linear-algebra

[fickle garden]

ok that makes more sense

thanks

[wintry steppe]

Can somebody help? I got this wrong

[vast torrent]

It's wrong but i cant find the mistake :/

@wintry steppe found it

Its in the step -3R1+R2->R2

Sign error

You could also noted that since you have x=z you can rewrite the system to have 2 equations in 2 unknowns

[spice ruin]

this might be a multivar question, but what is the distinction between an eigenvector and a standard basis vector being multiplied by a scalar?

[rapid nymph]

Hello I have a quick question, this is very early on into the course but Im not sure I understand the solution

so for 6b, I understand that I have to transform it into row ecehlon form

as I have done here, but Im not sure Im understanding the solution...

we have that b1 = -b3 and b2 = 2b1

and im not really sure where to go from there

the solution says its this

but Im not sure what it means by b2 can be anything and how the solution is a [b1, b2, -b1]

[lapis flicker]

Hello, if anyone could help i'd appreciate it. I have to express an equation in parametric form. This is the question:

My method of working was to solve for the implicit representation - in standard form which is: 3x -4y +18 = 0 and then use that to create the parametric equation. What i eventually came to was:

x = 2 + 3t
y = 6 -4t

However, this is wrong and i'm not certain why.

Or on a different note, would it be appropriate to let:

x = t

such that:

y = 9/2 + 3/4t ?

[nimble egret]

Yes

[stoic python]

mop:

[sonic osprey]

just by definition?

[wintry steppe]

huh?

uh aright

[pliant harbor]

All automorphisms are homomorphisms.

[clear spoke]

How do you prove that

$\lim_{x\to0}{\frac{\sin{x}}{x}} = 1$

[gray glen]

you don't

because that's not true

[terse mirage]

lmao

[stoic python]

<3 Hoodie <3:

[clear spoke]

Sorry, this

[gray glen]

depends how you define sine

[terse mirage]

^

also wrong channel tbh

[clear spoke]

English please

#multivariable-calculus ?

[terse mirage]

yea

or well

[gray glen]

I mean

#calculus

[terse mirage]

eh

yeah

just calc

[pliant harbor]

Suppose A is a n by n 0-1 matrix. Prove that the determinant of U = nJ+B+B*+2AA^T is divisible by 2^(n-1) where B=(i-1)AJ and J is the all ones matrix.

This is not the original problem, but I've reduced the problem to proving this, which should be easier. But now I'm stuck.

Perhaps we need to use the fact that U is the sum of 3 Hermitian matrices and is thus Hermitian.

[gleaming topaz]

If I'm asked to find the norm of the vector between l1 and l2 which is orthogonal to both lines and I know that the cross product of l1 and l2's direction vectors is (-1,2,1), is this a valid method for solving it? https://gyazo.com/43ef784e7f62a59403490f2fd6695b1d

[gleaming topaz]

<@&286206848099549185>

[vast torrent]

@gleaming topaz what does it mean for a vector to be in between two others

[gleaming topaz]

Well I'm looking for a line l3 which intersects both l1 and l2 and is orthogonal to them both and I'm asked to find the length of the line segment of l3 between l1 and l2

maybe I was a bit unclear but does that make sense?

[vast torrent]

I think i see

Well that would be the shortest line segment between the two lines

By some calculus argument somewhere

I would find the general form of a line segment connecting the two lines and then minimize the norm of the function using calculus. Someone feel free to chime in if there's a better way.

I don't know how you took cross products of lines

[gleaming topaz]

I took cross product of their direction vectors

to find the vector perpendicular to them both

[vast torrent]

I guess you took one of the 2 unit vectors on the line

Hmm

No

Direction vector won't characterize the line unless it goes through the origin

You need 2 vectors to describe the line, a displacement vector and a direction vector

So your idea would work if you accounted for that

[gleaming topaz]

Yeah but what I've done is find a vector between l1 and l2 in the same direction as l3's isn't it

[vast torrent]

What does it mean for a vector to be in between two others

In between two linew

Idk what that means

[gleaming topaz]

@vast torrent https://gyazo.com/16aa035a6087eaffb1ad9f3055593fe5 L1 and L2 are the red and blue lines right, I'm looking for a vector perpendicular to both l1 and l2 which gives the shortest distance between them

Lets say that the green line is the vector "between" them

[vast torrent]

Okay i looked around on stack exchange a bit

Each line has 2 vectors defining it

Displacement and direction, right?

$\mathbf s + t \mathbf{\hat{u}}$

[stoic python]

gfauxpas:

[gleaming topaz]

Well yeah a line has a point it passes and a direction vector

exactly

[vast torrent]

And the other line has

$\mathbf r + s \mathbf{\hat{v}}$

[stoic python]

gfauxpas:

[vast torrent]

You need to find the cross product of those two vector valued functions of a real variable

Then using calculus minimizes the answer in s and t

I'm looking if there's an easier way

Actually forget the cross product

Minimize the real valued function of the distance between them

[gleaming topaz]

You mean the norm of a general vector between l2 and l1?

[vast torrent]

The distance between two points

And if I'm right

The arg max of the distance function (s,t)

Will tell you which vectors to cross to find the vector between the lines

But I'd prefer someone check what im saying

[gleaming topaz]

I found this just now https://math.stackexchange.com/questions/1033419/line-perpendicular-to-two-other-lines-data-sufficiency

[vast torrent]

His method is harder i think

[gleaming topaz]

I'm not quite sure I understood yours if you don't mind elaborating

I could compare results

[vast torrent]

Take the eyclidean distance between two points x+tu-hat, y+sv-hat

Minimize that function to get the distance between those two lines

The vector with that distance in the direction of the cross product x×y is the one you want, or its negative

[gleaming topaz]

So you mean the length of PQ right? https://gyazo.com/75902d49b5423eaa90835f1a3bb3ad06

[vast torrent]

Distance between two points is sqrt(Δx²+Δy²+Δz²)

[gleaming topaz]

Yeah exactly the norm of PQ in this case

[vast torrent]

Though you will find it easier to minimize Δx²+Δy²+Δz² ,which has the same arg min by strict monotonicity of sqrt

Do you know that theorem

[gleaming topaz]

Nope, haven't started with multivar calc yet if it's from there

[vast torrent]

If f is strictly increasing, f(g(v)) has the same arg max and arg min as g(v)

The upshot is that you can optimize the inside of the square root instead of the whole thing with the square root

Square it then optimize

Do you know partial derivatives

[gleaming topaz]

Nope haven't done that yet

[vast torrent]

You need a very simple application of them here

Basically since you have 3 variables

You consider 3 seperate derivatives,in x, in y, and in z

And solve the derivative =0 simultaneously

3 derivatives =0 at the same time

Just like in calc i but three equations at a time

[gleaming topaz]

So in this case my (x,y,z) is (-1+s-t,-5+2s-t,-1+3s-t) ?

[vast torrent]

You want the distance squared function

Δx²+Δy²+Δz²

[gleaming topaz]

Oh so (-1+s-t)^2+(-5+2s-t)^2+(-1+3s-t)^2 ?

