linear-algebra

[pallid rampart]

No

I'm not sure what you mean

[wintry steppe]

well on the first row : -6x_1 + 6x_2 = 0

so -6x_1 = -6x_2

[barren void]

@barren void
So this means that T is onto because the range of T is all vector sp. of 2x2?
@tribal lodge

As I understand T is not onto

[wintry steppe]

x_1 = x_2

[pallid rampart]

Well that doesn't tell you anything

[wintry steppe]

what should I be looking at

[pallid rampart]

If one row is a multiple of the other row

Or one column is the multiple of the other column

If one row is -6 and 6, that doesn't mean the two rows must be linearly dependent

You can have something like

$$\begin{bmatrix}-6&6\0&1\end{bmatrix}$$

[stoic python]

Whoever:

[pallid rampart]

and the rows are linearly independent

[wintry steppe]

why is it linearly independent

thats what I don't get

-6x1 + 6x2 = 0

and x2 = 0

that's what we're looking at here right ?

[pallid rampart]

If the rows of a square matrix A are linearly independent, then the only solution to AX=0 is when X=0

So when you set up the equations, you can conclude x_2=x_1=0, then the rows are linearly independent

[wintry steppe]

okay I see

so in the case of the previous matrix

-6x1 + 6x2 = 0

and

5x1 - 5x2 = 0

there is no way to say x1 = 0 or x2 = 0

[pallid rampart]

x1 and x2 can be 0, but they do not need to be

and therefore the rows are not linearly independent

because there is another solution, namely x1=x2=1

so yes

[wintry steppe]

okay I see

thanks

[pallid rampart]

Now most of the time when they say obviously linearly independent/dependent, they're just replying on intuition

which hopefully you'll develop soon

[wintry steppe]

hopefully yeah 😂

[half storm]

I need to show that if the anihilator of W is invariant with respect to the tranpose of T, that W is T-invariant.

I can prove the converse.

Here's what I have so far
Let $x \in W$ Consider $T(x)$. We know that for any $g \in W^{0}$ that $T^{t}(g) := g \circ T \in W^{0}$ So $ (g \circ T)(x) = 0 \implies g(T(x)) = 0$

[stoic python]

JohntheDon:

[half storm]

<@&286206848099549185>

[forest quail]

wouldn't this be false because if u was a zero vector, it would just be a point?

my prof says it's true, but i can't figure out why

[wintry steppe]

your prof probably forgot about the case where u is 0

[forest quail]

ah okay

[pallid rampart]

The inner product should not be a function but a scalar

[brittle lark]

hmm

so how do i do this when there's functions

[pallid rampart]

So $\brk{\vec{v_2},\vec{u_1}}=\int_0^4 \vec{v_2}\vec{u_1}\dd{x}=\int_0^4\sqrt{x}\dd{x}$

[stoic python]

Whoever:

[pallid rampart]

This is how you should calculate the inner product

[brittle lark]

okay

$\vec{u_1}=1

Is right though?

idk how to use that bot

lol

[pallid rampart]

The expression for u_1, u_2, u_3 are all correct

But you didn’t evaluate a single inner product correctly

[brittle lark]

cool

[pallid rampart]

I meant that these parts are correct

[brittle lark]

yeah thats how i interpreted it

[untold wagon]

heya

im just learning linear algebra and need a bit of help with a matrix

[2 5 | 1 0]
[4 1 | 0 1]
..A.......I

i want to use elimination to find the inverse

how do i remove 4 at A(2,1)

or is the matrix singular

[wintry steppe]

A isn't singular

if you want to remove that 4, one thing you can do is add -2 times the first row to the second row

you can see that A isn't singular by either working out the computation you've started here, or by calculating it's determinant and seeing that it's nonzero

[brittle lark]

@pallid rampart so is this accurate now?

(Image uploading)

[untold wagon]

@wintry steppe thanks

[pallid rampart]

I don't think I can decipher that

[brittle lark]

lol uh

i made a mistake half way through

anyhow ill rewrite that

my prof will prob say the same thing

[brittle lark]

Should be much more readable now lol

As in what does it mean by associated normal system, ik it can’t be rref because thats c4

[pallid rampart]

Alright yes that is correct

The first image

But you don't need to scale them at the end

Because the question explicitly said you only need to find an orthogonal basis

Not an orthonormal basis

[gray dust]

@brittle lark you don't need to normalize. additionally this'd lose points bc the answer mustn't have fractions

[brittle lark]

wait so

okay let me fall back to smth i left out, when he says no fractions (he said this in a lecture) he only meant in the vectors not the numbers you multiply them by

also you're saying i dont need to normalize?

so just take the fractions out of the answer

[pallid rampart]

It is stated in the question

[gray dust]

normalize v=divide by ||v||

[pallid rampart]

That you can't have fractions

[brittle lark]

gotcha so i just remove the fractions from my answer and im good to go

(assuming i understood correctly)

[gray dust]

the issue is you don't get you did extra work

[brittle lark]

no i get it now

its not asking for me to normalize it

[gray dust]

yes only orthogonalize

[brittle lark]

yeah i got it now, i just misunderstood thank you lol

[gray dust]

yeah just keep in mind the difference of orthogonal vs orthonormal bases

[brittle lark]

did any of you catch the question i asked about c?

[dire delta]

I have two vectors A and B that are added together to create C

is the following basis enough to uniquely identify A and B (Rotational invariance assumed)

1. The opening angle of A and B
2. A + B
3. A^2 + B^2

The applications is I have two indistinguishable vectors going into an ML model and I would like my representation of the two vectors to not involve enforcing a lexiographic ordering for which vector is vector 1 and which is vector 2. It would preferable to describe the two vectors using only combinational qualities like those above.

[dusky epoch]

opening angle?

[dire delta]

The angle between the vectors

sorry I think thats physics terminology

[gray dust]

@brittle lark the matrix in c2 represents Ax=b. its associated normal system is A^T Ax=A^T b

[brittle lark]

so i find A^T*A

[gray dust]

& A^T b

[brittle lark]

so which value would the answer to that part be

from the equation

all i have is A

i dont have x or b

although A^T A cant be it because the rref of that gives me 0=1 so yeah...

[gray dust]

the matrix in c2 represents Ax=b
do you know what this means

[brittle lark]

some what my professor never really explained it just kinda started saying it in his prerecorded videos

...

[thick dock]

what kinda system are you working on?
Is it a wave equation of sorts?

