linear-algebra

[pulsar turret]

so is this just one line?

or well not infinite line but it stops somwhere

[dusky epoch]

what

$v = \sum_{i=1}^k t_i w_i$?

[stoic python]

Ann:

[pulsar turret]

yeah

[dusky epoch]

that's an expression of v as a linear combination of w_1, w_2, ..., w_k

with coefficients t_1, t_2, ..., t_k

is that the answer you were looking for?

[pulsar turret]

not really could you dive deeper into it

[dusky epoch]

there's nothing else that can be said without further context

[pulsar turret]

ok I am wondering if this vector v would be a one dimensional vector if w is just a point

[dusky epoch]

???

there is no "w"

there are w_1, w_2, etc

what are you trying to say when you say "one-dimensional vector"

[pulsar turret]

I mean you can't break it into smaller vectors

because like most basic vectors has an x and y component doesn't it?

[dusky epoch]

what do you mean by "break it into smaller vectors"

in no vector space is there a vector inexpressible as the sum of two other vectors

[pulsar turret]

ok wait let me draw a graph of what I think is hapening here

[dusky epoch]

because like most basic vectors has an x and y component doesn't it?
what's an "x component"? what's a "y component"? are you working in R^2, with the standard basis {(1,0), (0,1)}? then every vector has an x component and a y component.

you're not giving much context, so it's very hard for me to tell you anything constructive.

are you working on a problem right now?

are you trying to make sense of a theorem?

what is it, exactly, that you're doing?

[pulsar turret]

trying to make sense of the theory

[dusky epoch]

"theory" and "theorem" are two different words

can you show the theorem or lemma or result or whatever you're looking at right now?

in full?

[pulsar turret]

whats the difference?

[dusky epoch]

a theory is an entire corpus of knowledge

while a theorem is one statement, with a certain hypothesis and a certain conclusion.

in any case.

please show me the thing you're looking at, whatever it may be called.

[dusky epoch]

,rccw

[dusky epoch]

okay... so you're looking at the definition of a linear combination.

[pulsar turret]

yup

[dusky epoch]

i mean

[pulsar turret]

So I was just playing with it and i was wondering how v would change if you change w

[dusky epoch]

yeah, that's it really ¯_(ツ)_/¯ the two basic operations in linear algebra are addition and scaling, and when you do that to a bunch of vectors, what you get is called a linear combination

a simple example i guess

[pulsar turret]

could it also not be that w1 and w2 are on top of eachother?

[dusky epoch]

what do you mean by "on top of each other"

could they be parallel, or even coincident? sure, they can be whatever you want. just because i drew mine like this doesn't mean it always looks exactly like this

again, watch 3b1b's essence of linear algebra. his illustration skills are way better than mine.

[pulsar turret]

Ok I think I got it

Just overthinking

[pale shell]

What is the proper notation for span?

[cunning forum]

This was how my linear algebra book expressed it.

[sullen pollen]

Given this matrix. I need to find a orthonormal matrix M so that M^tAM is a diagonal matrix. How do I do that?

[empty copper]

,rccw

[cunning forum]

Thank you

If you need to make a orthonormal matrix have you tried to find the orthonormal basis?

[sullen pollen]

How do we do that

[cunning forum]

Is $M^t$ the transpose of $M$ or the inverse? Because I'm familiar with diagonalizable when it comes to $Q^{-1}AQ$ but not with $M^t$

[stoic python]

Jules Trayus:

[empty copper]

When $M$ is orthogonal, $M^\top=M^{-1}$

[stoic python]

Rijinaru:

[cunning forum]

Oh in that case we need to start by finding the eigen values and eigen spaces for $A_2$

[stoic python]

Jules Trayus:

[sullen pollen]

So the eigen values are: $det(A- dIn)$

[stoic python]

iiNathanz:

So the eigen values are: $det(A- dIn)$

[cunning forum]

The eigen values are values of $\lambda$ such that $\det(A_2-\lambda\cdot I)=0$

[stoic python]

Jules Trayus:

[sullen pollen]

Ah yes

So I need to work out the determinant and find values for lambda

[cunning forum]

Yup.

[sullen pollen]

okay i found: 1, 1 and 4

[cunning forum]

So now you'd want to find the eigen space

[pale shell]

Span(a,b)= R2 but what else could the answer be to span

Like if it is collinear or just one vector

[dusky epoch]

the span of any set is always a subspace of the space your set's from

in R^2, the only possible subspaces are:

• all of R^2 (dimension 2)
• lines through the origin (dimension 1)
• only the origin (dimension 0)

[pale shell]

Ohhh so it would either equal R1, R2, or R0?

[dusky epoch]

no

[cunning forum]

Nathanz you want to hop over to #help-4 and we can take this further there?

[sullen pollen]

Yes

[dusky epoch]

R^1 is what some people use to denote the real number line, which is distinct from and doesn't have any points in common with R^2
R^0 just doesn't make any sense

[pale shell]

So what is the notation then

[dusky epoch]

well a line is most commonly just specified as the span of a single vector

[pale shell]

Okay so span(a)=ca

[dusky epoch]

no

[pale shell]

Loll

[dusky epoch]

span(a) = {ca | c ∈ R}

[pale shell]

Omg

Ok

And if they are collinear

[dusky epoch]

who

who are "they"

[pale shell]

Span(a,b)={ca|cER}

A and b

[dusky epoch]

if you can't type ∈ please at least use the word "in" instead

also math is case-sensitive

[pale shell]

Yes sir

[dusky epoch]

and don't call me sir because i am not one

[pale shell]

How do you type that

Do you copy paste

Or is it a keyboard

[dusky epoch]

no, i have autohotkey.

[pale shell]

Okay so

Is that right

[buoyant viper]

Guys can you help me I don't understand espace vectoriel lecons please .I have a test tomorrow

[pale shell]

Span(a,b)={ca|c member of R}

[dusky epoch]

yes, if a and b are collinear, the span(a,b) = span(a) [= span(b)].

[pale shell]

Ohh ok

And for 0 it would just be equal to the zero vector

[dusky epoch]

span(0) = {0}.

[pale shell]

Yea

Thanks linear algebra is fun imo

Also did u take the test

[buoyant viper]

@pale shell. No I will take it tomorrow

[pale shell]

U said that you had an important test or something some time ago

[buoyant viper]

I understand matrice and system linear but I found espace vectoriel so difficult

[dusky epoch]

my exam is on the 22nd.