What do I derive with regards to here or is that what you mean? We could take it in PM if you'd like

[vast torrent]

The notation is $\pdv{f}{s}$ instead of $\dv{f}{s}$

[stoic python]

gfauxpas:

[vast torrent]

You take 2 derivatives. With respect to s, pretending t is constantl

And wrt t, pretending s is constant

And you want both to be 0 at the same time fir optimizing

Just like calc 1

But two equations simutaneously

[gleaming topaz]

Oh I see, so they'll give me a system of equations where they both equal 0 and I can solve for s and t?

[vast torrent]

You got it

[gleaming topaz]

I'll try and compare the results thank you

[vast torrent]

And you don't have to worry about testing second derivatives bc it's obvious from the definition of distance that there's going to be one minimum and no other extrima or saddle points

There is a second derivative test for partial derivatives but it's not necessary here

And it works a bit differently

[gleaming topaz]

Ok thanks I appreciate the help, I'll try it out

[vast torrent]

Also this answer will be the square of the distance, you'll take the sqrt at the end

[gleaming topaz]

Well this will give me s and t right so I can just insert that into the original equation for the distance

[vast torrent]

yes that should work too

[gleaming topaz]

Yeah

Same answer

Cool method, thanks

[vast torrent]

in fact I think you can even go back to the two vectors defining the plane, put in s and t

and take the cross product

for those particular vectors

so what was the other method?

a system of equations with dot products?

[gleaming topaz]

Yeah this one https://gyazo.com/5cf71624a1f6f352739b469a0006e851

[vast torrent]

I'd definitely forget that way on a test

another common notation for $\pdv{f}{t}$ is $f_t$

[stoic python]

gfauxpas:

[vast torrent]

for when you come across it in your travels

[gleaming topaz]

Is that from multivariable calculus?

[vast torrent]

yes

partial derivatives are from multivariable calculus, but this particular application was simple enough I had confidence you could handle it

[gleaming topaz]

Thanks for your time once again

[vast torrent]

np bro

or sis

what calc have you taken? just part i?

the deeper you go into calculus and linear algebra, they more they overlap

[gleaming topaz]

I've done half calc 1 I think, starting with integrals & diff eqs next week and when I'm done with that multivar calc

[cursive narwhal]

?

[rocky hill]

that shorthand notation is awful and I hate that it's used

[frosty vapor]

wym

is gud

[wintry steppe]

How do you solve this 3 variable linear system

[quartz compass]

quite a few ways

what have you been taught? elimination?

[proven sky]

ye

[wintry steppe]

yup

[hidden ember]

The matrix method

[quartz compass]

well, start eliminating then

[gleaming topaz]

Let Q be the quadratic form https://gyazo.com/0e3a183c5805a35917dbb9d38db5da52 How do I find the shortest distance from the surface Q(x)=1 to the origin?

<@&286206848099549185>

[vast torrent]

What's the usual convention for the operator that takes a matrix and stretches it out into a vector; column after column,or row after row?

Is there a common convention?

Nm found it on wikipedia, column by column is the standard

[pale shell]

Question

Why do vectors have arrows at the end if they are not moving infinitely in one direction

[nimble egret]

Well

It has to point in a direction

[cursive narwhal]

The arrow indicates the direction in space

[pale shell]

Okay

So what counts as the magnitute and what counts as the direction in a 2d vector

[nimble egret]

What do you mean?

The magnitude is more or less just how long the vector is

[pale shell]

Soo is that the x coordinate?

[nimble egret]

???

[pale shell]

Is the magnitude the x or like its actual length

[nimble egret]

The magnitude is the length

[pale shell]

I see thanks

So you would use the pythagorean thereom to find it?

[nimble egret]

Draw a triangle

[pale shell]

Yeah

[nimble egret]

With the x axis or y axis as one of the sides and the tail of the vector at the origin

It should become pretty clear that if you have a vector (x, y)

Then the magnitude is given by sqrt(x^2 + y^2)

[vast torrent]

The arrow is ultimately just a notation @pale shell

You can argue they're extraneous if the base of the arrow is at the origin

[pale shell]

Thanks

[vast torrent]

But typically we like writing the arrow in different places to show geometric things

In which case it's not obvious what the base is

The point itself is really the vector, the point at the tip of the arrow

I can go on if you want

[pale shell]

I see

Interesting stuff

So tell me

How do you imagine a vector in four dimensions

[vast torrent]

Most people can't imagine 4d space. I read somewhere that some people can but not 5d and up,idk if that's true

[pale shell]

Can you?

[vast torrent]

But since the point at the end of the vector is really the vector

No

[pale shell]

Hmm

[vast torrent]

(x,y) is a 2d vector

[pale shell]

I honestly am interested in linear algebra

[vast torrent]

And (x¹,x²,x³,x⁴) are vectors in 4d. Pretend they're subscripts not superscripts, it's harder to do subscripts on mobile

(x¹,x²,x³,x⁴,...,xⁿ-¹,xⁿ) is a vector in n dimensions

Linear algebra is really important

I have my qual on la tomorrow im so nervous

[pale shell]

Qual?

[vast torrent]

Qualifying exam

[pale shell]

Ohh

Good luck

[vast torrent]

I need to pass the quals in 3 subjects to get my MA

[pale shell]

Maybe you can teach me some more sometime

[vast torrent]

Complex calculus, real calculus,linear algebra

What level of education are you in

[pale shell]

Calculus

[vast torrent]

High school, college, university?

[pale shell]

High school

[vast torrent]

Ah, well, if you plan on going to college, you'll learn so much more than a high school class

[pale shell]

Of course

I want to be an engineer

[nimble egret]

Nice

[pale shell]

I am super interested in other dimensions and stuff

[vast torrent]

Now, in terms of of imagining higher dimensions

[pale shell]

Ye

[vast torrent]

The usually add another aspect to the graph that changes according to that aspect. Ill explain

Let (x,y,z) be 3 coordinates of position in space

(x,y,z,t) adds another dimension. So I'll pick one that can be described by real numbers, like time

Then a computer will have a 3d graph that changed as t goes from 0 to 60, then a 1 minute movie of a changing 3d image is 4d

Another way to imagine a dimensionis temperature

3 dimensions will give a shape, now imagine the shape changes depending on how cold or hot it is

Do that and time and your mind might be able to process a 5d set of points

Now connect the graph to a battery and have it change depending on your charge

(x,y,z,time,temperature,charge) is a 5d point

Some of these approaches don't allow for negative numbers for physical applications

You can go higher

And still have a physical interpretation

[pale shell]

Hmm

Very cool

Can a shape be a vector

Or is it just a line

[vast torrent]

A vector in R^n , which is what youre probably asking about,is really a point

The line connecting the tail to the head helps you understand those properties

But the vectors have to be straight, that's very important

There are more general definitions of vectors that you'll learn about

@gleaming topaz you can use lagrange multiplies

Know those?