[brittle lark]

its a quadratic equation

that im approximating

[gray dust]

to recap in a linear system Ax=b, A is a known matrix & b is a known column vector, x is an unknown column vector

[brittle lark]

so in my case what is b

my last column?

[gray dust]

the system's augmented matrix is (A|b), a matrix made by cramming A with b as the last column

[brittle lark]

okay i understand

so i have A and b

x is unknown

so when i do A^T A should I remove the b column?

right?

[gray dust]

what's A?

[brittle lark]

how do i use the bot thing in here to send that formatted?

[gray dust]

ok just look at the eqns in c1

[brittle lark]

yeah so it's like (-2^2, -2, 1) for the first row of A

[dusky epoch]

$ \bmqty{1 & 2 \\ 3 & 4 \\ 5 & 6} $

[stoic python]

Ann:

[gray dust]

yes & b is a column vector containing the values on the right side of each eqn

[brittle lark]

yep

[gray dust]

just to check, write the next 2 rows of A

(-2)^2 != -2^2 btw

[brittle lark]

yeah i recall that, i dont think it caused any errors in my math beyond what i sent in that picture, I simplified them all. but i just added the ()

(1, -1, 1), (0, 0, 1)

[gray dust]

ok now get the augmented matrix of

associated normal system is A^T Ax=A^T b

[brittle lark]

so my answer is [A^T A|b] or [A^T A|A^T b]

this is the part im confused at with what you gave me

in regards to the equation A^T Ax=A^T b

[gray dust]

to recap in a linear system Ax=b, A is a known matrix & b is a known column vector, x is an unknown column vector
the system's augmented matrix is (A|b), a matrix made by cramming A with b as the last column
i'll rephrase it as the augmented matrix of (matrix)(unknown)=(column) is (matrix|column)

[brittle lark]

so it's [A^T A|A^T b]

because A^T is also on the right side of the equal sign in that equation

[gray dust]

the whole right side is the column

[brittle lark]

okay

gotcha

[gray dust]

@brittle lark cool

[brittle lark]

I've learned i will never take an online class again because it's these little things that i cant raise my hand about and ask during class... thank god i found this discord

[gray dust]

ye stop by again if you ever need

[brittle lark]

that might be right now because the next and last question on today's assignment is off the walls crazy

granted there's a lot of explanation but it has literally nothing to do with his lectures

Please tell me im not the only one that thinks my professor just put some calc concepts into this assignment because he like calculus or smth...

[gray dust]

where are you stuck? @brittle lark

[brittle lark]

one second actually i figured out the next step i think

there's just so much information in the question which makes it hard to follow lol

so i have to find what that series converges to right? iirc the p-series test doesn't tell you what something converges to, just that it converges

its been a while since i took calc2...

[gray dust]

once you get ||t^2||^2 it's easy algebra to get the series' value

[brittle lark]

okay so i guess im confused as to what i should be plugging into the * formula

to get the value of ||t^2||^2

[gray dust]

do you know what ||v|| denotes?

[brittle lark]

normalizing

oh wait it'd just be t^2 for both f(x) and g(x) right

then take the square root of the result of the * equation?

[gray dust]

not quite, idk wym in normalizing

[brittle lark]

my professor called it "norm"

idk

[gray dust]

physicists call it magnitude

[brittle lark]

definitely not what my professor called it but we are probably talking about the same thing

[gray dust]

if you plot a vector in R^n it's what you'd call its "length"

[brittle lark]

ok

so is it kind of like calculating speed from velocity

[gray dust]

https://i.gyazo.com/e01285ea7839c8b58ad2d495d2de67e6.png ever saw smth like this in physics?

[brittle lark]

yes

[gray dust]

the vector's got "length", physicists call it magnitude, linalg calls it norm

[brittle lark]

ok

[gray dust]

take a vector in R^n v=(1,2,3). how do you find ||v||?

[brittle lark]

1^2+2^2+3^2

[gray dust]

not quite

[brittle lark]

also

sqrt

the sum

[gray dust]

note the steps you did. you basically did ||v||=sqrt(v dot v)

[brittle lark]

yes

so ||t^2||^2 = sqrt(t^2 dot t^2)^2

[gray dust]

how did your class introduce inner products?

[brittle lark]

using matrices

like

[gray dust]

[brittle lark]

<A,B> = tr(B^T A)

then ||A||=<A,A>^1/2

thats straight from my notes

[gray dust]

that's an inner product on a certain kind of vector space but did you cover just what it is in general?

[brittle lark]

not really

god i love online classes

[gray dust]

that's meh

ok think back to the dot product on R^n. it takes 2 vectors and gives a scalar while obeying certain rules

[brittle lark]

okay

[gray dust]

btw did you cover the defn of a vector space?

[brittle lark]

yes i am pretty sure

[gray dust]

axioms & such?

[brittle lark]

no

i have no idea what an axiom is

i think thats in the next/last unit

its a 1 month summer class

[gray dust]

this class is chaos

[brittle lark]

lol

[gray dust]

https://mathworld.wolfram.com/VectorSpace.html not familiar at all?

[brittle lark]

nope

might as well be chinese

oh

the math yes

vocab no

[gray dust]

or https://en.wikipedia.org/wiki/Vector_space#Definition just read the stuff in the chart

these are basically criteria that a set, along with given definitions of vector addition & scalar multiplication, must satisfy in order to qualify as a vector space

[brittle lark]

gotcha i read it it makes sense

[gray dust]

it's how we generalize what a vector space is beyond what you usually think of as a box of pointy arrows

[brittle lark]

wait theres a little more to the definition section

one sec

okay so axioms are like rules

[gray dust]

ehh you can take it that way for just this

again, the vector space axioms are criteria a set must satisfy in order to be a vector space

[brittle lark]

okay

[gray dust]

accompanying a vector space is a so-called field of scalars, a set from which scalars are picked, and in your linalg class you'll probably only ever look at vector spaces whose accompanying field of scalars is the set of real or complex numbers

[brittle lark]

so how does this tie back to the ||t^2|| stuff? I'm following what you're saying im just not sure where you're going with it

[gray dust]

ok think back to the dot product on R^n. it takes 2 vectors and gives a scalar while obeying certain rules

when you look at a vector space other than R^n, there are certain ways you can define taking 2 vectors and getting a scalar while obeying largely the same rules the dot product on R^n does

[brittle lark]

okay

[gray dust]

that's what an inner product is, a function that takes in 2 vectors from whatever vector space you're working with and gives a scalar, that has certain properties like the dot product does