[terse kestrel]

Hi, how do I find the values of a parameter such that given system of equations has an unique solution?

x + λy     = λ
-x + y + λz = λ^2```

[dusky epoch]

you want $\det\begin{bmatrix} \lambda & -8 & -8 \ 1 & \lambda & 0 \ -1 & 1 & \lambda \end{bmatrix} \neq 0$

[stoic python]

Ann:

[half robin]

How do I even begin with this .-.

I am following this ML course

and the first programming assignment wants me to do that

To implement finding euclidean distance in an efficient manner

That's what x and z are

I can't even find what a Gram matrix is

Some help please?

[cunning forum]

Are you familiar with what an inner product is?

[wintry steppe]

Hi. I'm 12, and i'm self studying linear algebra through a book by Ray Hoffman. I'm kinda stuck in the second chapter, because I can't remember the definitions and theorems. Is there an intuitive visualization of concepts like span, basis, and finite-dimension?

[cunning forum]

https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

[dusky epoch]

I'm 12,

discord's 13+, kiddo

[cunning forum]

🍌 🔨

[wintry steppe]

I mean, I turn 13 in March, so I guess it's fine.

Thanks for the link, by the way.

[half robin]

@cunning forum yeah I am familiar with inner products

[pale shell]

What the hell is eigne

[wintry steppe]

Having some trouble with this question

[echo quail]

how did you define addition of maps?

[wintry steppe]

T(u1 + u2) = T(u1) + T(u2)

[echo quail]

that's right

oh nope

f and g in Hom(U, V)

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

prove that f+g is linear

then prove that this new + is associative, commutative, give 0-map

construct opposite of f, such that f + (-f) = 0 and prove its in Hom(U, V)

then you construct and verify the rest of axioms for scaling and distributivity

[wintry steppe]

when you say f and g in Hom(U,V) are f and g linear maps?

[echo quail]

yes

Hom(U, V) is the set of all linear maps

[wintry steppe]

that take U to V?

[echo quail]

correct

[wintry steppe]

ok that makes sense

so essentially I prove the axioms of a vector space but with linear maps instead of vectors?

[echo quail]

yes

[wintry steppe]

why must I prove that (f+g) is linear?

[echo quail]

so, Hom(U, V) is closed under +

[wintry steppe]

oh I see

it shows that f + g is in hom(u,v)

[echo quail]

ye

[wintry steppe]

ok I get it, thanks @echo quail

[pale shell]

How to prove two vectors span r2

[feral grove]

using what information

[pale shell]

The vectors in component form

[feral grove]

this is trivial if you've proven things about the relationship between bases and dimension

i mean like in class

like what have you proven that you can use

[pale shell]

Idk

[terse mirage]

do you know what linear independence is?

[pale shell]

Yes

[feral grove]

i mean like 2 linearly independent vectors in R^2 span R^2

[pale shell]

But like he wants the system of esuation method

[feral grove]

so using systems of equations show every vector can be written as a unique linear combination of your two vectors

this is easy if you've proven this statement is equivalent to the only linear combination that gives you the zero vector being the trivial one

[pale shell]

So basically

I write out cv+cw=xy and then I write the equations but I keep solving them wrong to get the coefficients

[feral grove]

i mean ok then this is just a computation problem

which i mean it's tough to tell where you're going wrong

[pale shell]

Yeah

[sharp merlin]

How is this wrong

I have to do cross product of first two and then dot product of 3rd one times result right?

[nimble egret]

Volume

Negative volume?

Surely take the absolute value of the result?

[sharp merlin]

oh shit

thanks

[hoary zinc]

Hey guys, i'd be happy if someone could take a look at #help-6 for a minute

[fossil mortar]

How could I solve the bottom one thats marked on yellow?

So far I've separated them into "real" and "complex" numbers

rn I have: "2/(7e^6) * i/(e^i)"

[quartz compass]

you can't split it like that, that's not how exponent rules work

[fossil mortar]

how about now?

i think i mistyped the first time

[quartz compass]

ok better

[fossil mortar]

now im kinda stuck idk what to do next

[quartz compass]

you're trying to get it of the form r*e^{i theta}

[fossil mortar]

ye

[quartz compass]

so you have some things that are kind of like that with the i/(e^i) part, try to make that more like e^{i theta}

[fossil mortar]

could i move the bottom up? w negative exponent?

[quartz compass]

yep

[fossil mortar]

i see, but wouldnt that delete the "i"

[quartz compass]

you still have to deal with the i

i*e^{-i}

[fossil mortar]

ah true

[quartz compass]

how do you represent i in polar form

[fossil mortar]

seems like i didnt do much

uh

sec

oh as "y coordinate"

srry this is kinda hard for me to grasp

[quartz compass]

have you learned vectors before?

[fossil mortar]

not rlly

[quartz compass]

think of 'r' as being the length and e^{i theta} being the direction

[fossil mortar]

like i've studied a bit on my own

oh ok

yeah like r=radius and e^{i theta} is the angle

or something like that i remember from class

[quartz compass]

sort of, theta is the angle

e^{i theta} is more like a unit vector pointing in the direction

$e^{i \theta} = \cos(\theta) + i \sin(\theta)$

[stoic python]

Merosity:

[quartz compass]

you can think of these as its x and y coordinates

[fossil mortar]

huh okok

i see

[quartz compass]

so what is the 'length' of i

how do you find |x+iy| in general? Just do that

[fossil mortar]

1? because i have: e^{-i} which is re^{-i} where r=lenght?

[quartz compass]

|i| = 1

[fossil mortar]

i think its r=sqrt(a^2+b^2)

[quartz compass]

but this is entirely unrelated to |e^{-i}| = 1

[fossil mortar]

oh

[quartz compass]

these are two numbers multiplied together

so now we have |i|=1 what do we know about the angle?

you said it was in the y direction earlier, what angle is that

[fossil mortar]

90deg?

[quartz compass]

yep

[fossil mortar]

cause x=0?

oh pog

[quartz compass]

exactly

[fossil mortar]

so rn im only doing the "i" portion of "(i)*(e^-i)"?