I can think of another way but it's harder

[gleaming topaz]

No haven't learned that, I was given a solution but dont quite understand it, I'll post it and see if you could help me out maybe

[vast torrent]

The other way is to change the variables so that you form an ellipse in standard position such that the shortest distance will just be the smallest minor axis

For points (x,y,z), you want to find a matrix such that

<(x,y,z),A(x,y,z)> = 3x²+4y²+2xz+3z²

And to choose such an A such that it is symmetric

Can you do that?

It will have a good number of 0s

[gleaming topaz]

Well one such matrix A would be (3,0,1;0,4,0;1,0,3)?

[vast torrent]

Excellent

That's a symmetric matrix, what does the spectral theorem tell you

[gleaming topaz]

That it can be diagonalized by a matrix where the eigenvectors form a orthonormal base?

[vast torrent]

Yup

What does orthomormal mean in terms of a solid in 3d

[gleaming topaz]

The vectors are all perpendicular to eachother and are unit vectors right

[vast torrent]

Think determinants

[gleaming topaz]

The determinant is either 1 or -1 yeah?

[vast torrent]

Yup,so?

[gleaming topaz]

Sorry I'm not following, are you hinting towards something about the volume or something else?

[vast torrent]

The only transformations such a matrix could be

Is a product of rotations and reflections

Wait

That's not exactly right

The point is that it's an isometry

I think its right actually

[gleaming topaz]

Ignore the matrix A that's not right, but this is the solution I did not understand if you have any idea what happened? It's in swedish but maybe the math makea sense

Basically it says that the surface is given by 4(y1)^2+4(y2)^2+2(y3)^2=1 but I'm not sure what that means

[vast torrent]

How is that the right matrix

Theirs

[gleaming topaz]

Its wrong

[vast torrent]

Okay, well, i have to go, but i gave you enough to finish

[gleaming topaz]

Alright well thanks I'll see what I can do

[vast torrent]

Anyway my point was

Isometries will.rotate and reflect the ellipsoid

But won't change the lengths of its axes

Find a coordinate system where there's only u²,v²,w² terms in the quadratic

[wintry steppe]

Hey, so I'm currently studying linear transformations, more specifically, the change of basis applied on a transformation

My issue is when we're dealing with a transformation in between dimensions

So I kind of hit a roadblock here

I have this transformation T: R^3 -> R^2, which I have represented by the matrix m(t)=[(0,1,1),(0,1,-1)]

I now want this transformation, but in the basis S={(1,1), (1,2)}

My issue is, I can't get to the matrix that changes a vector from the basis E to the basis S

The basis E being E={(1,0,0),(0,1,0),(0,0,1)}

I'd be able to do it S were 3 dimensional, and if T were from R^3 -> R^2

But I'm just lost here

[slow scroll]

you're having trouble because it makes no sense to go from E to S as they live in completely different spaces with completely different dimension. You are most likely misunderstanding the instructions. Perhaps you want find out what the outputs of m(t) look like in the basis of S. In that case, you want to find a transformation A to compose on the left of m(t) that sends vectors in the standard basis to their representation in the basis of S. @wintry steppe

[wintry steppe]

I see, that must be it

I'll look over the instructions again and let you know if I have any more issues!

[tough socket]

hey here, i wanted to know what kind / group of graph can be described by an unitary adjency matrix

[lone quail]

If P is the projection matrix on a space U, is it possible for there to be two alternate forms of P?

[vernal salmon]

Is the minimum of a vectors absolute coordinates a norm?

Nvm it isnt

[dusky epoch]

[smoky lagoon]

i only just started my linear class this semester

but its tons of fun

so far

its like doing a puzzle

[gleaming topaz]

If I have a line that intersects a plane and want to find the angle between the line and the plane do I just take the dot product of the planes normalvector and lines direction vector divided by the norm of them multiplied by eachother?

Or is that wrong

[dark cedar]

linear algebra amirite?

[slow scroll]

@gleaming topaz the angle formed by the intersection of a line with a plane is not unique. It does however make an angle with the normal to the plane, and yea, the angle between the line and the normal is obtained using dot product

[wintry steppe]

in LU factorization, the U matrix can never have zeroes on its diagonal, right?

[pallid swallow]

if A=LU is invertible, then U cannot have 0s on the diagonal

[pliant harbor]

Yes, because U is a triangular matrix and det(A)=det(L)det(U).

[sullen pollen]

Hello! How do I find the Kernel (Ker) and the Image (Im) of the following lineair mapping R² -> R²: f(x,y)=2(x, -y). And if the invers of the lineair mapping exists, what is it?

[echo quail]

well, you can't get 0 unless x = y = 0, so Ker f = {0} and it's invertible, and the image is the whole codomain.

[pale shell]

Where can I find a set pf practice problems for each unit?

[sharp merlin]

how do u do this

[quartz compass]

do you know of any way to take two vectors and make a vector that's orthogonal to both of those?

[sharp merlin]

cross product

and then divide by length?

[quartz compass]

yeah you got it

[sharp merlin]

i did that

and got it wrong

@quartz compass

[quartz compass]

show your work, sounds like you made a mistake somewhere

or maybe they want you to simplify the square roots more

or rationalize the denominator even

[sharp merlin]

nah webassign doesnt care

nvm got it

How is this wrong

@quartz compass

oh wait forgot = 0

[sharp merlin]

How do u do this

@quartz compass

[floral barn]

Have you dealt with cross products?

[sharp merlin]

yes

@floral barn

[floral barn]

Nevermind, that might not be necessary

I'm not the greatest at this, but hopefully I can be helpful

I think what you wanna do is subtract one of the point vectors from the other

[sharp merlin]

so like the first part right

(9-0,-1-1, 0+9)

[floral barn]

I haven't read back that far

Yeah that'd work

[sharp merlin]

ok

[floral barn]

Then, you wanna find a vector normal to both that and your direction vector

So, dot product with (9, -2, 9) is 0 and dot product (2, 9, -6) is 0

[sharp merlin]

i would have to use cross product right

[floral barn]

Not necessarily

I haven't used cross product for this

n = (a, b, c)

[gleaming topaz]

Both methods work but cross product is easier/faster IMO

[floral barn]

I haven't worked with cross product so /shrug

If you know how to do it then great

[sharp merlin]

ok I did cross product

Got (-63,72,85)

[floral barn]

That's large

Hmm

One sec

[gleaming topaz]

That -63 is wrong

@sharp merlin

[sharp merlin]

how

oh oops

-69*

ok what next

[floral barn]

I believe those should be the coefficients of your plane equation

[gleaming topaz]

Now you've got a normal for the plane and to find the equation you need a point on the plane as well to find it's equation

Can you figure that out somehow?

[sharp merlin]

uh

i can just use one of them right

(0,1,-9) or (9,-1,0

[gleaming topaz]

Yeah that works

[floral barn]

I have a question as well, though I'm not sure if this is the correct place to ask

I'm dealing with subspaces and my question is ambiguous as to what set my scalars are in

So I can't tell if I'm closed under multiplication or not

[gleaming topaz]

@floral barn If it's not specified I would assume they belong to the real numbers? Or was that what you meant? Although I'm not entirely sure

[floral barn]

'Let P3(R) be the set of polynomials of degree at most 3 with coefficients in R.'