[brittle lark]

okay

[gray dust]

then you can view the dot product just as the (standard) inner product on R^n. and back in R^n, you find the length/norm/magnitude of v by ||v||=sqrt(v dot v)

if you have a general vector space upon which an inner product is defined, you can then also define a norm by ||v||=sqrt(<v,v>), <v,v> being the inner product of v with itself

[brittle lark]

so i plug in t^2 for f(t) and g(t)

since in my case v=t^2

then i take the sqrt of that

[gray dust]

if you rearrange, ||v||^2=<v,v>

[brittle lark]

yes

because ^2 cancels out the sqrt

[gray dust]

yeah. also if this seems long winded, it was to give you a peek at what a linalg class usually covers, so to give background & motivation for the computations you do in class

[brittle lark]

yeah no thank you lol it did clear up some confusion in other areas lol

[old flame]

If $U={(x,0,0) | x \in F}$ and $W={(y,y,0) | y \in F}$ under the context of sums of subspace, why is $U+W={(x,y,0) | x,y \in F}$

[stoic python]

Otoro:

[dusky epoch]

U is the span of {(1, 0, 0)} and W is the span of {(1, 1, 0)}

[old flame]

Sorry I haven't learnt about span yet

[dusky epoch]

are you asking why it isn't {(x+y, y, 0) | x,y ∈ F}

you could write it like that too

[old flame]

Yup

[dusky epoch]

it's just that every vector of the form (x+y, y, 0) can also be written in the form (x,y,0) and vice versa

[old flame]

Oh so the vector x+y just became another vector called x?

Just a problem with notations yeah ?

[dusky epoch]

x+y isn't a vector

[old flame]

Oh sorry, scalars I meant

[dusky epoch]

i mean sure if you wanna look at it like that

[old flame]

Okay thanks

[half storm]

Does this extend to the infinite dimensional case?

So if V and W are infinite dimensional, does there always exists a unique linear map between a basis for V and any subset of W?

[dusky epoch]

does there always exists a unique linear map between the a basis for V and any subset of W?
wat

i mean yea if you have a hamel basis for V then i guess this same existence & uniqueness of that linear map will still be true

[half storm]

Cool.

[dusky epoch]

but also nobody cares about bases for infinite dimensional vector spaces

[half storm]

I have to use this for a proof I'm pretty sure.

I mean they are importnat in some context right?

Like for fourier series

[dusky epoch]

not in the strict linear-algebraic sense

[half storm]

But I have heard that people are less interested in studying them.

[dusky epoch]

hamel bases only allow finite linear combos

if you wanna add infinitely many things then convergence issues get in the way

[half storm]

Do you have any idea how you would go about proving this for infintie dimensional spaces. It's clear that if V is finite dimensional, then it's a consequence of Theorem 2.6

But in the infinite-dimensional case right, where $\beta$ is infinite. It seems a bit more difficult to find what the linear transformation has to be.

[stoic python]

JohntheDon:

[half storm]

@dusky epoch

[dusky epoch]

what isomorphism

[half storm]

sorry, I meant linear transformation.

[dusky epoch]

recall the definition of a basis: for every vector for every $x \in V$ there exists a family of scalars ${c_v }{v \in \beta}$ such that $$x = \sum{v \in \beta} c_v v,$$ with the proviso that all but finitely many $c_v$ are zero.

moreover, this representation is unique.

[stoic python]

Ann:

[half storm]

From what you're showing me it seems that it's going to be very similar to the linear map that they give for the finite-dimensional case where
$T: V \to W$ $T(x) := \sum_{i = 1}^{n} a_iw_i$

Where $a_i$ are the set of scalars that you just gave i.e. $ x = \sum_{i = 1}^{n} a_ix_i $ where $x_i \in \beta$

[stoic python]

JohntheDon:

[half storm]

Don't you run into the problem of potentially taking a sum over infinitely many numbers.

Those convergence issues that you were talking about.

[dusky epoch]

no

the proviso ensures you don't

[half storm]

oh "all but finitely many are zero"

I see.

So you define the function basically in the same way. So you define the function $ T: V \to W$ $T(x) := \sum_{(c_v \neq 0) \land (v \in \beta)} c_vf(v)$

[stoic python]

JohntheDon:

[half storm]

They solved a system of equations.

If I type a lot of this out then I'll be repeating a lot of what they are saying

But basically you were supposed to find a matrix represenation of that linear map between $P_3 \to M_{2 \times 2}$ yea

[stoic python]

JohntheDon:

[half storm]

that matrix that they have right $\rho_{c}$, it's a 2x2 matrix and we know that for any 2 x 2 matrix, we can express it as a linear combination of the matrices in the set $C$.

[stoic python]

JohntheDon:

[fickle agate]

oh ok so did they basically figure out the a(matrix_1 in c) + b(Matrix_2 in c) + c(matrix3_in c) + d(Matrix3_in d) = matrix [20 45 -24 69]

[half storm]

Yup

That's exactly it.

When you solve that, those numbers that they have should pop out.

👍

[fickle agate]

i'll clear my msgs (johnthedon has an unanswered q above)

[half storm]

no problem.

[stoic python]

JohntheDon:

[delicate zealot]

I have a question about expression vectors in different coordinates systems. I'm not sure that I am doing the problem correctly. If I'm not can you point out my error and if I'm right can you explain why?

Naturally this is all surprising and confusing when seeing something like this for the first time and I'm just trying to wrap my brain around it as best as I can

[umbral smelt]

I've checked that, and I think you already do that correctly

@delicate zealot

[delicate zealot]

Whaaaaatttttttttt

Why?

[umbral smelt]

Uhh, why do you think you did that wrongly? lol

I don't really know the terminology for this in English, so sorry in advance

[delicate zealot]

Well I just don't understand why the process works I guess

That hasn't clicked yet

[umbral smelt]

But I think you want to change alpha into the coordinate bases w.r.t gamma right?

[delicate zealot]

yeah

[umbral smelt]

So yeah, the coordinate bases w.r.t to some bases is basically we just take the scalars as it's vector (I know this is poorly worded)

So, if we want to change the elements of alpha into the coordinate bases w.r.t to gamma, you need to change it into a linear combination of gamma first

[delicate zealot]

OHHHHHHHHHHHHHHH there's a theorem that says that as long as the basis represents the same vector space then you can express one vector in a certain basis as a linear combination of the basis vectors of a different coordinate system

[umbral smelt]

then, after you found the scalars that satisfies the linear combinations, you take the scalars, and you get the vector w.r.t to gamma base

because it's a linear combination of gamma bases

[delicate zealot]

yes yes yes yes I think that this is starting to make sense now as to why that process works

I don't give a rip if I can do the work if I don't understand what it means so thanks for helping me work though that 🙂

Brain took awhile to see why it worked

gotta understand the why before you can feel good about the how

[umbral smelt]

You're welcome!

