garnet]

$rank(A)=rank(A^2)=1, rank(A^3)=?$

[stoic python]

Nguyễn Thành Trung:

[undone garnet]

I think the solution for this problem is 1 >= rank(A^3) >= 0

am I right?

ah no

I think rank(A^3) = 1

because if rank(A^3)=0 implies that A^3 = 0 => trace(A) = 0 because A is nilpotent matrix
But A^2 = trace(A).A = 0, rank(A^2) = 1 contracdict

[noble swallow]

I think it's true in general that if r=rank(A^m)=rank(A^(m+1)) for some m, then rank(A^n)=r for all n>=m

[undone garnet]

how?

I know that

rank(A^m) >= rank(A^(m+1)) >= rank(A^(m+2)) >= ...

[noble swallow]

You can prove it I think. When we take powers of an endomorphism, the kernel can only get bigger or stay the same. If the latter happens, it keeps staying the same

For instance take ker(f)=ker(f^2)
Then ker(f^2) is a subset of ker(f^3), but we can also prove the other inclusion

[undone garnet]

hm..

I think that we always have

ker(A) <= ker(A^2) <= ker(A^3) <= ... so on

do I misunderstand?

[noble swallow]

No that's correct, but assume for a moment ker(A)=ker(A^2)

Then let us take a vector v in ker(A^3)

A^3(v)=0

[undone garnet]

I think I know what you're gonna do

😄

[noble swallow]

Nice lol, you proceed then

[undone garnet]

let me try

A^3v = A^2Av = 0 => Av in ker(A^2) = ker(A)

=> A.Av = 0 <=> A^2v = 0 => v in ker(A^2)

=> ker(A^3) <= ker(A^2) => ker(A^3) = ker(A^2)

is it right?

[noble swallow]

Perfect

[undone garnet]

and if we keep continue then we have

[noble swallow]

The same thing can be done in general for m,m+1,m+2

[undone garnet]

ker(A) = ker(A^2) = ker(A^3) = ... = ker(A^something)

[noble swallow]

Yes right

[undone garnet]

nice

I've learned a lot

thank you

[noble swallow]

You're welcome

So that reflects on the dimension of the rank, for your question

[undone garnet]

I'm a bit confused

is ker(A) means null(A) right?

because in rank nullity theorem they said rank(A) + null(A) = n for A is mxn

[noble swallow]

Uhm null(A) in that case would be dim(ker(A))

But normally I saw null being used to denote the null space of A

So if I'm not wrong normally null(A)=ker(A)

rank(A) + dim(null(A)) = n for A mxn

[undone garnet]

$A^2 = A$, show that $rank(A)+rank(I-A)=n$

[stoic python]

Nguyễn Thành Trung:

[undone garnet]

this's how I do

A^2 = A <=> A(A-I) =0

rank(A) + dim(ker(A)) = n

<=> rank(A) + rank(A-I) = n

<=> rank(A) + rank(I-A) = n

is it right?

just a bit confused that

I don't know how to show rank(A-I) = dim(ker(A)) though I know A-I in ker(A)

ah well

we also have

A in ker(A-I)

[noble swallow]

Ok yeah I agree

There's also an important theorem for which since A(I-A)=0, then V equals the direct sum of Ker(A) and Ker(I-A) if I'm not wrong

That would as well solve it I guess

[undone garnet]

so

AB = 0 then ker(A) + ker(B) = V?

right?

dim

dim(Ker(A)) + dim(Ker(B)) = dim(V)?

[noble swallow]

No it's a theorem about polynomials

I don't remember its name though

[undone garnet]

well

quite tough

well

I gotta go to bed

bye

[noble swallow]

Bye, sleep well!

[wintry steppe]

Hi

Can anyone help me?

[sonic osprey]

[sonic osprey]

Take a screenshot

[stone drum]

These images aren't helping us understand your problem.

[sonic osprey]

I told you to take a screenshot

[wintry steppe]

Sorry

I did

But they all just deleted for some reason/:

[sonic osprey]

I'm confused what you're trying to do

We're not here to do your homework for you

[stone drum]

Um. Is this a test?

In the video, it seems to say "Quiz"

In which case, no. We will not help you on a test.

And asking is actually grounds for being banned.