[quartz compass]

so now you write it as i = e^{i theta}

what's theta

yeah just the "i" portion

[fossil mortar]

oh i see

[quartz compass]

e^{-i} is taken care of really

[fossil mortar]

e^i90

[quartz compass]

although we can adjust it too

no, in radians

[fossil mortar]

oh

e^1/2pi

btw why radiants

just curious

[quartz compass]

because it's gross

lol

[fossil mortar]

xD

[quartz compass]

wrong

[fossil mortar]

oh

[quartz compass]

fix it did you see your mistake

[fossil mortar]

e^i(1/2pi)

[quartz compass]

better but

write e^{i*pi/2}

or use parenthesis at least

[fossil mortar]

oh okok

[quartz compass]

usually people like theta to be between 0 and 2pi

so your other angle of e^{-i} could be fixed but, well it's up to whoever is asking the question it doesn't matter to me

[pallid rampart]

Hmmmmmmm

[quartz compass]

well ok first, what's your final answer

[fossil mortar]

ok so, i have (e^(pi/2))*(e^(-1))

oh and the other part

the 2/(7)(e^(6))

that i just kinda left there

[quartz compass]

that's fine as it is

[fossil mortar]

im guessing thats my R

bc it has no i

[quartz compass]

although you used parenthesis wrong

[fossil mortar]

oh how come

[quartz compass]

it's 2/(7*e^6)

[fossil mortar]

ohh ok

[quartz compass]

as you wrote it, it's like e^6 is in the numerator

or just ambiguous

[fossil mortar]

ok so r=2/(7*e^6) and theta= (e^(pi/2))(e^(-i))

theta= e^(pi/2)-i

[quartz compass]

no, the angle is in the exponent

e^{i *theta}

that whole thing is not the angle

[fossil mortar]

oh so i have 2 angles rn right?

[quartz compass]

that's like a direction vector

[fossil mortar]

have to multiply them

[quartz compass]

there is only one angle

[fossil mortar]

and then take that one?

[quartz compass]

yeah you have two angles you need to combine yeah

[fossil mortar]

oh okok

theta= (i)(pi/2)-(i)

so theta=pi/2

or did it get lost

cause im not sure where the "i" were

[quartz compass]

yeah you are warmer but still wrong

[fossil mortar]

e^{i*pi/2}

[quartz compass]

$e^{i \theta} = e^{i \pi/2}*e^{-i}$

[stoic python]

Merosity:

[fossil mortar]

e^{-i}

ok

[quartz compass]

$e^{i \theta} = e^{i (\frac{\pi}{2}-1)}$

[fossil mortar]

jsut wanted to paste them here

wah

[stoic python]

Merosity:

[fossil mortar]

i understand the first one

this: $e^{i \theta} = e^{i \pi/2}*e^{-i}$

[stoic python]

Mekbot:

[fossil mortar]

but where did the -1 come from

on the second one

only thing i dont get

[quartz compass]

try to see if you can figure it out

[fossil mortar]

ok

[quartz compass]

combine the exponents

[fossil mortar]

oh

factoring

the i out

[quartz compass]

yeaH

[fossil mortar]

pog

[quartz compass]

so what's theta = ?

[fossil mortar]

(pi/2)-1

[quartz compass]

perfect

[fossil mortar]

damn that was kinda hard

so does there always have to be an "i" multiplying the exponent?

[quartz compass]

yeah

[fossil mortar]

like lets say in the rectangular form: a+bi

[quartz compass]

that's where the angle will be

[fossil mortar]

I have something like: 24

so there is no i, bc i=0

[quartz compass]

then it is r=24 and theta = 0

[fossil mortar]

i would still write 24e^(i)

[quartz compass]

if it was -24 you'd put r=24 and theta = pi

$-24 = 24 * e^{i \pi}$

[stoic python]

Merosity:

[fossil mortar]

yeah makes sense

-24 would mean 180deg

[quartz compass]

no you would put 24e^{i0}

not i

[fossil mortar]

oh ok

[quartz compass]

that would be 1 radian

[fossil mortar]

true

[quartz compass]

I think the most straightforward way to do this if you're given z is to find what |z| is first

[fossil mortar]

mmm

[quartz compass]

then divide that out and just figure out the angle by putting it into the form e^{i theta}

[fossil mortar]

wym by that

absolute value?

[quartz compass]

so if z = x+iy then |z| = sqrt(x^2+y^2)

[fossil mortar]

oh

didnt know that

[quartz compass]

you don't actually have to evaluate this directly

like we ignored the real numbers automatically

there are harder ones though, like for instance 1+i

in some ways, it may be easy or hard depending on what you know

[fossil mortar]

huh

[quartz compass]

what's r and theta for 1+i?

[fossil mortar]

so r= sqrt((1^2)+(1^2))

so r=2?

[quartz compass]

it's the pythagorean theorem, really

you're missing a + sign

[fossil mortar]

and the theta

[quartz compass]

r is not 1

[fossil mortar]

oh

2

and theta?

i know tangent is

[quartz compass]

r is not 2

[fossil mortar]

op/adj

[quartz compass]

yeah use tangent to find theta good

[fossil mortar]

r=sqrt(2)

[quartz compass]

correct

[fossil mortar]

i thought i'd heard

it was

arctangent

tan^-1

but i dont recall why

[quartz compass]

same thing

[fossil mortar]

or maybe im mistaken

does it change the soh cah toa order?

[quartz compass]

doesn't change anything

[fossil mortar]

like if tangent is o/a

is tan^-1 a/o?

[quartz compass]

x+iy you have real part is x and imaginary part is y

[gray dust]

one may just plug into arctan for Re(z)>0, otherwise atan2 works nicely

[quartz compass]

so tan(theta) = y/x

[fossil mortar]

oh

[quartz compass]

just like normal

[fossil mortar]

what does Rokabe mean?

by that

[quartz compass]

eh

if you're programming I would teach you this function

[fossil mortar]

so in the 1+i example

oh im not programming yet

[quartz compass]

if you already have it in rectangular form, you can just get the correct quadrant to fix what angle it has to be in

[gray dust]

if $x>0$, then $\theta=\arctan\br{\frac yx}$, otherwise use $\theta=$ atan2$(y,x)$

[stoic python]

RokettoJanpu:

[fossil mortar]

oh

whats "a" in the atan2

just the "a" from (a+bi)?

[quartz compass]

no, just short for 'arc'

[fossil mortar]

oh

why the comma

[gray dust]

i guess it's a bit much to introduce the atan2 function at this time, mero can help you sort through the sign casework w/o it

[quartz compass]

it's part of the true inverse to the coordinate transformation, x=r cos theta and y=r sin theta

r =sqrt{x^2+y^2} and theta =atan2(y,x)

[fossil mortar]

damn

idk if im ready for that tbh

[quartz compass]

yeah don't worry about it, just focus on the complex numbers for now

[fossil mortar]

so in the 1+i example

[quartz compass]

doing it by regular trig like you were taught is good enough

[fossil mortar]

theta would be

tan(y/x)?