[latent ledge]

Suppose V and W are finite-dimensional and $T \in\mathcal{L}(V,W)$. Prove that there exist a basis of V and a basis of W such that with respect to these bases, all entries of $\mathcal{M}(T)$ are 0 except that the entries in row j , column j , equal 1 for $1 \leq j \leq\dim range T$.

[stoic python]

Konoha:

[latent ledge]

this is rank null theorem problem, but I not certain to say that j,j entries are 1

say v1,v2,...,vn satisfy Tv1=w1,...,Tvn=wn where w1,...,wn is part of the basis of W, and Tu1,..Tuk maps to the null of W

[half forge]

can someone help me with 2 and 3

[half storm]

@half forge Well, they gave you multiple choice so there is something you can do. You can take each of the vectors and see if they satisfy they aformentioned conditions yea?.So say if i were to take the first vector $ a = (1 , 1 , -6 )$. I check to see if it is in the span of the aforementioned vectors i.e. there exists $ a,b,c$ s.t. $ a (1,1,1) + b (1, -1, 0) + c( 4, 0 , -5) = (1,1,-6)$ and then check to see if $(1,1,-6)$ is orthogonal to $(1,1,1) \text{ and } (1,-1,0)$ i.e. $(1,1,-6) \cdot (1,1,1) = 0$ and $(1,1,-6) \cdot (1,-1,0) = 0$.

If the vector statisfies all three of those statements then it's the one that you want.

If it doesn't move on to b) c) d) until it does satisfy all three of those statements.

Of course, you could solve directly i.e. without checking all of the vectors.

[stoic python]

JohntheDon:

[half storm]

I can go ahead and tell you that the first one (1,1,-6) doesn't work because $(1,1,-6) \dot (1,1,1) = 1 + 1 - 6 = -4 \neq 0$ so $(1,1-6)$ is not orthogonal to $(1,1,1)$. So it's not the vector we are looking for.

[stoic python]

JohntheDon:

[half forge]

oh so whicj is the answer?

[half storm]

d) is. But you should probably figure out how to solve for it yourself.

[half forge]

okay got it

what about 3?

[half storm]

I'll tell you how to do it but I won't give you the answer this time. You need to check if each of those vectors are in the span of (1,1,1) (1,-1,0). If it's in the span, then you check to see what it's distance from (4,0,-5). i.e. use the distance the formula between two points.

[half forge]

okay got it

how do you do this one

[wintry steppe]

why does it say that the components of $\begin{bmatrix} 1 \ -2 \ 1 \end{bmatrix}$ and $\begin{bmatrix}0 \ 1 \ -1 \end{bmatrix}$ add to 0? am i missing something here

[stoic python]

polynomial:

[half forge]

for which one?

[wintry steppe]

i'm not talking to you

i'm asking a question

verbatim

Every combination of v = (1,-2,1) and w = (0,1,-1) has components that add to 0.

@dusky epoch

[spice storm]

it means they are orthoganal

[dusky epoch]

uh

what

Every combination of v = (1,-2,1) and w = (0,1,-1) has components that add to 0.

this is false

[wintry steppe]

yeah exactly

i am so confused

[dusky epoch]

v+100w has no zero components

is that verbatim the statement you got hit with

[wintry steppe]

yup

1 sec

[dusky epoch]

can i see the entire problem

[wintry steppe]

#6 is the solution

[dusky epoch]

so it's a blank, not a zero

[wintry steppe]

no.

it means fill in the blank, ann...

[dusky epoch]

oh

wait no

[wintry steppe]

yes.

[dusky epoch]

okay no aight

i just got it

it took me this long bc i lit just woke up

but i just got it

[wintry steppe]

got what

[dusky epoch]

every linear combination of v and w is in the subspace {(x,y,z) | x+y+z=0}

cause v and w are

[half storm]

@half forge Do you know the taylor explansion of the matrix exponential is.

This one is easy to look up

Then you just differentiate term by term.

[dusky epoch]

aight uh.
one of these convos has got to move

[wintry steppe]

ann

what does that have to do with it, though

like

the components do not add to 0

for every combination

[dusky epoch]

actually they do

[wintry steppe]

oh wait

i mean

i guess i get what you mean

but

like

that is very bad wording

?

[dusky epoch]

it's not the worst wording i've seen

but it's kinda wonky yes

wonky enough to confuse my brain in "not yet fully woken up" mode

[wintry steppe]

🙂

what does it mean

for a vectors components to add to 0

is there any particular significance to it

or was that just the way to solve the problem without algebra

i.e. by noticing that they add to 0 after you combine them

[dusky epoch]

it means that the vector's components, when added, give 0

[wintry steppe]

yes but is there any more significance to it

[dusky epoch]

it means that the vector lies in the plane {(x,y,z) | x+y+z=0}

no there isn't

[wintry steppe]

ok

[dusky epoch]

beyond the ability to say that the vector is orthogonal to (1,1,1) i suppose

[spice storm]

the answer zero will come significance when you learn orthogonality. @wintry steppe

[wintry steppe]

.. i know what orthogonality is.

this is my second treatment of linear algebra..

[weary isle]

Why max((A-B)+B) <= max(A-B)+max(B) ?

[dusky epoch]

what are A and B in this context

@weary isle

👻

[weary isle]

@dusky epoch I have figured it out with the help of others. Try to think A and B in terms of triangle side length

[dusky epoch]

ok you didn't answer my question

[weary isle]

10 <= 6 + 8

[cursive narwhal]

She asked you what A and B are

[dusky epoch]

this does not answer my question of "what are A and B in this context"

what part of "what are A and B in this context" do you not understand

did you even read the message where i asked you "what are A and B in this context"

[weary isle]

@dusky epoch @cursive narwhal

[dusky epoch]

aight that's a lot of symbols

[cursive narwhal]

can you like

[dusky epoch]

are A and B supposed to be functions or what

[cursive narwhal]

just post the original thing from the start next time?

[dusky epoch]

oh wait. i forgot. i don't get the privilege of getting direct answers to my questions.

[wintry steppe]

lol

rip ann

maybe you should try asking a third time

third time's the charm as the saying goes

(/s)

[weary isle]

How to prove max(sum(k*x)) <= max(x) , 0 <= k <= 1 and sum(k) = 1 ?

[dusky epoch]

i know you hate responding to these questions in a simple and clear manner, but what is x?