[broken hawk]

not just that, we’ve already told them so

https://discordapp.com/channels/268882317391429632/540211747613704221/618535958312124456

@stone drum

[stone drum]

Yeah. Ok.

Banned.

[thin bloom]

Can someone link me to a proof of why the pivots column of a matrix leads to one forming the basis vectors

I cant seem to find it

[sonic osprey]

basis vectors of what

[thin bloom]

basis vector of the column space

of the matrix

wait

I think khan academy has one

I'm unsure if its a proof tho

[prime knoll]

Does anyone know of a good linear algebra book? Preferably a PDF

[native lodge]

What level and flavor? Intro? Advanced?Applied? Pure?

[prime knoll]

Well I'm trying to learn it for programming, if that helps

[native lodge]

So intro applied might be suitable for you

[prime knoll]

I'm not new to linear algebra though, I do have a pretty good understanding of matrices

[native lodge]

Linear algebra and it’s applications by Gilbert strang can be a good for you as well

I would think an intro to applied would still serve you well

[prime knoll]

Alright, I'll look at those

Where should I look if I'm interested in learning about quaternions?

Alright I'll read that one then

Oh hell yea I just found the old one I was reading in my history

I didn't think I would be able to

Is this one good for my purposes? http://148.206.53.84/tesiuami/S_pdfs/Linear Algebra Done Right.pdf

"Linear Algebra Done Right"

[wintry steppe]

Hi, is there an applied linear algebra textbook in a PDF that I can reference to?

[prime knoll]

bro

scroll up

[native lodge]

LA Done Right is a pure approach

[prime knoll]

What does that mean?

[native lodge]

I don’t know how useful that will be for you

Pure meaning lots of theory and not too much talk about numerical examples or applications IRL

[prime knoll]

Oh alright

[gray glen]

what's the downside tho

[wintry steppe]

hey can anyone have a quick look at this just to make sure I have everything right

ty

[wintry steppe]

<@&286206848099549185> ^ ty

[plucky sapphire]

It'd be better to take screenshots and post here. I doubt anybody would even bother to download that pdf for you.

[wintry steppe]

k

ty

You can't be serious

wat wrong @wintry steppe

[native lodge]

Pretty big document to ask us to look at lol

[wintry steppe]

Can be made shorter

ye will do

I just write everything down and shorten it after

I just need to if it looks crank

[wintry steppe]

What is it for

Since you have no citation for any of the theorems that arent proved

[rigid cypress]

[wintry steppe]

which one?

is it if dim$(V) < \infty$ then $(V^)^ = V$

[stoic python]

quite so:

[wintry steppe]

Theorem 2

I mean iits obvious

I'm just saying

Idk what you're writing this for

can I just put it as definition

my own notes

so I get it q_q

Ah

Then you can do whatever you want

From the long exposition for definition of vector space

I thought you were giving it to students

I need to practise writing properly lmao

lmao

[rigid cypress]

these are personal notes?

[wintry steppe]

ye

[rigid cypress]

ahh

[wintry steppe]

When you are writing more seriously

You arent expected to cite or reference something so simple

Some things are taken as obvious

k ty

any other obvious problems?

ty for helping btw n.n

No looks fine

And now you know integrals are linear

Take this knowledge with you to measure theory and probability please

yeet ty

[idle grotto]

I am having issues with the greatest integer function, is it appropriate to ask my questions in this channel?

[dusky epoch]

#prealg-and-algebra

or #precalculus maybe

[untold quiver]

Is it a good idea to take linear algebra and calculus 2 together at same semester ?

[earnest zinc]

@untold quiver i took calc 3 and linear alg same time

was completely fine just to hw, no outside time should be needed besides that

just do the hw**

[untold quiver]

I am a computer science major so I don’t need cal 3 or cal 4

[valid crag]

theres no calculus 4

and you need calculus 3 yes

dont underestimate math

[untold quiver]

What ?