[quartz compass]

no

[fossil mortar]

actually, arctan(y/x)

[quartz compass]

yeah

[fossil mortar]

aight i think im ready-er

for tmorrow

ty

one last thing if i may, a friend sent me this, but im pretty sure he shouldt be able to do that right?

where he gets 3(e^6*e^(ipi))

and multiplies the 3

on both

[quartz compass]

I have to go

[fossil mortar]

shouldnt it just multiply one of them and never turn into a 9

oh dw

imma go to sleep then, thank you for the help, really appreciate it and i learned quite a bit

[feral grove]

this is just scalar multiplication and vector addition yeah

LADW can be tough if you haven't read a book like it before

you might have to reread chapters multiple times

it's a really good book, but it's dense and written like a "real" math textbook

uh not that i'm aware of

nah

most of them should be fine w/o a solution set, if you have any questions you can come here

1.7 and 1.8 are probs the most challenging from this set

that's ch.2

ch.2 is also probably the most simplified chapter of the whole book, so once you get through 1 2 should be pretty easy to work thorugh

also note an intro to lin alg class covers up to about halfway through ch.5

if you want to go more in depth you could after ch.4 read ch.9, then do ch.5 and ch.6

ch. 7/8 are pretty abstract, they're good, but probably harder than the rest of the book for intro to lin alg

[tranquil junco]

oh are we shilling linear books

[feral grove]

i just told him to read ladw not ladr

[tranquil junco]

are you a math major?

[feral grove]

i assumed he was when i made this recommendation

if you can read ladw it means you're ready to work through higher level books as well

[half robin]

I am following this ML course
and the first programming assignment wants me to do that
To implement finding euclidean distance in an efficient manner

Posting again cause didn't get any response yesterday .-.

[empty copper]

What have you tried?

[pale shell]

How to visualize linear transformations?

[empty copper]

3Blue1Brown is king of visualizing linear algebra

Watch his video series on linear algebra to see what I mean

[pale shell]

Also how do you draw one tho?

[dusky epoch]

draw what

[pale shell]

Draw R2 after a linear transformation

[dusky epoch]

well you could draw a grid like what 3b1b does i guess

[pale shell]

Yeah but would you just map out where î and j hat go so then you could fill in the rest of the lines like that?

[dusky epoch]

i mean... sure?

[pale shell]

Ok

[wispy delta]

so based on my understanding the spectral theorem allows us to find an orthonormal basis of eigenvectors

but are there any applications where we specifically require the basis be orthonormal?

e.g. for computing large matrix powers diagonalizability seems to be sufficient

[wintry steppe]

I remember it being helpful in computating projections and stuff

https://math.stackexchange.com/questions/518600/why-is-orthogonal-basis-important seems to answer your question

And iirc some theorems required orthonormal basis but not sure

[wispy delta]

ah that makes sense, thanks

[pale shell]

What do determinants do

[real wedge]

Please help me with these questions. Im not sure at all how to approach them

@pale shell easy way to find if your matrix is invertible

And if the matrix is invertible then you know a whole bunch of things about it

[feral grove]

sure yes the invertible matrix theorem is a thing but also you should prove every step of it

you want to find n linearly independent vectors all of whom have a as a root for 23

24 you have n+1 vectors, so if that set is l.i. then it forms a basis, argue why it can't form a bases

basis

[lone quail]

Why is the dimension of the nucleus=n-r where n is the dimension of the starting space in a linear function and r is the rank of the representative matrix of said linear function

I did it a long time ago and am revising but totally forgot

[brittle juniper]

so you have $E$ a finite dimensional vector space with $\dim E=n$, $F$ another vector space and $f:E\to F$ a linear map with rank $r$
\ \
you take $S\subset E$ a supplementary subspace of $\Ker f$ (I assume you know how to justify the existence of such a $S$), and your goal is to show that $S$ is isomorphic to $\operatorname{Im}f$
\ \
so you look at the linear map
$$\fun gS{\operatorname{Im} f}v{f(v)}$$
and show it's a bijection (and that's not too hard!)

[stoic python]

Tuong:

[woven shore]

can someone help me

I need to know how to solve 15xb>55

and have no place to start

[gray dust]

this isn't linear algebra. go to #❓how-to-get-help

[woven shore]

no one will answer me D: and ok

[pale shell]

Why does nobody care about geometric meanings

[feral grove]

is this referring to your question about determinants

[pale shell]

Ye

[gray dust]

linear transform, scaling factor, det

[pale shell]

The answer was basically a cool number

[feral grove]

determinants are generally introduced poorly

might be good to read ladw's ch.3 on it

[gray dust]

i can't tell if ladr or ladw gets rec'd more often

[feral grove]

well ladr doesn't get recc'd for intuition about determinants ever

so that's something

[gray dust]

3b1b focuses on visual intuition, not computations

linear eqns can be used, but not solely used, for data analysis, and linear eqns are perhaps a very big motivation for learning linalg concepts

[pale shell]

Does anyone know why the zero test for linear dependance works?

[gray dust]

wdym zero test

[feral grove]

you mean like for a set of vectors to be linearly independent, the only linear combination of them that equals the zero vector is when they're each multiplied by 0?

[pale shell]

Yes

[vast torrent]

what's your definition of linear dependence

[gloomy arrow]

In my school calc 2 is a prereq for linear algebra, why would that be?

[feral grove]

no material from cal 2 is necessary but most lin alg classes expect a level of mathematical maturity and use the calc 2 pre req as an attempt to achieve that

[nimble shuttle]

Hey guys! my exam is coming up soon and my teacher mentioned something about problems that combine both arithmetic and geometric sequences and series. However I don't understand what he meant by that as I have never encountered a question like this. have any of you encountered questions like this, and if you have can you please @ me or private message me as I do have a few questions. Thanks 🙂

[cursive narwhal]

Try Brilliant.org if you want some hard problems on that stuff

[native river]

Hi, Why is the dot product of the zero vector and (v,v^2,...,v^n) linearly dependent?

[gloomy arrow]

@feral grove Oh okay that makes sense

[vast torrent]

@native river that question doesnt make sense. The dot product gives you a number

[nimble egret]

Are you asking why a set containing the zero vector is linearly dependent?