Does this mean its scalars are in R?

[gleaming topaz]

I would interpret it that way yes

[floral barn]

Alright, great

I'll include a bit of an asterisk about if they were in C

[sharp merlin]

How do u do this one

@floral barn

[floral barn]

Looks like you wanna find what's perpendicular to the line, then find values that satisfy the plane equation and add the point

I think

Should be fairly straightforward

[sharp merlin]

I got 7(x-1)+54y+12z+84=0

how do i convert it to parametrized form

[floral barn]

Are you on a strict time limit here?

I set up a system of linear equations

6x - y + z = 0

6x + y - 8 = 1

Matrix and row reduce, yada yada

You should be able to express two of the variables in terms of the third plus a constant

Then you can add the point, I think

[alpine echo]

https://math.stackexchange.com/questions/3514173/transpose-of-a-matrix-and-the-product-of-a-and-a-transpose

Can someone please take a look at this question? The question is basically "What's the geometric meaning of the transpose of a matrix and the product of AA^T?"
@dusky epoch

[dusky epoch]

uhh

yeah i got nothing

i mean i could tell you that the (i,j) entry of AA^T is the scalar product of the i'th and j'th rows of A but that's about it

sorry @alpine echo

[alpine echo]

Ah, alright no problem, is there anything you suggest I can do to understand this?

Like the 3b1b way?

[dusky epoch]

can't think of anything atm

never tied any geometric picture to it myself

[elder robin]

Does it look like I did this correctly? It feels right but I haven't done anything like this before.

[dusky epoch]

your a and 9 look weirdly similar

[elder robin]

ya, sorry for trash handwriting

[dusky epoch]

it doesn't look like you made any algebraic fuckups

[elder robin]

I kind of just plugged and played with formulas from my notes, which I hate doing bc I never know if it's right lol

I could derive the dot product thing, that would prob help intuition

[sullen pollen]

hello, how do i prove that: f: R² -> R³: f(a,b)=(a,b,a+b) is a linear mapping

[cursive narwhal]

? (a,b,a+b)?

@sullen pollen

[sullen pollen]

i dont know xD i need to prove that for every a and b this is a linear mapping

is linear mapping and transformation the same?

[dusky epoch]

yes

and you need to use the definition of a linear map

[cursive narwhal]

Yea linear mappings and linear transformations are the same thing.

What's the definition of a linear map?

[dusky epoch]

@sullen pollen are you still here?

[sullen pollen]

yes i am

if f(a+b)=f(a)+f(b) and f(ca)=cf(a), a, b vectors and c a constant

[gray dust]

ok, do you know how to run with that?

[sullen pollen]

not exactly

f(c*a) = cf(a) i understand a little bit

[gray dust]

$\fxn{f}{\bR^2}{\bR^3}{(x,y)}{(x,y,x+y)}$

[stoic python]

RokettoJanpu:

[gray dust]

this is the defn of f

i'll make you show additivity first

let $a=(a_1,a_2),b=(b_1,b_2),\quad a,b\in\bR^2\\$show $f(a+b)=f(a)+f(b)$

[stoic python]

RokettoJanpu:

[sullen pollen]

so a + b = (x1+x2 , y1+y2) and evaluated in f it gives: (x1+x2, y1+y2, x1+x2+y1+y2)
when we evaluate a en b in f it gives: (x1, y1, x1+y1) and (x2, y2, x2+y2). When we add them together it gives the expression above and then we proved that

[gray dust]

ok i worried x1,x2,y1,y2 would throw you off since i used x,y in the defn of f

ok i'm gonna assume the field of reals. now show me homogeneity

let $a=(a_1,a_2)\in\bR^2,\quad c\in\bR\\$show $f(ca)=cf(a)$

[stoic python]

RokettoJanpu:

[sullen pollen]

ca = (ca1 , ca2), so f(ca) = (ca1, ca2, ca1+ca2) = (ca1, ca2, c(a1+a2)) = c(a1, a2, a1+a2) = c * f(a)

[gray dust]

all done!

[sullen pollen]

ooooo now i see how the definition lets me show how to prove it in exercises

thank you very much!!

[gray dust]

no problem man

[arctic osprey]

Hello, I have a question: How would I check if two vectors or two lines lie on the same line? I am not asking if two vectors are colinear, I am asking if they lie on the same line on 2D plane.

[gray dust]

fuzzy question. there always exists a line that runs through the tips of any two chosen vectors

[sullen pollen]

If the kernel of a lineair transformation is (0,0) (assume R²), is the image always R²?

[dusky epoch]

assume R^2

does that mean "assume the domain is R^2" or "assume the codomain is R^2" or what

[sullen pollen]

uhm the transformation is R² -> R²

[dusky epoch]

then yes.

[sullen pollen]

thanks!

[pale shell]

How

[dusky epoch]

how what

[pale shell]

How do I do it

[dusky epoch]

have you ever converted vectors between polar form and component form before?

[pale shell]

No

[dusky epoch]

then read up on that

[pale shell]

I tried to

Its confusing

[dusky epoch]

can you show what you've read and which part of it is confusing?

i can try to clear that up, but i need to know what needs clearing up

[pale shell]

Well it is telling me to use sin and cosine

[dusky epoch]

can you show what you've read and which part of it is confusing?

[pale shell]

To convert the form

[dusky epoch]

can you show what you've read and which part of it is confusing?

[pale shell]

On khan acadamey

[dusky epoch]

can. you. SHOW. what. you've. read?

[dusky epoch]

okay, so it's giving you stuff pertaining to this problem in particular

[pale shell]

Yes

[dusky epoch]

they could've done a better job presenting it, imo

first convert each vector to component form, then add.

[pale shell]

Okay so what exactly is component form though?

[dusky epoch]

$\vec{v} = (8 \cos(140^\circ), 8 \sin(140^\circ)) \ \vec{w} = (4 \cos(40^\circ), 4 \sin(40^\circ))$

[stoic python]

Ann:

[dusky epoch]

...uh

yknow

when you give your vector in terms of x and y coordinates instead of a direction and an angle

[pale shell]

Oh

And a direction and an angle is polar form?

[dusky epoch]

yes

[pale shell]

So polar form would look like a matrix?

[dusky epoch]

...

wh

no?