My school require cal 2 for computer science

[valid crag]

if youre in a good enough Uni/College you will learn cal 1 to 3

[untold quiver]

Cal 4 is differential equation

[valid crag]

no

Differential equations are just differential equations

no such thing as calculus 4

[frosty vapor]

erm

ive heard in some places they call diffeq cal 4

its weird

[valid crag]

very weird indeed

[untold quiver]

Hardest math of all

To me linear is the hardest

[valid crag]

Hardest to you

linear algebra isnt so bad

its actually pretty useful for computer science

[jagged saffron]

for : Show that a subspace of a finite-dimensional vector space is finite-dimensional.

can i just simply show that given a vector space V of dimension n, the dimension of any subspace w of V is at most n

therefore it has to be finite dimensional as well

[valid crag]

[jagged saffron]

would this be just |x|

i was thinking let $f_{x_1}(x) = 1 <-> x = x_{1}$ and 0 otherwise would be a basis for $x_{1},..x_{n} \in X$

[stoic python]

Victoria:

[sleek helm]

I don’t believe that works

Let B be a basis

Oh wait sorry read that wrong

But no in general this shouldn’t work

Let K be of dimension larger than |x~

|x|

Then each constant function to a basis element

Is linearly independent

But this is already bigger than |x|

But these functions I’ve described aren’t enough either as we can consider functions that aren’t aren’t constant

Let me know if you want another hint @jagged saffron

[jagged saffron]

Uhh

Say i have a function that maps f(x_1)= k_1 respectively for 1..n

Then its just the linear combination k_{1}f{x_1}+....

Isnt it?

[idle grotto]

x2 + y2 + 6x + 2y + 6 = 0 , would anyone be able to walk me through completing the square?

[sleek helm]

Ryan this is both the wrong channel

And someone is already talking

Be respectful

#prealg-and-algebra

Anyway

Anyway Victoria

Is k_1...k_n a basis for k?

You need to be able to make every function

[jagged saffron]

K_n are arbitrary values

In K

I can generate any function with the basis i described can i not?

Do you have a counterexample cause im not sure what is wrong with it rn

We are considering k^x

[sleek helm]

Can you re-describe your basis

[jagged saffron]

Ok so

$k^x={f | f:X->K}$

[stoic python]

Victoria:

[sleek helm]

Yeah

[noble swallow]

Mm, I think you answer is correct Victoria. Moreover, in the question K is probably meant to be a field. But in general, if K was a vector space of finite dimension over some field, then dim(K^X)=dim(K)|X|

[jagged saffron]

Let our basis be $ f_{x_1}..f_{x_n}$ where

[stoic python]

Victoria:

[jagged saffron]

<-> means iff

also oh okie

Thanks

[sleek helm]

Oh, sorry

I was assuming K was a vector space

An arbitrary one

Yeah this works

[noble swallow]

If I am not wrong, also, in general if X is a finite set, A^X can be given the same structure of A^|X|

[sleek helm]

I think in general the answer is |X|^dimK though

Functions from X to the basis

Where the basis is 1 here

But maybe we can do better

[noble swallow]

Mm, no because it would then be isomorphic to K^|X|, which has dimension dim(K)|X|

[sleek helm]

But yes that’s true mat

What’s the basis there?

Is there a citation for that?

[noble swallow]

About the dimension?

[sleek helm]

The thing I was saying is true is the structure one

Yeah dimension

Oh

No i see it

My suggestion isn’t linearly independent, you just need the same collection that Victoria described but let 1 vary of the basis of K instead

Giving dimK * |X| yeah

[noble swallow]

After all it's like taking
|X|-uples of elements of K. In case K has infinite dimension, probably instead isomorphic to K itself?

[sleek helm]

Assuming choice

I think that works

because your vector space is defined by its cardinal basis

And that shouldn’t change the cardinal

[noble swallow]

I see, I was thinking a similar thing

[sleek helm]

I’m only half sure that works though, functional analysis has a tendency to mess w intuition

[hasty summit]

Hello

im not sure if this goes here

but

How can I find out at what point a line crosses the x axis?

[sleek helm]

Not linear algebra

But if you have y=mx + b

Then the “x-axis” means y=?

[hasty summit]

yeah

[sleek helm]

Typo lol

no it was a question

What does y equal

[hasty summit]

wait im wrong

y=0

[sleek helm]

Plug it in

[hasty summit]

im doing a velocity graph

[jagged saffron]

Doesnt the zero matrix exist in every subspace

For 5,q2) why does the intersection of u,w is empty

Im trying to show if v is in both u and w, then 2v is not uniquely expressible as it can be 2v+0 or v+v or 0+2v

But i was thinking, doesnt 0 matrix exist in both u,w always?