[native river]

Yes

[nimble egret]

What's the definition of linear dependence?

[pale shell]

Whats up linear algebra peep

[native river]

A vector can be express as a linear combination, such that -cv= c2v2+c3v3+...+cnvn

[nimble egret]

Is that the definition of linear dependence?

[pale shell]

Every day i am becoming more and more advanced at linear aventa

Algebra

[nimble egret]

Considering I can set all those coefficients to 0

And get a valid equation

Are you sure that's the definition?

[native river]

How will you define linear dependence?

[nimble egret]

c1v1 + c2v2 + .... + cnvn = 0

Has a solution

With at least one of c1, .... , cn not zero

[native river]

Okay

[nimble egret]

From the definition

You should be able to see why a set containing the zero vector is linearly dependent

[unique stone]

i like nutting omg

give me a damn linear algebra question

or i will fuck your wife

till her pussy bleeds

i like licking period blood

so its all good

[native river]

@nimble egret can you please explain to me?

[unique stone]

its obvious dawg

if you have a fucking 0 vector

it makes the set linearly dependent

[nimble egret]

Well

Suppose v1 is the zero vector

And we set c2, ... cn = 0

So we have c1v1 = 0 right?

What value of c1 satisfies this equation?

@native river

[native river]

c1=0

[nimble egret]

Sure

That's one

Anymore?

Remember v1 is the zero vector

[native river]

The reals

[nimble egret]

Yes

Since all real numbers are a solution

Clearly there is a non-zero solution right?

Which satisfies the definition of linear dependence

[native river]

@nimble egret Thank you

[nimble egret]

np

[real wedge]

@feral grove thanks for the help, i think i understand how to do 24 now

Im still kind of confused in general about telling whether or not a set of polynomials are linearly independent or not

the normal basis {1,x,x^2,...} makes sense why theyre lin. ind. but other times its confusing to me how you can figure that out

[nimble egret]

Can't you just treat them like vectors and do the normal approach

Well

They are vectors

[real wedge]

Idk... yeah theyre "vectors" but not vectors like <2,4,1>

[nimble egret]

I mean you can just represent them like that for the sake of checking linear dependence anyway

[real wedge]

oh how do u do that?

[nimble egret]

1 + 2x + x^2 as <1, 2, 1>

[real wedge]

Ok

so for example how would u show that {x,x-2,x^2+x} is linearly dependent or not?

[nimble egret]

It's basically the same as

Is <0, 1, 0>, <-2, 1, 0>, <0, 1, 1> linearly dependent?

[real wedge]

i cant see that

why is that

I put three vectors there

[nimble egret]

Missed one

[real wedge]

oh lol

cx^2+(a+b+c)x-2b = 0 u can set it up like this?

[nimble egret]

Hmm

What's that?

[real wedge]

I did ax+b(x-2)+c(x^2+x) = 0

trying to see if that means a=b=c=0

to check if theyre lin ind.

then i re-arranged that equation

do u see what i mean?

[nimble egret]

I see

Yeah

You can do that

[real wedge]

then

cx^2+(a+b+c)x-2b = 0

does that equation above imply a=b=c=0?

[nimble egret]

Oh well

Ahh

Clearly c and b are 0

And it then follows that a must be 0

[real wedge]

right

[nimble egret]

Yeah, that works

[real wedge]

Cool

so I can just do that for any polynomial stuff i guess

[nimble egret]

Yeah

If you can reason about it properly

[boreal crescent]

hey

is someone up to help a poor lad

[dusky epoch]

[boreal crescent]

i know its false

need a counterexample

cant think of a function that obeys that rule

i proved the opposite of that statement so i know for a fact that its false, just dk how to show

[dusky epoch]

what do you mean by opposite

[boreal crescent]

i proved that

[dusky epoch]

so what you are saying is that you think it's impossible for two sets to be subsets of each other?

$A \subseteq B$ and $B \subseteq A$ aren't negations of one another, you know.

[stoic python]

Ann:

[boreal crescent]

i mean yea whatever

i know (f) is false though

cus (e) is true

ohhhhhhhhhhh

[dusky epoch]

no, you aren't getting away with "yea whatever" on this one, because (f) is true

[boreal crescent]

i see what you're saying

[wintry steppe]

This doesn't look like linear algebra, this looks like set theory.

[boreal crescent]

just cus one is true doesn't mean the other is false

gotcha

[wintry steppe]

You're confusing subsets with proper subsets?

[boreal crescent]

its in my linear algebra homework sir, maybe i am mistaken

wait (f) is true?

im so confused

this was the set of questions

c is false and i showed it with y = x^2, d is true and i proved it, and e is trie and proved it

[dusky epoch]

@wintry steppe no that is not what the problem is saying

[boreal crescent]

any hints on how to prove (f) @dusky epoch ?

[dusky epoch]

these are all set inclusions. prove them as you would prove a set inclusion.

let y in B. show that y is in f[f^-1[B]].

[boreal crescent]

hmm

[dusky epoch]

or maybe it is actually false and i fucked up.

[boreal crescent]

see this is confusing. if its false will it break during it?

cus c was false

and i just showed a counterexample to p=disprove it

i didnt do it abstractly, idk if the derivation will break midway

[dusky epoch]

try the derivation anyway

see what breaks

[wintry steppe]

If its true you can show that:

if let y in B. show that y is in f[f^-1[B]].
and
let y in f[f^-1[B]], show that y is in B

If its false, at least one of those statements must be false

[boreal crescent]

yea so i showed that your bottomm statement is true

i guess ill try the top one

cus the only example i can think of that will break the top rule is a relation

not a function

for example x = y^2 breaks the top rule

[wintry steppe]

This looks like set theory to me, so idk what you mean by function in this example

Oh you literally mean that

Well yeah that's a valid example

[boreal crescent]

yea but x = y^2 isnt a function

[wintry steppe]

?

[boreal crescent]

??

its a relation, it aint a function boss

[wintry steppe]

I'm just thinking of this in terms of set theory, so I'm trying to give hints

[boreal crescent]

gotcha yea

this is confusing cus my course is trying to merge the two

its intro to lin alg so i guess they're teaching a bit of evreything

[lone quail]

Why is the dim(ker(f))=n-r ? (where n is the dimension of the starting space in a linear function and r is the rank of the representative matrix of said linear function)

(I’m searching for the proof or just the reasoning behind it)

[brittle juniper]

Take a supplementary subspace of ker(f) and show it's isomorphic to im(f)

that's all

[lone quail]

Why would the fact that its isomorphic show that?