@weary isle

...ok, so now that you've edited it: what's k now?

it's not clear what you're even trying to say let alone how to prove it

and can you please post your clarification as a new message instead of making a nonsensical edit to your original

@weary isle?

if you don't want me to help you then please say so explicitly

yes

[weary isle]

@dusky epoch both k and x are sequences of values

[dusky epoch]

okay... so then why max(sum(k*x)), and not just sum(k*x)?

not that you can't take the max of a single number - you just get the number itself - but the wording is just weird to me

also, will i have to ping you for every single reply? @weary isle

[weary isle]

@dusky epoch wait, I am still reorganizing my thought

[dusky epoch]

this answers neither of my questions.

[weary isle]

see the last 2 lines of the proof

@dusky epoch

[dusky epoch]

aight what the fuck am i looking at lmao

[half storm]

I mean for the second statment you need to show that 0 (i.e. the zero polynomial is unique). The uniqueness of an additive identity is importnat.

But you've shown what is the most important part.

The set {1, t, t^2, t^3, ..., t^n} is a basis yup.

And if the aforementioned set is linearly independnet then it indeed does form a basis for P^2. There is a theorem that says any linearly independent set that has the same number of elements as the Hamel Dimension of the space is a basis for it.

So yea that works.

Yup

That's the standard basis.

Yea, I've heard about it. I'm doing the same thing and I understand LA alot better than I did when I first took the class.

[half storm]

Yea these are kind of hard and require some algebraic shuffling.

You have to be clever with the algebra

I think that the first thing you probably want to do is use the fact that you have distributivity over scalar additon right, so
$0v = (0 + 0) v = 0v + 0v$

[stoic python]

JohntheDon:

[marble lance]

That's exactly right, but if you're still very early in a LA class, it might be expected that you don't work so loosely with associativity of vector addition and that you keep the brackets where you have three terms, and show how you move the brackets

[marble lance]

That's a fine proof.

[spice storm]

@wintry steppe I found the proof very lacking. Where is your let Statement? What properties did you use?

[formal stump]

Not sure if this is the right channel but does anyone know how to get the answer to this?

[gray dust]

try #prealg-and-algebra

[formal stump]

thanks

[celest slate]

so

basically like

well, linear algebra involves infinite spaces

those too, but I mean like

you can go an infinite amount in any dimension

1,2,3...infinity

R3

see, in my head, it’s often easiest for me to get intuition by looking at small subsets of the space, and examining what happens to that subset

and doing that is a lot easier to express in abstract algebra

already, learning basic set theory has cleared up tons of confusion

because when I imagine an example in my head, now I can actually write down what that example is

I couldn’t do that before

and I still can’t always do that, which is why I want abstract algebra so I can

I was saying things like “Take N, where N is an arbitrary number. Now if you choose a certain value of N...”

and obviously that makes no sense

[quartz compass]

if anything it will make your life worse

linear algebra is more concrete than abstract algebra

[celest slate]

now I can say “Define N to be the set of numbers (n,2n,....). Now, imagine taking the number n.”

And obviously now it’s a lot more evident exactly what I mean, and a lot easier for me to avoid confusing myself

Concrete isn’t good when I often wind up getting an intuition that’s hand-wavy and not concrete

[quartz compass]

hey I was gonna say that

[celest slate]

ik I did shorthand

[quartz compass]

you basically just said concrete isn't good because it's not concrete @celest slate

[celest slate]

I meant “Let N be the set of numbers n in R such that n^2 < n^4. Now choose a number n from N, and consider the equation n^3....”

yes

general mathematical writing

exactly

is that not... I was under the impression a lot of that writing comes from abstract algebra

well yes but

that sort of set and group writing

[quartz compass]

lol set and group writing

[celest slate]

the knowledge for how to do that would be best explained in an abstract algebra course, right?

[quartz compass]

what's your goal

[celest slate]

see this is exactly my point

I can’t express my goal

Yes

[quartz compass]

what do you like

[celest slate]

My goal is, when I get confused and ask a question, to be able to understand the answers I get

[quartz compass]

why do math and not, biology say

[half storm]

That style of writing you're talking about comes from set theory

[celest slate]

yes

that specific example does, John

I think other styles of writing similar to that are from group or ring or stuff theory right?

[half storm]

Abstract Algebra uses this notation because set theory is the basis of all mathematics

[quartz compass]

I think if you learn any of that you'll just make your problems worse

[celest slate]

I know

[quartz compass]

you seem to have no idea what you're talking about currently, trying to learn abstract algebra on top of that will just make you have more jargon piled onto your nonsense

[celest slate]

I meant um

[half storm]

Abstract algebra itself studies properties on sets and operations on them these are algebraic structures.

[quartz compass]

abstract algebra is a topic, not just notation

[celest slate]

I might say “variables come from high school algebra. Matrices come from Linear algebra. Groups come from....”

how would you finish that sentence

I know it’s a topic but it introduces notation is my point

wait what

[quartz compass]

notation is introduced in every topic

if there's something you don't understand, you look at the notation in the topic you're in already

[celest slate]

yes I know

[quartz compass]

you don't hop to another topic

your entire conception is like this though

so don't say you know when you don't

[half storm]

The most notation you can get from a group is something like this $ \langle G, * \rangle $

[stoic python]

JohntheDon:

[celest slate]

but I can’t always express my confusion in the notation of the current topic

[quartz compass]

yep

[celest slate]

Until I learned set theory, I couldn’t express my confusion from linear algebra whatsoever

basic basic set theory I mean

I want to learn more advanced set theory

wdym proofs

like, real analysis?

I can do proofs, at least I think I can

as long as I know the foundational concepts for stuff

what book

can you give me an example of what a proofs based problem might look like, and I can tell you if I know how to do it

sure I mean

hang on a sec

let me go get you one of my old proofs

[quartz compass]

you're putting the cart before the horse

there are proofs in all areas of math

[celest slate]

\

Theorem:

Let $a,b,c$ be natural numbers. Then, $(a+b)+c) = a+(b+c)$.

\

Proof:

\

The proof is by induction on $b$, with $a,c$ being fixed. We need to check two things:

\

1. When $b=0$, the property holds.

\

2. When the property holds for $b$, the property also holds for $b'$, where $b'$ is the successor to $b$.

\

I will frequently be using the properties that $a+b = b+a$ and that $(a'+b) = (a+b)'$.

\

Because for all $a$ it is known that $a+0 = a$, then $(a+0)+c = a+c = a+(0+c)$, so the base case is true.