[pallid swallow]

hmm yeah

there may be a problem there

[jagged saffron]

But if the intersection contains only zero matrix, then this holds right?

[pallid swallow]

and vector spaces can't be empty

yeah

[hasty summit]

@sleek helm

so

got y=8x-24

what do i do now

[sleek helm]

That’s a typo for sure, it means what you think it means

Go to #prealg-and-algebra Fernando

And @ me

[indigo cradle]

hi guys

how may i find a group of markov matrices that give the same steady state

with respect to this challenge problem

im supposing that im suppose to work backwards but i have no clue beyond that

[pallid swallow]

hmm

I'm pretty sure it has something to do with diagonalising your matrix

[indigo cradle]

i think u mean diagonalising a matrix to find its steady state but its asking for matrices with a particular steady state

@pallid swallow

[noble swallow]

(1,-1) is regardless an eigenvector, the condition determines the other eigenvector (up to multiples), which turns out to be (3/2, 1)
So if the matrix is [(x,y),(1-x, 1-y)] with 0<=x,y<=1, then we get y=(3/2)(1-x) with hence the condition 1/3 <= x <= 1 @indigo cradle

[late rampart]

If the eigenvalues of A are k1, k2, ..., k(n), are ALL the eigenvalues of A^2 necessarily k1^2, k2^2, ..., k(n)^2 ? Do A and A^2 always share the same eigenvectors?

^ I'm asking this because I'm trying to understand why trace(A^k) is the sum of the eigenvalues to the power k.

[pallid swallow]

Well, yeah

because if eigenvector v has eigenvalue k, A^2v=k^2v

[winter reef]

Isn't it alao becasue then in some basis you have an 'identity' matrix with eigenvalues on the diagonal?

then you can easily take it to any power

[broken hawk]

that said, if v is an eigenvector of A², that doesn’t mean it’s one of A. consider rotation by 90 degrees in ℝ². the matrix representing that map has no eigenvectors at all, but its square, rotation by 180 degrees maps every vector to minus itself, and so every vector is an eigenvector with eigenvalue -1

(of course this is mostly a consequence of ℝ not being algebraically closed, said rotation matrix {{0, -1}, {1, 0}} does have eigenvectors over ℂ and the eigenvalues are ±i, which indeed squares to -1)

[late rampart]

@broken hawk thanks a lot! That solves it.

[broken hawk]

(I believe in ℂ the implication should hold both ways but don’t quote me on that)

[late rampart]

I know that if the matrix is not invertible though, every eigenvector of A^2 is one of A. I mean that one I can see why it would be true.

[winter reef]

ive had a very similiar question on final LA exam lol

[late rampart]

xD. Yeah it would be a good exam question haha.

I was thinking of looking at the Jordan normal form representation of the matrix. That would probably show it off.

[broken hawk]

wait what is the claim? if A is not invertible, then every eigenvalue of A² is one of A?

[late rampart]

No.

[broken hawk]

I mean that’s already false cause of a missing square somewhere

[late rampart]

Wait I'm asking in general. But I know it's true if A is not invertible.

Eigenvector also

Not eigenvalue

Sorry I corrected it.

[broken hawk]

I’m not sure how invertibility comes into play though, consider the matrix $A = \begin{bmatrix} 0 & -1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}$. This is not invertible but otherwise it should give the same issue as with the 2D case

[stoic python]

Sascha Baer:

[late rampart]

Well assuming we are in a closed field though.

[broken hawk]

oh

[late rampart]

Invertibility comes into play through minimal polynomials.

Oh nevermind. I was wrong about that

Lmao

For some reason I thought that if A satisfies some polynomial which doesn't have a constant coefficient then so should A^2. That's not generally true though haha.

Well it is true if the minimal polynomial of A does not have any term of odd degree.
For example if the minimal polynomial of A is of the form: x^2-1, then the minimal polynomial of A^2 divides x-1, which means A^2 necessarily shares the same eigenvalues.

[undone garnet]

if A = P^(-1)DP

can we prove that det(A-D)=0?