[pale shell]

oh

Uhh

[dusky epoch]

what do you even mean "look like a matrix"

[pale shell]

It would be inside a matrix

[dusky epoch]

[pale shell]

Omg

Inside the brackets

[]

Okay give me an example of a vector in polar form

[dusky epoch]

,,,,,,,

there's no real agreed-upon notation for polar form

because from experience it's rarely used for vectors

[pale shell]

Okay so look

[dusky epoch]

because it's hard to add them in that form

[pale shell]

It is saying that I have to convert from a magnitude and an angle to xy coordinates

How do I do this

[dusky epoch]

i mean you COULD write something like [r, θ] for a vector with length r and angle θ but that's not a matrix just two numbers in square brackets

anyway

the vector with length r and direction θ is (r cos(θ), r sin(θ))

khan has GOT to have a video that explains this

[pale shell]

I was literally watching them and it said nothing on this

But I like how they sometimes have you do some research to get the answers

[dusky epoch]

I was literally watching them and it said nothing on this
surely they have something on "conversion to and from polar form"

or sth

[pale shell]

Okay so how do I convert from a magnitude and angle to component form though

[dusky epoch]

the vector with length r and direction θ (in component form) is (r cos(θ), r sin(θ))

[pale shell]

Okay thanks

[pulsar turret]

Can someone explain to me what a subspace is? like for example V is linear dependent on v and w. So V = all linear combinations of v + w, so "For All a,b element of R: av + bw = V". So the subspace of V would be some like only even numbers a and b?

[dusky epoch]

holy fuck that's gotta be some sort of bad wording x3 combo

are you translating this from another language @pulsar turret

[pulsar turret]

yeah, kind off

my english isn't that bad so I could've be more compact

[dusky epoch]

it's not about being compact

also how can there be a "kind of" in response to "are you translating this from another language"?

either you are or you aren't

[pulsar turret]

it's not a definition I am reading

[dusky epoch]

like for example V is linear dependent on v and w.
this already doesn't make any sense

[pulsar turret]

it's explained in my own words

[dusky epoch]

@quasi vale are you gonna try to respond to rolly's bad wording with an entire paragraph of more bullshit or sth? seeing you type an entire essay in the discord message box is kind of stressing me out

[quasi vale]

I have a very big question but you were helping him out so I copied it and went out from the channel

[dusky epoch]

anyway... yeah, nobody here knows what you're talking about, rolly.

[pulsar turret]

Maybe i ment linearly independt on v and w?
\

[quasi vale]

ill ask it later when you are done with him

I think he means that v and w are linearly independent vectors?

[dusky epoch]

that also doesn't make any sense

there's no such thing in linear algebra as being "linearly (in)dependent on something"

[pulsar turret]

you lost me

let me rephrase it then

[dusky epoch]

yeah, please rephrase it, hopefully in a way that's actually understandable this time

[pulsar turret]

if V is a vectorspace, and W is a subset of V. W is only a subspace if W itself is also a vectorspace. Now my question is how does V and W look like to eachother?

[dusky epoch]

how does V and W look like to eachother?

............ what

what does that even mean

[pulsar turret]

like, let's begin with maybe what is V and W?

[dusky epoch]

you yourself said it

V is a vector space, and W is a subspace of V

[pulsar turret]

yeah I'm not sure what it is tho

[dusky epoch]

what do you mean "not sure what it is"

[pulsar turret]

like what does it represent

[dusky epoch]

what does WHAT represent

[pulsar turret]

a vectorspace

[dusky epoch]

are you struggling with the fact that the idea of a vector space is fairly abstract at first glance

[pulsar turret]

I know something is a vectorspace if some conditions are met.

yeah I guess

[dusky epoch]

i mean...

idk. would it help if i gave some examples of vector spaces that are easy to visualize

i guess the most prominent ones would be R^2 and R^3

whose elements are vectors in the pre-linear-algebra sense

i mean there's a reason why vector spaces by themselves are so abstract. bc there's many things out there which are vector spaces

like solution sets of certain differential equations, or spaces consisting of functions meeting certain criteria

not that you need to actively think about all that when doing linear algebra

in fact the opposite is true since the point is to abstract away all the specifics and focus only on the structure that arises when all you consider is the behavior of your objects when they're added and scaled

[pulsar turret]

yeah so R^2 is a vectorspace

[dusky epoch]

you might consider giving 3b1b's Essence of Linear Algebra a watch some time

[pulsar turret]

I am doing it right now

but I guess I need some time to think about it

[quasi vale]

Watching a vid atm on independent/dependent vectors. To check if three vectors are independent, we can do that by c1v1 + c2v2 + c3v3 = 0, where c1,c2,c3 are constants and they all should be 0 for linear independence. There's another method. If the determinant is non zero, then the vectors are linearly independent and if the determinant is zero, the vectors are linearly dependent. Can we think of the determinant as the volume of the parallelepiped and if the vectors are linearly dependent, then they don't form a 3D shape but instead a 2d one(assuming the other two are independent), and hence the volume is zero. If the determinant is non zero, then there is some volume and the vectors are linearly independent and form a 3D shape.

[pale shell]

What nind of equation is this

L={a+t(b-a)|t∈ℝ}

Where a,b are vectors

[vast torrent]

it's an equation defining a set

= means "We are defining L to be equal to"

[pale shell]

Also does it matter if it is b-a or a-b for that specific equation

[sullen pollen]

What is the standardbasis for the vectorspace of 2x2 matrices?

Isnt it just the unity matrix

[pale shell]

Is this correct to represent the line to pass through these two vector points

,rotate

[dusky epoch]

please don't write t's like that

but yes it is

[storm carbon]

where did they get the matrix [3 2, 2 6] from

nvm im an idiot

[pale shell]

Also is this how you would write the parametric equation

[dusky epoch]

please don't write t's like that

[pale shell]

Sorry

But for these vectors is that right

To pass through those two vector point

[dusky epoch]

fix your t's then we'll talk.

[pale shell]

Srsly?

Fine

[quasi vale]

Watching a vid atm on independent/dependent vectors. To check if three vectors are independent, we can do that by c1v1 + c2v2 + c3v3 = 0, where c1,c2,c3 are constants and they all should be 0 for linear independence. There's another method. If the determinant is non zero, then the vectors are linearly independent and if the determinant is zero, the vectors are linearly dependent. Can we think of the determinant as the volume of the parallelepiped and if the vectors are linearly dependent, then they don't form a 3D shape but instead a 2d one(assuming the other two are independent), and hence the volume is zero. If the determinant is non zero, then there is some volume and the vectors are linearly independent and form a 3D shape.

[pale shell]

Uhh

Are the ts good now?

Ann tuong prix

Also what does your name mean

[vast torrent]

,rotate

[vast torrent]

it's not a rule of math, it's just good practice for your ts to look different than +s. It

[pale shell]

Okay also did I get the parametric equation right

[vast torrent]

uh I dont know what the original problem was

but as to your determinant question yes. If a parallelotope is of dimension less than the space it's in, it has 0 volume

[pale shell]

Sorry I had to write the equation of the line that went through those vectors but with parametric equations

[vast torrent]

I don't understand what you did

[pale shell]

I subtracted them

And I multiplied that by t

[vast torrent]

you wrote + instead of - then

[pale shell]

I added the negated version

[vast torrent]

bad practice

[dusky epoch]

you haven't fixed your t's.

