/sci/ - Science & Math » Thread #14361291

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Anonymous Sun 03 Apr 18:19:02 2022 No.14361291 View Reply Original Report

Quoted By: >>14361385 >>14361395 >>14361414 >>14361444 >>14361449 >>14361458

Are complex numbers 2x2 matrices or vectors in R2?

Anonymous

Anonymous Sun 03 Apr 2022 18:47:34 No.14361385 Report

Quoted By:

>>14361291
The canonical way to view complex numbers is as the quotient field $\mathbb{R}[x] / (x^2 + 1)$

Anonymous

Anonymous Sun 03 Apr 2022 18:50:29 No.14361395 Report

Quoted By:

>>14361291
matrices because of the multiplication

Anonymous

Anonymous Sun 03 Apr 2022 18:55:52 No.14361414 Report

Quoted By: >>14361449

>>14361291
complex numbers are scalars.
they can also be represented as vectors in R2 (and other ways, like with 2x2 real matricies)

Anonymous

Anonymous Sun 03 Apr 2022 19:07:05 No.14361444 Report

Quoted By:

>>14361291
>Are complex numbers 2x2 matrices
they are a class of special 2x2 matrices equipped with normal matrix addition and multiplication...
OR, they are ordered pairs of real numbers where (a,b)+(c,d)=(a+c,b+d) and (a,b)(c,d)=(ac-bd,bc+ad)
these are equivalent

Anonymous

Anonymous Sun 03 Apr 2022 19:08:17 No.14361449 Report

Quoted By: >>14361453

>Are complex numbers 2x2 matrices
they are a class of special 2x2 matrices equipped with normal matrix addition and multiplication...
OR, they are ordered pairs of real numbers where (a,b)+(c,d)=(a+c,b+d) and (a,b)(c,d)=(ac-bd,bc+ad)
these are equivalent

>>14361291
>>14361414
>R^2
WRONG. C is not like R^2.

Anonymous

Anonymous Sun 03 Apr 2022 19:09:30 No.14361453 Report

Quoted By: >>14361460

>>14361449
>represented as vectors in R2
>represented
as in mathematical representation.

Anonymous

Anonymous Sun 03 Apr 2022 19:12:01 No.14361458 Report

Quoted By: >>14361466

>>14361291
it depends on which level of "is-ness" you attach to isomorphisms

Anonymous

Anonymous Sun 03 Apr 2022 19:12:31 No.14361460 Report

Quoted By:

>>14361453
C is not R^2, by no means.

Anonymous

Anonymous Sun 03 Apr 2022 19:15:02 No.14361466 Report

Quoted By: >>14361485 >>14361516

>>14361458
C is trivially NOT isomprphic to R^2 by any stretch. It is, however, isomorphic to the space of 2x2 matrices of form [a, -b; b, a].

Anonymous

Anonymous Sun 03 Apr 2022 19:18:53 No.14361485 Report

Quoted By: >>14361491 >>14361494 >>14361503

>>14361466
you sound pretty sure of yourself.

i can think of things i can do with an R2 representation that i can't do with C, but not the other way around.

enlighten us. what's wrong with using R2 as a representation of C?

Anonymous

Anonymous Sun 03 Apr 2022 19:20:31 No.14361491 Report

Quoted By:

>>14361485
>things i can do with an R2 representation that i can't do with C
for instance, you can optimize non-holomorphic complex functions. this is how complex least squares is analytically formulated. the derivatives of real functions don't have as many constraints as the derivatives of complex functions.

Anonymous

Anonymous Sun 03 Apr 2022 19:21:19 No.14361494 Report

Quoted By: >>14361520

>>14361485
>i can think of things i can do with an R2 representation that i can't do with C
Privide an example, because you are dead wrong. C can do EVERYTHING R^2 can do--as well as things it cannot do.
>what's wrong with using R2 as a representation of C?
R^2 lacks a field structure

Anonymous

Anonymous Sun 03 Apr 2022 19:24:06 No.14361503 Report

Quoted By: >>14361520

>>14361485
complex derivatives are not directional so you don't need that niggerlicious total derivative bullshit

Anonymous

Anonymous Sun 03 Apr 2022 19:26:59 No.14361516 Report

Quoted By: >>14361522

>>14361466
>C is trivially NOT isomprphic to R^2 by any stretch
C is isomorphic to R^2 e.g. as vector spaces or as smooth manifolds

Anonymous

Anonymous Sun 03 Apr 2022 19:28:14 No.14361520 Report

Quoted By: >>14361527

>>14361494
>R^2 lacks a field structure
i see you're trolling. to be a representation of C, you equip R2 with a multiplication rule. the addition rule is the same as the vectorspace addition rule, so you don't need to supply a new one of those.

>>14361503
gradient based complex optimization is used all the time

Anonymous

Anonymous Sun 03 Apr 2022 19:28:58 No.14361522 Report

Quoted By: >>14361540

>>14361516
C has structure that R^2 specifically does not have. This makes them no isomorphic.
>they are both vector spaces!!!!
"yes"

Anonymous

Anonymous Sun 03 Apr 2022 19:30:18 No.14361527 Report

Quoted By: >>14361529

>>14361520
>you equip R2 with a multiplication rule.
you equip an ordered pair with a multiplication rule. once equipped, it is no longer the same as R2.

Anonymous

Anonymous Sun 03 Apr 2022 19:32:07 No.14361529 Report

Quoted By: >>14361557

>>14361527
>once equipped, it is no longer the same as R2
that does not follow at all. defining a function on a vectorspace doesn't magically nullify the vectorspace axioms.

Anonymous

Anonymous Sun 03 Apr 2022 19:34:29 No.14361540 Report

Quoted By: >>14361560

>>14361522
What's your definition of "isomorphic"?

Anonymous

Anonymous Sun 03 Apr 2022 19:38:51 No.14361557 Report

Quoted By: >>14361560 >>14361565

>>14361529
A is "isomorphic" to B when A and B share identically the same algebra. R^2 lacks structure that C definitely has. So they arent the same.
At the same time, the ordered pair thing and the special 2x2 matrix thing CAN be shown to all satisfy the same field axioms. So these may be said to be isomorphic.

Anonymous

Anonymous Sun 03 Apr 2022 19:39:53 No.14361560 Report

Quoted By:

>>14361540
>>14361557

Anonymous

Anonymous Sun 03 Apr 2022 19:41:16 No.14361565 Report

Quoted By: >>14361567

>>14361557
Define what it means to "share identically the same algebra". GIve a precise definition please.

Anonymous

Anonymous Sun 03 Apr 2022 19:42:00 No.14361567 Report

Quoted By: >>14361582

>>14361565
no

Anonymous

Anonymous Sun 03 Apr 2022 19:45:53 No.14361582 Report

Quoted By:

>>14361567
fuck off with your handwavy arguments then

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