[dusky epoch]

no

because det(A-D) need not equal 0

let A = diag(1,2,3,4,5,6) and D = diag(2,3,4,5,6,1), for example

A - D = diag(-1,-1,-1,-1,-1,5), whose determinant is certainly not zero.

[alpine gulch]

hey can anybody help me with this, it's part of some practice questions and i'm not too sure how to do it https://i.gyazo.com/3316deda404f7430d2fb6f4b08142944.png

[sleek helm]

i'll offer a first step, what happens if you take z1,z2 to be real numbers

[surreal horizon]

hey guys, i coudl really use some help on this

i know to find the domain you gotta focus on the bottom portion of the fraction

[half ice]

,w zeroes of x^2 - 10x + 24

[stoic python]

Display results online and refine query

Requested by Kaynex#7520

[pallid swallow]

so, what does that mean?

[half ice]

Oh, you put an interval in

[pallid swallow]

@surreal horizon that the roots of the denominator are 4 and 6

The function can't be evaluated when x=4, 6.

[surreal horizon]

so what should i put :/ i understand all that but what of the domain

[heavy glacier]

so I thought that det(A^k) = (detA)^k but I seem to be running into an issue with this matrix

unless I made an error, I'm getting A^3 = I_3

and det(I_3) = 1 since its triangular

[sleek helm]

Your matrix multiplication is correct

Maybe you got diet A^k wrong

Det*

[heavy glacier]

oh nvm I see it now

detA is 1 by a very easy cofactor expansion

now it makes sense

[sleek helm]

Yep

[merry shuttle]

I've just started learning lin alg, so this question might be kind of basic. We're given the following matrix A and vector u, and the question I'm trying to figure out is how I would modify A to make a new matrix called B so that Bu is the projection of u on the y-axis

im just stumped on how i would go on about to modify A

[slow scroll]

im not sure what you mean by "modify A", but understanding this might help:

$\begin{pmatrix} 1&0\0&0 \end{pmatrix} \begin{pmatrix} a\b \end{pmatrix} = \begin{pmatrix} 1\0 \end{pmatrix} a + \begin{pmatrix} 0\0 \end{pmatrix} b$

[stoic python]

kxrider:

[merry shuttle]

okay what i mean is that A is supposed to be adjusted to a new matrix B that is supposed to be [0 b] (vertically)

[slow scroll]

like, adjusted by row operations?

[merry shuttle]

i guess, all ive really learned that could help was transposing but it doesnt help me here

[slow scroll]

well, you know what matrix you need: $\begin{pmatrix} 0&0\0&1 \end{pmatrix}$. You could apply transformations that switch the columns of $A$ and the rows of $A$

[stoic python]

kxrider:

[slow scroll]

maybe think about the matrices that you could multiply to the left/right of A that do this

[thorn tangle]

Ho Ho i made it through the first 3 weeks of linear algbra ;p

anyone else just getting to upper & lower triangles ?

[pallid swallow]

well, if there's a question, just ask

[wintry steppe]

hello could anyone give me a hand with finding a model that shows tax in terms of price

[latent minnow]

what does it mean for S to be directly proportional to p?

[pallid swallow]

Also, this is closer to precalc than linear algebra

[wintry steppe]

ah sorry its listed under linear variations in my course

it just means it varies directly

so if x varies directly/ is directly proportional to x

it would look like y=cx

c being a constant variable

[latent minnow]

constant variable is kind of an oxymoron but yes c is a constant

so in terms of the variables in your question...

[wintry steppe]

like for the 2nd half of the question

i found the tax was .07

290 x .07

=20.3

+the orginal 290

[latent minnow]

just take it one step at a time

[wintry steppe]

ok

yeah i feel as tho its super simple

and im just way over thinking it

[latent minnow]

construct an equation involving S and p using the fact that S is directly proportional to p

[wintry steppe]

s=p.07 ?

[pallid swallow]

S(p)=0.07p might be slightly cleaner way of writing it

because some people would misinterpret . as multiplication and some computers might misinterpret 07 as octal

[latent minnow]

well being directly proportional only gives you a c, not 0.7 or whatever

[wintry steppe]

oii

that worked

ty

and ty for not just spoon feeding it