\

Now, suppose that it has been shown that $(a+b)+c = a+(b+c)$ for some arbitrary $b$. Then, we need to show that $(a+b')+c = a+(b'+c)$. We can do this by the following chain of reasoning:

$$(a+b')+c = (a+b)'+c = ((a+b)+c)' = (a+(b+c))' = ((b+c)+a)' = (b+c)'+a = a+(b+c)' = a+(b'+c)$$

Hence, by induction, the statement holds for all natural numbers.

[stoic python]

Notchmath:

[celest slate]

besides a stray parenthesis on the first line I just noticed, there’s a proof I did

so in terms of “intro to proofs”, how does that affect things

[quartz compass]

affect what things

[celest slate]

like

[quartz compass]

next time you get stuck trying to figure something out, just ask about that specific thing and see what the person here tells you to do for that and ask what prereqs they'd recommend maybe

[celest slate]

does that give you an idea of where I am with proofs?

would you still recommend I take an intro to proofs class first?

[quartz compass]

you said you have no goal so it doesn't matter

we can't really recommend anything if you have no goal

[celest slate]

I said I did have a goal

[quartz compass]

you didn't earlier when I asked

[celest slate]

yeah I did

[quartz compass]

you said you couldn't express it

[celest slate]

My goal is, when I get confused and ask a question, to be able to understand the answers I get
@celest slate

[quartz compass]

that's too vague

[celest slate]

that’s what I said earlier

[quartz compass]

that could apply to biology

[celest slate]

oh

[quartz compass]

has nothing to do with math

[celest slate]

in math I meant

[quartz compass]

why do you like math

[celest slate]

and I don’t mean conceptually understand, I mean understand the language of math they’re written in

[quartz compass]

what's so great about math

[celest slate]

I like the logic behind it, I like coming up with creative ways to approach problems, I like connecting seemingly unrelated ideas, I like the fact that it’s highly conceptual and less grounded in reality

[half storm]

These are decent reasons. If you like to think about abstractions then math can be a thing for you.

Of course there is logic in and of itself.

[celest slate]

abstractions yes those

I like to think about things in abstractions

[half storm]

You could just study this or other forms of philosophy.

[celest slate]

but I don’t have the language to express abstractions

[half storm]

Yea, I understand. I say study whatever you feel.

[quartz compass]

if you have the abstraction in your mind, you can make the language yourself

I give you permission

[wintry steppe]

if we have $\begin{bmatrix} 1 \ 1 \ 0\end{bmatrix}$

[stoic python]

polynomial:

[celest slate]

my understanding is that, while to fully express abstractions I would need almost every topic, the one which will most help me express an abstraction is Abstract Algebra

[wintry steppe]

and we reflect that over a line with direction $\begin{bmatrix} 1 \ -1 \ 2\end{bmatrix}$

[stoic python]

polynomial:

[wintry steppe]

then do we get $\begin{bmatrix} -1 \ -1 \ 0\end{bmatrix}$

[stoic python]

polynomial:

[celest slate]

polynomial would you mind waiting a second?

[wintry steppe]

but you're not asking questions

you're just discussing math

why can't you do that in #math-discussion

[celest slate]

yes I am

I was told to come in here

[quartz compass]

just ask in a questions channel @wintry steppe

[wintry steppe]

can you tell me merosity

i think it is

(it is what we get)

[quartz compass]

I didn't read your question

and I'm not gonna

[half storm]

I mean there are a lot of abstractions in a lot of different areas of mathematics lol.

[celest slate]

right

in terms of representing groups

[half storm]

I mean abstract algebra has abstract in it's name and it is full of "abstractions".

[celest slate]

n-dimensional spaces

[quartz compass]

abstract algebra just has the word abstract in it lol

[celest slate]

finite n-dimensional spaces

[half storm]

This is linear algebra ; a subset of abstract algebra .

[celest slate]

no like

wait what is

[quartz compass]

it should probably have a different name, like group, ring, and field theory

[celest slate]

yes

[wintry steppe]

this should really be in #math-discussion

[celest slate]

but groups rings and fields are what I want to be able to express

[quartz compass]

ok then learn it

[celest slate]

so why would abstract be bad to go to now

[wintry steppe]

@wintry steppe how so

[quartz compass]

you can safely ignore polynomial

[half storm]

Well, you can't discuss abstract algebra without discussing those things. So if you do want to study those algebraic structures, then you should study abstract algebra

[wintry steppe]

@wintry steppe good logic so if i was talking with someone about dog food in #discrete-math

@wintry steppe and someone came in asking a question

you'd also tell them to not interrupt right

sound logic

[quartz compass]

you were saying you want to learn abstract stuff, so you decide "I'll learn abstract algebra" - that's bad @celest slate

[half storm]

^

[quartz compass]

if you want to learn about commutative rings, then you say "I'll learn abstract algebra" - that's good

[half storm]

Right but if you want to study those things such as groups rings and fields, then abstract algebra is fine.

[celest slate]

but in particular

[quartz compass]

like there are genuine ideas to focus on

[pallid rampart]

At a higher level, all the fields of math are pretty abstract

[quartz compass]

you are entirely notation oriented here

which is not productive

[celest slate]

the concepts that I wind up having in my head, I’m pretty sure they’re all groups and rings and fields

and sets

[quartz compass]

yeah, intro to proofs is probably your best bet

just describe your concepts in english and through examples or whatever and see what people say about them as they are

if they fit to already established notions, great

if not, who cares

[celest slate]

they say they’re rings and fields and groups

so I want to learn those

[quartz compass]

sure whatever, I don't believe you

[celest slate]

well

[quartz compass]

just go learn abstract algebra if you're so hell bent on it when you don't seem to have any idea what it really is

[celest slate]

more accurately

they give me answers

and I try to look up the terms they used

[quartz compass]

just learn it

[celest slate]

and those terms are rings and fields and groups

[wintry steppe]

please explain

how this is related to linear algebra

[quartz compass]

why are we having this conversation if you know the answer

[wintry steppe]

you're just talking about education

[celest slate]

because he told me I shouldn’t and I don’t understand why

[quartz compass]

the rules aren't so strict as you believe @wintry steppe

they don't have to answer you

[wintry steppe]

so what?

that guy

@wintry steppe

said its rude

even though you're the ones talking in a questions channel

[quartz compass]

so stop interrupting or I'll delete your messages for being rude @wintry steppe

[wintry steppe]

about something offtopic

lmao

aren't you a big man

[quartz compass]

[limber sierra]

this is a questions channel?