I read your answer above but i thought it was for another question xd

[brittle juniper]

the dimension of the space is the sum of the dimensions of the supplementaries

If $\Ker f\oplus S=E$ then $\dim\Ker f+\dim S=\dim E$

[stoic python]

Tuong:

[lone quail]

Kk ty but thing is my book is using the fact that dim (kerf) = n-r to prove the above

So i cant really use this

(Even though i understand where this is going)

[brittle juniper]

that's weird because this thing is just a consequence of the incomplete basis theorem

[lone quail]

Yeah, we do it slightly differently, it says its a consequence of Rouchè Capelli theorem but I don’t see how that fits

You know that theorem?

(I do linear algebra in a different language so some names may not match)

[boreal crescent]

how do i even start

i know i have to prove A implies B, B implies C, C implies A

so uhh yea im lost lol

[brittle juniper]

you can start by noticing C implies A is obvious

the rest is really just playing around with the definitions

[boreal crescent]

im sorry, i just started learning lin alg a week ago at uni, could you explain how i would write C implies A in a 'mathematical' way?

[brittle juniper]

think about what it means for T not to be injective
T not injective means there exists two distinct vectors that have the same image by T

[boreal crescent]

ohh ok so since there exist an infinite number of vectors that have the same image, T isn't injective

im stuck on A -> B

so there are multiple vectors in Rn with the same image. how does that mean that there will be an infinite number of vectors in Rn such that they can be transformed to the zero vector

is it cus you can multiply it by a scalar and it will still become 0? since if it was injective multiply it by 0 would result in another new vector and that cant have the same image as the original

no doesnt sound right

[brittle juniper]

T not injective means there exists two distinct vectors x1 and x2 such that T(x1) =T(x2), i.e. such that T(x1-x2)=0

now have a look at the line generated by x1-x2

[boreal crescent]

for any two artbitrary vectors though right

[brittle juniper]

they aren't really arbitrary

they're just two vectors which existence comes from the definition of "not injective"

[boreal crescent]

right

how does that show there are an infinite number of possibilities for T(x1-x2) = 0 though

[brittle juniper]

when α is a real number, look at T(α(x1-x2))

[boreal crescent]

itll still be 0, since T is linear you can pull alpha out\

oh is that it?

[brittle juniper]

for all real number α, you have T(α(x1-x2))=0, does that makes infinitely many vectors that are mapped to 0?

[boreal crescent]

uhhh no i guess

maybe do something like assumed c = x1 - x2 and take x1 to be a constant vector. For any random vector c in Rn, there will exist a x2 such that x1-x2 = c, and therefore its infinite

does that make more sense

no

idk bro, i dont get it

[brittle juniper]

it does, x1 and x2 are distinct, x1-x2≠0, {α(x1-x2) | α real} has as many elements as there are real numbers

[boreal crescent]

gotcha

so for B --> C

assume T(x + z) as the transformation

that equals T(x) + T(z)

we know there are an infinite number of x so that T(x) = 0

so T(z + x) = y has an infinite number of vectors z + x that map to y since there are infinite number of vectors such that T(x) = 0 right?

@brittle juniper

[gray dust]

you basically got it

might be a good idea to preface with "for any y in im(T), there exists at least one element z in R^n so that T(z) = y" bc you introduced z outta nowhere

[boreal crescent]

gotcha yea ill write it in a better way, just wanted to confirm logic

thanks boys, this has been real helpful 😄

[lone quail]

@brittle juniper thank you anyways

[pale shell]

DIM

[cursive narwhal]

SUM

[pale shell]

Do u whaat the fuck aeigen os

[gray dust]

nani

[cursive narwhal]

I dream of a day when you'll learn how to construct proper sentences

[pale shell]

Does this result mean that they don’t span R2

[gray dust]

not much work is needed as you think

observe (2,4) is 2*(1,2)

[pale shell]

I know

Its just if there are vectors that aren’t so clear like that

[gray dust]

what's kinda noteworthy is that your work leads to 0=y-2x, which says the two vectors span the line given by y=2x

[pale shell]

Ohhh

Thanks

[gray dust]

you're welcome

[gilded bobcat]

so what guarantees the fact that we can always remove one of w's and not the u's?

[pulsar turret]

Can someone explain in own words what a generator is and give an example? I don't understand in the way it is worded here

[dusky epoch]

"here"?

[pulsar turret]

in my tekstbook

[dusky epoch]

well we can't see your textbook, einstein! it's not hooked up to a webcam!

[pulsar turret]

it's in dutch so it would be useless to give it to you,

[dusky epoch]

the notation should still be the same

and i'm pretty sure i could reason out what the dutch means

it ain't chinese

[pulsar turret]

ok, if you need I can try to explain a few words if needed

[dusky epoch]

just post it already

[dusky epoch]

,rccw

[dusky epoch]

so... generator = spanning set

[pulsar turret]

is that a question or answer?

[dusky epoch]

i mean as far as i can tell this just says "we say S is a generator for V if span(S) = V"

[pulsar turret]

yeah

so thats it?

[dusky epoch]

uhhh yeah? what else do you want to hear from me

[pulsar turret]

so, S is a subset of V and we only say S is a generator if span(S) = V?

[dusky epoch]

"S is a generator of V" is DEFINED as meaning "span(S) = V"

[pulsar turret]

ok, then we get introduced to linear independence

and I think I understand it

but then we get to basis, and it says it is a basis if it is linear dependence and if it is a generator.

[dusky epoch]

er

no

a basis is a set that is linearly independent, and a generator, simultaneously.

[pulsar turret]

yeah thats what I mean

[dusky epoch]

ok but the way you said it was wonky

[empty copper]

,rccw

[stoic python]

Couldn't find an attached image in the last 10 messages

[empty copper]

you sniped me

[pulsar turret]

can you tell me why they are important or what they do, the basis?

so I atleast have an idea of what I'm doing?

[dusky epoch]

well

if you have a basis for your vector space, every vector in the space can be expressed as a linear combination of your basis, and said expression is unique

the former is guaranteed by the basis being a generator and the latter by its linear independence

so yknow. that lets you get a better grip on your vector space as a whole

and also any two bases for the same vector space have the same size

and that size is called the dimension of your space

which is another important quantity in linear algebra

[pulsar turret]

ah oke. thanks

[alpine echo]

Why do things tend to move towards the eigen-span in a ODE (is it generally true for any DE system)?