[vast torrent]

it's unclear to the reader what you did

[pale shell]

Ok I get with the t’s

Ill write them neater

[pale shell]

Ok ok fine

[pulsar turret]

Take a chill pill ann

[vast torrent]

it took me weeks to get into the habit of making ts that way when I decided it was a good idea to do so, Ann

if it's a habit, it's hard to break

it's worth breaking, but it can be difficult

[pale shell]

Okay

[vast torrent]

so you have a system of parametric equations. Let's say you didn't have internet access. How would you check your equations?

without asking us

[pale shell]

Uhh

I don’t know

[vast torrent]

you see if your line goes through your two points you started with

[pale shell]

Well my graph isn’t exact tho

[vast torrent]

use the algebra

$x(t) = -2 + 4t; y(t) = 2 + t; v= (2,3); w = (-2,2)$

[stoic python]

gfauxpas:

[pale shell]

Yeah

So I plug it in?

[vast torrent]

is there a t that gives you v?

you can plug it in but here it's easy enough you should be able to do it in yoyur head

[pale shell]

T=0

[vast torrent]

that's w, yes

[pale shell]

So

Ohh whoops

[vast torrent]

t = 1 gets the other one

[pale shell]

T=1

Yes

So

Does that mean they are right?

[vast torrent]

yes

[pale shell]

Oh so if there is some t to get one vector and some t for another, it is true?

[vast torrent]

I don't want to say a blanket statement for all lin alg problems

in this particular kind of problem, it is the right way to check it

because the question says

the line has to go through those 2 points

and if there are ts that make the line go through those points, well, then the line goes through those 2 points

[pale shell]

Hmm

Makes sense

But the reason why I checked here was because

I tried an online one and I gave me a different equations for them

It was like a solver

[vast torrent]

because parameterizations aren't unique

[pale shell]

Oh

[vast torrent]

the parameterization can change the speed and direction your point traverses the graph of the curve

I say curve not line because in calculus the same thing comes up

paramaterizations are not unique

[pale shell]

How do parametric equations even work like I just learned how to put them into the x= and y= form

It didn’t really go into deph

[vast torrent]

they make a function describe each coordinate of points of a graph, essentially

a graph in 3d is a set of point (x,y,z) and if you assign a function to each coordinate x(t), y(t), z(t) that's a system of parametric equations

I'm glossing over some details

[pale shell]

Ohh

I get it

[vast torrent]

🙂

[pale shell]

So it kind of divides the x and y coordinates into two separate equations

[vast torrent]

in a sense

[pale shell]

So if you can plug in a value for t so that it gets both sets of points, then those points are on that line

[vast torrent]

or in general a curve

[pale shell]

Thank you so much

[vast torrent]

there's a caveat that the functions you choose have to be strictly monotonic but I'm not sure you know what that means

[pale shell]

No

[vast torrent]

basically the functions you choose have to keep going along the graph only once, not changing direction, not doubling in on itself, not stopping and then continuing

[pale shell]

I see

Thanks

[vast torrent]

np

[sullen pollen]

if a matrix is in (reduced) row echelon form, and we transpose it, will it give us the (reduced) column echelon form?

[dusky epoch]

yes

[sullen pollen]

and what information does that give about the basis of a subspace of R^n spanned by the rows/columns of A?

[pale shell]

Could someone help me visualize the span of 2 R3 vectors and why they would make a plane

[nimble egret]

Can you visualise vector addition?

[pale shell]

Mhm

[nimble egret]

You can just take two pens and point them randomly them in the air

[pale shell]

Ok

[nimble egret]

And then visualise the pens scaling up and down

And the plane formed by the addition of these two "vectors"

[pale shell]

Omg

Thankssss

Wait

If they all cross the origin

How could it be a plane

[nimble egret]

What do you mean?

[pale shell]

Don’t all the vectors cross the origin?

[nimble egret]

We often don't really care about the absolute position of a vector

But rather it's relative direction and magnitude

[pale shell]

Oh

[nimble egret]

We often represent vectors as a point

From the origin

Cause it's convenient

[pale shell]

Ohhh

I thought of a better way to visualize everything

Imagine each additional one as a control knob

So aV would just make a line

Then av+bw would move that line to make a plane

Then you move the plane around with another vector

[nimble egret]

The third vector you get from adding the the two vectors

Traces out the points of the plane

[pale shell]

Mhm

[nimble egret]

And as you vary your coefficients, you get a different vector

Until you trace out all the points of your plane

[pale shell]

Like if you can imagine one scalar controlling a vector to create a line

[nimble egret]

Yeah

If you imagine the vector as a pen again

The tip of the pen traces out a point on your line

And as you extend or shrink the pen

You trace out more points

And you can visualise that it traces out a line

[pale shell]

So the red lines would be two arbitrary vectors that have been scaled

And the points inside the blue lines will be all the possible vectors produced by the additions?

[nimble egret]

You can visualise it like that sure

[pale shell]

Is it accurate?

[nimble egret]

Yeah

[pale shell]

That really helps me

So if we consider two planes intersecting

[nimble egret]

2 linearly independent vectors in R2 will span R2

[pale shell]

Then all the possible points will create a solid- R3

Is that correct?

[nimble egret]

What do you mean?

[pale shell]

av+bw+cu

I guess so three vectors

Will that create a solid?

[nimble egret]

If they are linearly independent

It will span all of R3

Which I guess can be considered a solid?

[pale shell]

Sir I have another problem

I can’t exactly visualize four dimensions

How should I consider spans in higher dimensions

[nimble egret]

I don't really visualise the 4th dimension myself

I just consider it algebraically

[pale shell]

O is there a way to figure out spans algebraically

Without kind of visualizing it

[nimble egret]

Linear dependence and stuff

[pale shell]

Ok

[nimble egret]

Like we stated

Two linearly independent vectors span a plane right?

And in the case of R2, all of R2

Take e.g. [1, 0] and [0, 1]

I think it's pretty clear to see that

a[1, 0] + b[0, 1]

Can equal all possible [x , y] depending on what we pick for a and b

[elder robin]

Can anyone point me in the right direction on how to solve this problem? I know I need to find the orthogonal line that goes from point A thru the line L (90 degrees at L intersection), but I'm not sure how to do it.

And once I find the line I should be able to find the length pretty easily.

[nimble egret]

Do you know how to calculate projections?

[elder robin]

Ah, that's probably what I was missing. I remember doing it in lecture but I'm not sure how

iirc I should decompose A into pieces with 1 parallel to L and one orthogonal

[nimble egret]

Something of that type yes

Basically pick some random point on L

Find the vector joining it to A

This gives you the hypotenuse of the right triangle you want to form

Use projections to form the rest of the triangle

[elder robin]

Thank you

[wintry steppe]

How do I prove that a vector space doesn't have a finite basis?

[sonic osprey]

show that for any finite set of vectors, there's some vector that cant be written as a linear combination

[wintry steppe]

but say I'm given the vector space of all continuous real functions

[sonic osprey]

ok

[wintry steppe]

I'm thinking, I show that it's not finite dimensional

so thus its infinite dimensional, and as such doesn't have a finite basis

its obvious that the continuous functions in R doesn't have finite basis, so isn't finite dimensional

idk how to formally show this though

[sonic osprey]

do it exactly like I said

show that for any finite set of vectors, there's some vector that cant be written as a linear combination

[elder robin]

@nimble egret so I picked the point (7,6,5) and found that the vector joining L and A is (-6,-8,8). Now how do I find the projections? Do I find the angles between using the joining vector and L / A?