???

[wintry steppe]

isn't it?

it's not #discussion-education

[limber sierra]

its a general subject channel where questions happen to be allowed

[celest slate]

oh

[wintry steppe]

yeah but they're not talking about linear algebra and want me to go somewhere else even tho i want to talk about linear algebra

[half storm]

Notchmath it seems to me that you are more intersted in things like math logic and symbolic logic.

[celest slate]

I thought you meant I shouldnt do it until I got to after analysis and linear algebra

[thorn robin]

completely unrelated
but why do you two have the exact same pfp? 😂

[celest slate]

oh okay

[quartz compass]

lol he wants me to notice him @thorn robin

[celest slate]

intro to what

[quartz compass]

I'm his senpai lmfao

[wintry steppe]

good one merosity

hilarious

[quartz compass]

imitation is the highest form of flattery, every day is a compliment

you could have picked anyone, but you chose me

[wintry steppe]

nice try

[quartz compass]

sweet kid

[celest slate]

well

and I’m not approaching any of this with the goal of number theory, at least not moreso than “I do all the cores, and eventually once I have the cores I can expand further to more courses, including number theory”

no no no not at all

I don’t want to cherry-pick

The reason I want to stop analysis for now

It seems more focused on reconceptualising what I already know into a more abstract form

which I mean, yes that’s very helpful

but in terms of allowing me to ask questions outside of what I already know, it’s not

Rudin’s abstract algebra book?

what’s rudin in this context

Okay I guess

here’s what I need to do

Can you briefly explain exactly what analysis (in Rudin) entails, what abstract algebra entails, and what topology entails

[limber sierra]

Idk if rudin has non-analysis books

he has a memoire

does that count

[celest slate]

so I can understand exactly what the course itself is

well what do you mean more general settings

[median forum]

metric spaces should be between topology and analysis

[celest slate]

so

[median forum]

topology is basically a generalization of geometry
adding structures that entail geometric and distance properties in sets Id say

Id say analysis + topology = func anal

[celest slate]

but what’s analysis

you keep defining analysis as basically “things”

[median forum]

Im only saying this, because the motivation for it was euler trying to classify shapes, wasnt it?

analysis is a qualitative way of studying calculus

[celest slate]

ohhhh

[median forum]

a more advanced one

[celest slate]

well Tao was just sort of proofs-based arithmetic and set theory, as far as I got

well my point being

what exactly is after that

not exactly

generally

[median forum]

thats because analysis is the firtst proofs subject for a lot of people

[celest slate]

oh

ugh

yeah algebra it is

[median forum]

continuity, convergence, integration..

[celest slate]

I’m more curious about the finite than the infinite

finite & discrete

[median forum]

thats a weird statement

[celest slate]

oh of course

but I mean in terms of

what I do first to help develop my intuition for higher level mathematical concepts

wdym it’s a weird statement

[quartz compass]

seems fine to me

[median forum]

I personally dont know how to even separate or classify my liking of finite vs infinite things

[celest slate]

like, I’m more interested in the integers than I am in the reals

[quartz compass]

graph theory, combinatorics, stuff you can do on a computer

[celest slate]

is what I mean

that sort of thing

[median forum]

I feel like thats way more specific than saying one likes finite things, merosity
but I get it now ig

[celest slate]

more interested in the set of numbers between 0 and 100, and how it relates to other sets of numbers, than I am in the set of numbers between 0 and infinity

[quartz compass]

ok that sounds weird to me lol

[celest slate]

more interested in the unit square and its properties than I am in the infinite plane

[quartz compass]

but sure, whatever floats your boat

[median forum]

hah
so you didnt have the right idea either

[celest slate]

what’s weird about it lmao

I like being able to... to grab it

[thorn robin]

he likes bounded sets

that's an analytic property

he's an analyst after all

[median forum]

true

well

[celest slate]

to feel like I can grab one end, grab the other end, and play with it like putty

[median forum]

bounded is a metric property

[celest slate]

I can’t

well

you misunderstand

like

you can’t fold the plane into a torus

well you sort of can but it’s weird

idk

[median forum]

or lie groups ig

[celest slate]

I mean like imagine it as a physical object I can interact with, and see what happens as I do different things to it

yeah

but also, imagine taking a square and folding one corner onto another corner

[median forum]

that sounds like isometries

oh nvm

[celest slate]

I like

I like numbers as things in space, I guess

not arbitrary coordinates but as actual things

in my head

[median forum]

actual?

[celest slate]

like

aargh i can’t explain myself

like I like to imagine the square that represents 5, and the square that represents 7, stick them on top of each other, and see what I get

or line

or cube or whatever

my point is

you can’t do that with the reals, you can’t do that with infinity

[median forum]

those are statements I dont know how to parse

what is that supposed to mean?

[celest slate]

like

I might imagine 5 as a 5x5 grid of points

and imagine 7 as a 7x7 grid of points

and slide those grids around on each other and see what happens

things like that

[median forum]

a lot of what youre saying is already generalized in maths

[celest slate]

yes

where

where do I learn those things

[median forum]

youre only talking about very specific things

[celest slate]

fine

[median forum]

those could be seen from the perspective if any area

[celest slate]

just that one

where would I learn that thing

[median forum]

you learn how to generalize that and find underlying structures

thats what youll find areas about

[celest slate]

what course would you recommend I take

[median forum]

hardly is there a specific example of something

[celest slate]

so I could mathematically express and generalize that

[median forum]

I mean

firstly, like people have said

you need the foundations

[celest slate]

I know all of them

I know

but which foundation would you recommend I take first

[median forum]

uhh

it could be any from basic diff equations, group/ring theory or basic analysis in the line

[celest slate]

it does when I can’t stop conceptuaising things that way

wdym

um, up through linear algebra

arithmetic, pre-algebra, geometry, algebra, trig, calculus 1-3, differential equations, linear algebra

statistics

maybe

I got an A in diffeq, but idk that I really understood a thing I was doing

it was... a weird class

[median forum]

but which diff eq?
highschool des?
never had a course that general

ah

so basic ODEs

gotcha

like learning what exponentials are useful for

dang
imagine having that structure into hs

after hs I didnt know shit

[celest slate]

I’m confused

what are we talking about

[median forum]

mb

[celest slate]

I um

I don’t know

it was that weird

I was doing stuff for sure

[half storm]

lol were you solving differential equations?