I think my question is ill-defined, let me find an example.

https://www.youtube.com/watch?v=i8FukKfMKCI&list=WL&index=160&t=1226s

[dusky epoch]

i'm gonna disappear soon but what do you mean by "things"

[alpine echo]

In this video

About 13:00

And many other DE examples

[alpine echo]

he starts at an arbitrary point with arbitrary populations and eventually, it settles on the eigen span

[slow scroll]

can't watch the video rn, but the concept is kind of similar to the way differentiating exponentials works. If you have some differential operator A and some vector v such that Av = av for some a in R, then there had better be some exponential function at play (e^{at} where a is the eigenvalue).

It turns out that the eigenvectors span the space of solutions to the ODE, and when you parametrize the solutions in this way, its clear that solutions either converge or diverge from eigenvectors (depending on the sign of the eigenvalue). @alpine echo

[alpine echo]

@slow scroll thank you, but I don't understand this second part

It turns out that the eigenvectors span the space of solutions to the ODE, and when you parametrize the solutions in this way, its clear that solutions either converge or diverge from eigenvectors (depending on the sign of the eigenvalue).
Specifically, I don't get what you mean by, "when you parameterize the solutions in this way"
Can you please, provide a simple example?

[vast torrent]

a simple example are the lotka-volterra predator prey equations, do you know those

oh actually

those are more complicatied

because theyre not linear

I meant something like

$\begin{cases} \dot{x} = ax + by \ \dot{y} = cx + dy \end{cases}$

[stoic python]

gfauxpas:

[vast torrent]

let's make it symmetric just to guarantee it's diagonalizable

$\begin{cases} \dot{x} = 2x - \frac 1 2 y \ \dot{y} = - \frac 1 2 x + \frac 1 3 y \end{cases}$

[stoic python]

gfauxpas:

[vast torrent]

$\dot{\begin{bmatrix} x \ y \end{bmatrix}} = \begin{bmatrix} 2 & - \frac 1 2 \ - \frac 1 2 & \frac 1 3 \end{bmatrix}\begin{bmatrix} x \ y \end{bmatrix}$

[stoic python]

gfauxpas:

[vast torrent]

so what exactly is your question, why you can diagonalize this and get the solutions?

[lone quail]

To find the axis if a parabaloid, can i find the intersection between the quadric and a plane (which is a conic), then find the centre of said conic and find the equation of the straight line passing through the centre of the conic and parallel to the eigenspace of the eigenvalue 0?

(Advanced linear algebra xd)

[vast torrent]

Maybe? Why not just do it all at once

@lone quail how would you know which plane to use?

[lone quail]

Any plane would be fine i guess

Because the centre of the conic which comes out is still a point in the axis no?

[vast torrent]

I dunno, seems convoluted to me

[lone quail]

(This is non canonic form)

[vast torrent]

Just diagonalize a 3x3 symmetric matrix

[lone quail]

Yeah and then?

[vast torrent]

The eigenvectors unitized will be the directions of the axes and the lengths of the axes will be the eigenvalues

[lone quail]

That wont work because the quadric is rototranslated from the origin

[vast torrent]

Or whatever the right term is for the "fake axis' of a paraboloid

[lone quail]

Meaning that the axis dont pass through the origin

[vast torrent]

So complete the square first and shift it?

Does that work

[lone quail]

You cant really complete the square when you have like 9 terms

Xd

[vast torrent]

Whats the quadratic form in questuon

[dusky epoch]

,rccw

[lone quail]

This is the generic form of a quadric

You can take the non linear part which js a 3x3 matrix and diagonalise it the issue is that the eigenspaces which come out are actually parallel to the axis

So my idea was to find a point passing through the axis by intersecting a plane with the quadric and finding the centre of the conic which comes out

(You can find the centre very easily using partial derivatives)

[vast torrent]

I saw online an approach using homogeneous coordinates but I've never actually tried it

A 4x4 quadratic form on [x,y,z,1]

[lone quail]

Yeah but you can just consider a 3x3 matrix because the rest is linear

[vast torrent]

Well you're saying that's not good enough for your problem

[lone quail]

You do find eigenspaces

But they are parallel

So any better idea in finding maybe a point passing through the axis?

As intersecting a plane seems a bit convoluted as you say

[vast torrent]

https://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections

Id try it using wp's way with homogeneous coordinates

And then come back and @ me with it because id love to see it in practice, ive only seen it mentioned in passing

[lone quail]

Kk ty

I’m gonna try it tomorrow its a bit late today xd

[vast torrent]

Gl and @ me :]

[lone quail]

Ty

Will do

[vast torrent]

Ahah, @lone quail,found it on the page

If you set the first 3 rows of the matrix in homogeneous coordinates

And dot them with a to-be-determined vector (x0,y0,z0,1)

And equate to 0

That's precisely the coordinate at which all degree 1 terms vanish simultaneously

So you make a substituion x'=(x-x0) and so on

And the center will be the origin in the new system

The same result.as completing the square in 3 variables

Well you dont need to make the substitution i suppose

You can just translate

[lone quail]

Kk thanks!

[vast torrent]

problem though

this change of coordinates will change the constant turn

term

WP has a formula for the new constant term for conic sections in 2 dimensions, but we want an equivalent term for 3 variables. maybe I'll try to find a formula when I have more energy

[alpine echo]

so what exactly is your question, why you can diagonalize this and get the solutions?
@vast torrent I'm not very sure, how to put it. But, why do the "initial condition vector" eventually settle in on the eigen span

[pale shell]

Does anyone know any begineer subspace proofs?

[vast torrent]

@alpine echoi think its related to the so called stable manifold theorem but im busy atn

[alpine echo]

Stable manifold theorem
I'm relatively new to linear algebra 🥺

[vast torrent]

k so

let me look it up and see how simple it is to explain

the idea is that

the end behavior of trajectories settle into these subspaces as t goes on

and the stable manifold theorem gives conditions on which these subspaces have eigenvectors have bases

[alpine echo]

One sec, I've got a rookie question

Aight, that was stupid, sorry, continue

[vast torrent]

no worried

worries

so that's what the theorem says. under certain conditions, the "end behavior" of these trajectories are subspaces, and the eigenvectors span those subspaces

as an easy example

let's say for certain initial conditions the trajectory ends up at 0

and let's say that you end up at zero any time you start anywhere on the vertical axis

and if you dont start at the vertical axis you never end up at 0

then the set of all initial conditions that lead the trajectory to 0 is the vertical axis

which is definitely a subspace

an eigenvector that spans it would be (0,1)

[alpine echo]

Alright....🤔

[vast torrent]

but this theorem says

that the solution space can be split up into these subspaces

each subspace corresponding to the end behavior of a trajectory

[vast torrent]

and each "end point" of the trajectory spanned by eigenvectors of the system

these end points are called fixed points

and they might be infinite

okay dinner time

I'll be around maybe later

[alpine echo]

Yeah, cool thanks a lot.