Then with the angles I use trig to find lengths, which I can use to find the exact point?

[nimble egret]

You don't need angles

Well actually hmm

My approach would have been to find projection side and use Pythagorean theorem

But angles might work as well, right angle trig anyway

[wintry steppe]

@sonic osprey so you're saying I must show that there is a vector that can't be written as a linear combination (span) of a finite set of vectors?

[elder robin]

Ok I'll try it out, ty

[wintry steppe]

@sonic osprey so there is a real valued continuous function that can't be represented by the span of a finite set of real valued continuous functions?

[sonic osprey]

yes

[wintry steppe]

so could I say that since R is uncountably infinite, there is a polynomial function (hence, continuous) that is nx^y, but for any linear combination of (x^y, 2x^y, ... , mx^y) I can always set n = /sum(1,m) +1

and since nx^y is a vector in the real-valued continuous functions that can't be represented by the span of finite number of real-valued continous functions, the vector space doesn't have a finite basis

@sonic osprey how does that look?

[sonic osprey]

bad

because you've only shownthat nx^y can't be written as a linear combination for this specific basis

like I said

you need to show that for any finite set of vectors, there's some vector that cant be written as a linear combination

[wintry steppe]

but just showing that nx^y can't be written as a linear comb. shows that it is not finite dimensional

and thus doesnt have a finite basis

[sonic osprey]

no thats just wrong

Because youve only shown it cant be written as a linear combination of (x^y, 2x^y, ... , mx^y)

[wintry steppe]

right, idk how to generalize it to show that for any finite set of vectors, there's some vector that cant be written as a linear combination

how would you start to prove it? I'm stuck

[sonic osprey]

Your idea will work, just not in the way you've stated it

[wintry steppe]

ok so I show that a subset is not finite dimensional

subspace*

[sonic osprey]

the other way to think about this is to find an infinite set of continuous functons are linearly independent

[wintry steppe]

ok so the set of polynomial functions {x^0, x^1, x^2,..., x^n} where n is uncountably infinite

the sum of these can only equal zero if all coefficients are zero

so linearly independent

[sonic osprey]

yeah that works t

[wintry steppe]

what does this show?

[sonic osprey]

this shows that there's a set of infinite vectors that are all linearly independent

And its a fact that if you have some set of vectors that are linearly independent, then the dimension of vector space is at least as large as that set

[wintry steppe]

ohhh right

but isn't that only for finite dimensional vector spaces?

@sonic osprey I have this lemma in my notes that is exactly what you are saying

but it has the qualification that V is a finite dimensional (f.d.) vector space

[elder robin]

Ok, I found a vector (p) which is the correct length of the orthogonal one, but now how do I get the point at which this vector comes out of L to hit A?

Sorry for the messy handwriting

I know the correct point is (5,4,3), but I can't figure out how to get there

[nimble egret]

Draw it

[elder robin]

Yea, I've been using an online graphic thing to visualize everything

nvm, got it

(-4, -6, 10) is the vector relative to point A, so if I do A - (-4,-6,10) I get (5,4,3) 🙂

-4 in x direction, -6 in y direction, 10 in z direction from point A

[sonic osprey]

@wintry steppe its true for infinite too

try to adapt that proof

[pale shell]

Guys

If we have a linear combination of three indépendant vectors in R4, what is the span?

[empty copper]

... a 3-dimensional subspace (?)

[slow scroll]

@real plaza anything specific you don't understand? The idea is that the row operations (switching equations, adding an equation to another, and scaling equations) doesn't change the solutions.

There is another interpretation in which each elementary row operation has a corresponding matrix which applies that operation. Does that ring a bell?

[nimble egret]

Well the left sides and right sides are equal in each equation

[slow scroll]

its just adding something on both sides. If y = x and a = b then y + a = x + b

They are taking the equation y = x and replacing it with another (true) equation y + a = x + b

[nimble egret]

If a = b

[slow scroll]

so, one key idea is that you still have the other equation, a = b
So you go from
y = x
a = b
to
x + a = y + b
a = b

and the statement y = x is still there, since you can "undo" it by replacing row 1 by row2 - row1.

and not every operation has this property. Namely, if you were to instead replace row 1 by 0 times row1, then you lose that information that was there before.

nope, but there are examples of operations that will give you "extra values." Namely, if i take a row and replace it by another row:
y = x
a = b becomes

y = x
y = x
Then instead of finding x1 x2 x3 that satisfy y = x and a =b, they now only have to satisfy y = x, a weaker constraint.

The key idea is really invertibility. At any step along the way during elimination, you can undo what you have done to get the original information. Similarly, if you start with the solutions, you can find the original system by doing the row operations that solved it, in reverse.

surprise
and you keep 7=7 because you need that information to see that 5=5. Thats the whole idea here

linear algebra provides a good framework for what is going on. When matrices are introduced, you'll see that a system can be represented by a matrix A and a vector b (where b is the column of right hand sides) in the equation Ax = b.
Any elementary row operation has a corresponding invertible elementary matrix.

So the question is: if you have some elementary row operation matrix E, does the equation Ax = b have the same solutions as the equation EAx = Eb, the system you get by applying the transformation to both sides?

Kind of, you have to be careful not to take something for granted: not all matrices are invertible.

And it turns out, since that operation is invertible, you multiply both sides by the inverse of E: E^-1
E^-1 E Ax = E^-1 E b
=> Ax = b

doing Ax = Eb and not EAx = Eb would be more like a mechanical error in the elimination process. If you have worked with gaussian elimination on matrices, it would be like performing the row operation on the left matrix and forgetting to do the same on the augmented part.

or like saying y = x implies a*y = x

But unlike the algebra we're used to, there are nonzero matrices "X" such that you can have
XA = XB but not A = B for matrices A and B i.e. cancellation doesn't work in general. delicate point

[final wing]

<@&286206848099549185>

[feral grove]

!15m

[final wing]

I tried using definiteness to prove the condition but im not sure if that's all that needs to be showed?

[slow scroll]

Don’t worry too much about it. I just dumped a ton of information on you lol. In short, solving a system is like solving a normal linear equation ax=b, except we have to be more careful about how we go about it.

“Learning about matrices” isn’t exactly that straightforward since to understand what they are, u should really understand what a vector space is.

Soon enough tho (probably before introducing vector spaces) you’ll shift from using linear equations to describe systems to using matrices. The conversion between a system and its matrix representation is pretty simple

I taught myself what I know using mostly a book called “linear algebra done wrong” by Sergei Treil. It’s a pretty theoretical book, but it doesn’t outright avoid determinant and matrices like Axlers “linear algebra done right”. I agree that it can be daunting. I think that its not a bad idea to use multiple resources, except that differing terminology and ordering of topics can possibly make things confoosing.