[wintry steppe]

Just checking my answer

2 and 6 are independent ? They both have no free variables in ref

[celest slate]

I don’t think I know exactly what that means

wdym by “differential equation”

the term itself wasn’t clearly defined to me

[median forum]

6 isnt

[stoic python]

slimvesus:

[celest slate]

I think I did some stuff like thag

[median forum]

oh nvm
6 is

[celest slate]

or 0

[median forum]

mb

[wintry steppe]

thx fractal

[half storm]

Are you asking are they a linearly indepenent set?

[celest slate]

I think I was mostly confused because it felt just like more calculus

[half storm]

Cause 7 isn't.

[celest slate]

and also there were drones and mathematica

I wasn’t really

I felt like I wasn’t doing anything

but I got over 100% so

I clearly was doing something

[half storm]

lol ODE's does kind of feel like that though

[celest slate]

Calc 1 through DE all felt the same tbh

the drones were only a small thing on occasion

[half storm]

lol I spent most of my time in that class wondering "how the fuck did someone figure out this algorithm to solve this"

[median forum]

I mean
tbf notch
a lot of things there sound like hs math

[celest slate]

wdym

[half storm]

It was mostly just recognizing what the equations looked like and then use the standard procedure to solve it.

[median forum]

basically john just gave you what it is

unintentionally

[celest slate]

sure but that was also the case in calc 2 and stuff

the issue is I genuinely can’t remember a single equation that I can tell you with 100% certainty was from DE as opposed to Calc III or even Calc 2

[median forum]

its hard to give motivation to study maths
a lot of it is either self given or by experience

[celest slate]

can you give an example

and I’ll see if I know how to solve it

oh

well but that was calc I sort of

[stoic python]

slimvesus:

[celest slate]

you wouldn’t need to know anything beyond calc 1 to do that

[median forum]

ay''+by'+cy=0

[stoic python]

slimvesus:

[median forum]

a,b,c constants

damn slim
we thought of the same

but yours is not homogenous

yikes

[celest slate]

sorry I actually have to go suddenly

something came up

sorry

[median forum]

well technically neither, cause yours is specific but not homogenous

[wintry steppe]

If two vectors u = [1, 2], v=[2,4] are linearly dependent, then they span is one dimension less than how many components they have ?

u and v would span R^2 but they are linearly dependent so they just span R?

[median forum]

wym how many components?

[wintry steppe]

components like [x, y, z]

3 components

[median forum]

I dont think there is an answer to your question

you can find it in in n

[wintry steppe]

ah interesting okay

yeah for sure

if they span a two-dimensional subspace of R^3

would you say they span R^2 or R^3?

lol

[median forum]

you can have all kinds of planes passing through zero in R3

not sure there is any youd call R2

and the kind of isomorphism is very strong might I add

[wintry steppe]

why

[median forum]

they span the plane x,y

[wintry steppe]

if I do c * [1,0,0] and v * [0,1,0] cant I reach everything in r^2 ?

[median forum]

x,y,0

pretty muh
it is embedded in R3
same structure as R2, but not "the same"

[wintry steppe]

oh okay I think I get it

like you said

[1,0,0] and [0,1,0] both span a two-dimensional subspace of R^3 (in this case the xy plane)

oh okay

and [1,0] and [0,1] together span R^2

I haven't

Should I ?

Yeah it's linear algebra this semester followed by differential equations next semester

[median forum]

oh jeez

doubt it

[wintry steppe]

The last thing we study is diagonalization and eigenvectors

[median forum]

des + linalg is v crammed

[wintry steppe]

yeah I know linear transformations

[celest slate]

Fractal

can I PM you about that question

[median forum]

isomorphism of vector spaces that is

[celest slate]

so as not to clutter?

[median forum]

sure

[wintry steppe]

oof that was discrete math class which I barely remember lol

Injective means that for every input there will be a different output, surjective means one input can map to multiple outputs

[quartz compass]

got surjective backwards

multiple inputs to the same output is possible

[wintry steppe]

oh okay so surjective can have one input map to multiple outputs, it just has to have all outputs covered

lol right

[half storm]

@wintry steppe You're trying to say that if a map is surjective. The the elements of the codmain can correspond to more than one element in the domain.

So for any output you can have multiple inputs

And that's right.

[quartz compass]

personally the reason I care about surjective and injective is because together you get bijective which means it's invertible

and usually you like to do and undo things

[wintry steppe]

wait so isomorphism

is this ?

S(T(x)) = x

T(S(x)) = x

oh okay I had no idea it was called isomorphism

okay I gotcha

its nice to have a link with bijective, making connections here lol

[celest slate]

so

Fractal verified for me it was indeed deq I took

and I also remember why it was so weird

it was mixed with calc III during the first semester of the year

and during the second semester, there was like nothing

[wintry steppe]

Sure, I'll try

[celest slate]

halfway through the second semester all the other students graduated and I was the only one left

and the course was designed around that earlier end date

but it meant for half a semester I was doing literally nothing

[wintry steppe]

So you want a linear transformation like T(x, y) right ?

[celest slate]

that’s why I couldn’t remember what I did

[wintry steppe]

alright

[celest slate]

because I didn’t

I was taking DE and I wasn’t aware of it

I thought I was still in calc iii

[median forum]

isomorphism of vector spaces <=> bijective homomorphism <=> bijective linear transformation <=> invertible linear transformation

[celest slate]

who are you talking to

slimvesus

oh

good

[median forum]

preserves the operations of vector spaces

[half storm]

Me neither

But you could all them that

[median forum]

I mean

they are groups

[half storm]

Yea it would just have to be additive I think

[median forum]

it would be a homomorphism of groups, but not of vector spaces

[stoic python]

slimvesus:

[median forum]

homomorphisms/isomorphisms in general are to be specified

[half storm]

Same.

[median forum]

I mean

modules have their own ig

in fact, I forgot how isomorphisms worked for a second and I only recalled it because theyre bijective homomorphisms

[wintry steppe]

is it just T([x, y]) = [x, y, 0] ?

@wintry steppe

[half storm]

That isn't an isomorphism.

[median forum]

well ye

unless

you make it

R2->R2×{0}

[wintry steppe]

f

oh

[median forum]

well

[half storm]

Oh I see what you're saying.

[median forum]

then R2->R2×{0} is right

ye
cause vector spaces and their isomorphisms specific af

spoiled things

imagine not only having a continuous, differentiable, smooth, analytic, isometric symmetry, but LINEAR

[wintry steppe]

I get that v3 is a multiple of v2

oh so

they automatically create a plane with v1 then

and if I say w = [8, 4, 7] - w is not in the subspace because it's not a multiple of v1 nor v2

so it goes in a different direction than both of them