It makes sense, when we look at it,
$X(t) = P e^{Dt} P^{-1} X(0)$ this way. Cause, solving like this, will definitely yield, a combination of $e^{\lambda_nt}$
(Oh wait, I'm actually not able to reason it this way either 🤔

But, I've got no insight as to why that should be the case without looking at it this way

[stoic python]

Aravindh_Vasu:

[alpine echo]

Got another doubt, does "as t goes on" correspond to applying the companion matrix again and again?

[rotund jetty]

Really stupid question: Why isn't $\mathbb{R}$ a vector space over $\mathbb{Z}_2$?

[stoic python]

Nicholas:

[nimble egret]

In what context

[rotund jetty]

For my question, is it because the elements of $\mathbb{Z}_2$ are equivalence classes not integers, so they can't be multiplied by real numbers?

[half ice]

They are any choice of scalars

[nimble egret]

^

[rotund jetty]

yours

those are arbitrary scalars

[stoic python]

Nicholas:

[half ice]

Unless I'm missing something, this does work as a vector space

Wait, mb.
What's a(1 + 1)?

Where a is a real vector?

And 1 is the element from Z2?

[rotund jetty]

0?

[half ice]

Or is it a + a = 2a?

[vast torrent]

2a=0

[rotund jetty]

well that was the equivalence class thing lol

but yeah that's just a consequence of overloading notation

[vast torrent]

Because 2=0

[rotund jetty]

so that does work?

[half ice]

2 ≠ 0 in the reals

We are overloading lol

[vast torrent]

We're talking about R over Z2

[rotund jetty]

which is why this is an example of overloading notation

yeah^

we've multiplied an equivalence class by a number

[restive shuttle]

so i have a matrix, each column is a vector. But two of the values in the matrix are variables. I have to find variables for this 3x3 matrix such that they do not span R3. How would I go about this? I have learned only span and indepedence so far. And I am having trouble understanding what the relationship between independence and span is, particularly when it says a set of vectors spans or doesn't span a r^n.

[vast torrent]

Okay so let's use [0] and [1] for the scalars

[half ice]

a + a is the addition between two vectors, and is done in the reals

[vast torrent]

[0]a=0 for sure

[half ice]

Unless a is its own additive inverse, this can't be 0

[vast torrent]

But is a+a=0? We have a+a=[2]a=[0]a=0

[rotund jetty]

ok so I'm actually working towards a question where it's asking me to prove that in any vector space over Z2, every element is it's own inverse. That lead me to "well in R things aren't their own inverses" and then this

so a being its own inverse is a good thing lol

[vast torrent]

We've proved that if R over Z2 is a vector space, a=-a for all a in R

That doesn't rule out being a vector space

[half ice]

If every element is its own inverse, the vectors have to be Z2ⁿ

[rotund jetty]

well yeah I was thinking that under standard R, x + x =/= 0, which is why I was thinking it wasn't a vector space, then I realized I was dumb

[vast torrent]

The vectors are real numbers

[half ice]

No you have it exactly. Two of the same real can't add to make 0

Unless it is 0

[vast torrent]

Why not

Better question

If this is a vector space

Is it a 0 dimensional one

Is there an element that's not a scalar multiple of 0?

[rotund jetty]

1?

[vast torrent]

So that means the only solution to

[c]1=0

Is [c]=0

[rotund jetty]

and that's true

[vast torrent]

Okay

[rotund jetty]

is it not?

we're still overloading notation so much

[vast torrent]

So we didn't collapase the space into a point

We're doing pretty good, put scalars in []

Vectors on their own

[rotund jetty]

oh yes nvm 1 is a vector yeah

[restive shuttle]

so if my set of vectors doesnt span r^n

what does that mean in terms of independence

[rotund jetty]

tysm for all the help lipschitz and kaynex :D

[restive shuttle]

Find all values z1 and z2 such that (2,−1,3), (1,2,2), and (−4, z1, z2) do not spanR3

is my question

[vast torrent]

Wait tho

Um

Wp says

Apart from the trivial case of a zero-dimensional space over any field, a vector space over a field F has a finite number of elements if and only if F is a finite field and the vector space has a finite dimension.

But that doesn't seem true here

R over Z/2Z has how many elements?

Still continuum many, no?

[half ice]

@restive shuttle
So Rⁿ is an n dimensional space. Every basis of Rⁿ has n vectors in it.

If there's too many vectors, they're lin dependent. If there's too few, they can't span the entire space.

[restive shuttle]

so how might i approach this

i brought it into echelon form

[vast torrent]

Why did you bring it into echelon form

Not saying you shouldn't

[restive shuttle]

we havent learned basis or anything else

[vast torrent]

But you should have a reason to

[restive shuttle]

we only learned independence and span

[nimble egret]

Reason through your every step

[vast torrent]

Your step might be correct, but you need to have an approach

Not just throw techniques at it randomly

So what are you trying to do?

[restive shuttle]

trying to find z1 and z2 such that the 3 vectors do not span r3

[vast torrent]

What does that have to do with row reduction?

[restive shuttle]

im not sure but the answer key has a solution that appears somewhat in my echelon form

and im not sure why

[vast torrent]

Well let's figure out why

[restive shuttle]

answer key: z1 = t, z2 =/= (t-32)/5

[vast torrent]

Lets make a matrix with the 3 columns the vectors in the problem statement

[restive shuttle]

yes i did that

this is my reduced form

[vast torrent]

Yes but it you don't know why you're putting it into row echelon form

Then you're just pushing random buttons hoping something happens

That's not math

[restive shuttle]

ik, im just really confused

my prof through a bunch of theorems at me

and the concepts arent connecting

[vast torrent]

So don't rush ahead

Our matrix

We'll call it M

Do you know what the dimension of a matrix is?