[feral grove]

treil is the best intro to lin alg book imo

the only problem w/ treil is how he deals with dimension, but yeah treil is better than that

[slow scroll]

That’s not surprising. There is a lot of debate about “where to start” and how to teach it when it comes to linear algebra in particular. What’s weird about Treil’s dimension?

[feral grove]

he mentions dimension late in the book and he doesn't do super rigorous stuff with it

late being not first chapter

your book feels like a book about linear algebra for like engineers

which tbf is most intro lin alg books

[slow scroll]

I am taking what is supposed to be a theoretical math major LA course this semester and my first homework is basically putting matrices in RREF lul

[half ice]

They're taking it easy on you

Hopefully it won't stay that way

[final wing]

lmao

[slow scroll]

I doubt it will. I think everyone has to go through the process of reducing matrices... at least once... I guess....

[final wing]

we started off in first year with LADR

[feral grove]

because the goal isn't to teach linear algebra, but to teach how linear algebra can be used in computation

[final wing]

Started off rough

[feral grove]

ladr is an eh book

[slow scroll]

Wow lucky tho

[feral grove]

i think so

[final wing]

I really like LADR tbh

I looked LADW

[feral grove]

that determinant section just makes me cringe

in ladr

[final wing]

im skipping chapter 10 though

[slow scroll]

No. LADR is just really... different

[final wing]

yeah

need to know determinants properly

[feral grove]

ladw has a really nice construction of the determinant

because most books kinda just say this is the thing, here's the formula, done

yes

[final wing]

We did the first 5 chapters of LADR+matrix stuff in 4 months

and we're currently on inner products

[feral grove]

treil does a great job of motivating linear algebra (again outside of dimension) and giving the actual math of it

ladw will be in a different order

probs ch.1/ch.2 flipped

could be the same after that

yeah but make sure you understand all of it

[slow scroll]

Random question, but how does axler get the eigenvalues of a matrix without determinant?

Ur good

[feral grove]

he does the determinant

[slow scroll]

Yea it’s been a while tho

[feral grove]

he just does it badly

it is

3b1b is very good

i'd reccomend it

just remember all his videos deal exclusively in R^n

ladw will deal with more abstract vector spaces

over more abstract fields

yes but like

what about C^n, or function spaces

or polynomial space

R^n is a vector with n coordinates of real numbers

[half ice]

In an engineering course, not much time is taken on a vector space. You see what makes a subspace, and move on. The course is largely based on solving systems of equations, computations on matricies, using the determinant to solve problems.

[feral grove]

i.e. big bad

[half ice]

Oh and eigenstuff

The basics

[feral grove]

still computation, you probably don't even mention characteristic polynomials

[half ice]

Btw C^n is a list of n complex numbers.

Much like R^n is a list of n real numbers

We do mention characteristic polys, and use them to get the eigenvectors and values. We use these for diagonalization. We don't mention Cayley-Hamilton, or use diagonalization for anything

[feral grove]

ok, i guess that's more than i thought

[half ice]

That's the upper end. Basic change of basis, but no mention of an orthonormal basis. No finding them

No using diagonalizations for anything D:

[feral grove]

oof

cayley-hamilton is a really nice result

that feels reasonable to teach too

C is just better R

[half ice]

When you start considering the topology of these spaces, C is always better than R

[feral grove]

fundamental theorem of algebra

[sonic osprey]

The solution to equations has nothing to do with topology

[feral grove]

i was typing that while kaynex was typing the topology thing

[half ice]

You get a vector multiplication that works pretty well on C

[feral grove]

also asuna low tier waifu

this seems like overkill

[half ice]

Lol forget I mentioned it. I didn't have any good examples on the mind.

[feral grove]

i mean yes

[sonic osprey]

There are no topological properties that C has that R doesn't that makes C algebraically closed and R not

terrible sentence but

Sure, but it's hard to say that this is the reason that C is algebraically closed

There are plenty of spaces that contain R that have such a subset that are not algebraically closed

[half ice]

I think we got two conversations mixed up here

[sonic osprey]

So it's hard to say that this a topological property inherent to the algebraic closure of R

[half ice]

I mentioned that C^n has better topological properties but am likely talking shit because it may not I didn't think through my stuff there

[sonic osprey]

Yeah, I was trying to say that it's not topological properties that C has that makes it nicer

It's algebraic closure, which is not really a topological property

[half ice]

Especially if we're considering vector spaces because we're pretty locked down

[feral grove]

bottom line, C is better R

[sonic osprey]

Better is ambiguous but sure

[feral grove]

i mean i agree that's mostly a meme statement

[half ice]

We never focused on recognizing vector spaces. We didn't talk about what a linear transformation does to a vector space. A basic talk about linear independence and dimension but we're not tested on it and don't see it in the nullspace/column space

[feral grove]

never heard of it

oh mit ocw

it's probs fine then

i think it's applied tho

it's hard to find a good rigorous linear algebra class

[cursive narwhal]

Try Algebra by Technion

on youtube

[drowsy goblet]

Is this any close to being correct?

[dusky epoch]

you didn't include units

[gray dust]

and why did you multiply V with s^2?

[drowsy goblet]

0.054 m/s^2

because I am given s and not s^2?

so I made s^2

[gray dust]

no, you take the product Vs not Vs^2

as given in the directions

[drowsy goblet]

so my new answer is 1.8m/s

One of the s in the denom cancelled

[gray dust]

cool

[drowsy goblet]

is that actually right though?

[gray dust]

👍🏾

[drowsy goblet]

Nice

I went from acceleration vector to a velocity vector, didn't I?

[gray dust]

yes

[drowsy goblet]

Look at that, didn't even have to use calculus

[gray dust]

wait til you do vector calc in kinematics

[drowsy goblet]

thank you for the help

[gray dust]

you're welcome

[pulsar turret]

could you say that N is a subset of N?

it's not wrong is it?

N is here all natural numbers

[cursive narwhal]

What's the definition of the subset relation?

[dusky epoch]

any set is a subset of itself @pulsar turret

[cursive narwhal]

Suppose A and B are sets. Then, what does $A \subset B$ mean?

[stoic python]

Abhijeet Vats:

[pulsar turret]

ok thx

[cursive narwhal]

Oof nice lol

[pulsar turret]

@cursive narwhal A is a subset of B

[cursive narwhal]

Yea but what does that say about the elements of A?

[pulsar turret]

all the element in A are also in B

[cursive narwhal]

So if we're asking ourselves if $A \subset A$, you're saying that all the elements of A are in A

[stoic python]

Abhijeet Vats:

[cursive narwhal]

That's a tautology

[pulsar turret]

yeah

[cursive narwhal]

So, since it's always true, every set is going to be a subset of itself.

This has nothing to do with sets of numbers, by the way. This is really just something in the context of set theory

[pulsar turret]

yeah, I was just wondering if there was a rule that says you can't do it or something, you know like deviding by zero

thanks for the help

[cursive narwhal]

Sure

[pulsar turret]

what does this represent. vector v = bigsum(ti wi) for i=1 to k.