Recent questions in Upper level algebra

nainamourya181
2022-08-25

If (R, +, .) Is a ring and a belongs to R then S = { x belongs to R : ax = 0} is a subring of R

Bilgehan Yurtcan
2022-08-01

can and how you solve the set of inequality and find the intervals for variables in inequalities in the document which I uploaded using excel.

Bilgehan Yurtcan
2022-07-31

How can I solve set of inequality and find the intervals for variables in inequalities in the document which I uploaded using excel.

Upper level algebra
Answered

Michelle Mendoza
2022-07-11

What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler, but I don't claim to be an expert) and have taken abstract algebra (most familiar with group and ring theory) and have briefly skimmed through linear algebra books covering this material, but I don't quite understand the "big picture" idea, i.e., why is this useful in application? One person once told me it is the "most straightforward and useful algorithm for solving systems of linear equations, once you get beyond 3 variables or so," but maybe I'm missing something, since I usually don't see anything like what this person described to me in the linear algebra books I have. Most textbooks I've seen tend to have a more theoretical focus on this topic.

Also, any suggested texts which have good coverage on this topic would be very helpful.

Upper level algebra
Answered

ScommaMaruj
2022-07-08

I find myself completely lost proceeding to the following section of the text.

"Let $R$ be a relation, $A$ a mathematical object, and $x$ a letter (i.e., a "totally indeterminate" mathematical object). In the assembly of letters and fundamental signs which constitutes the relation $R$, replace the letter $x$ wherever it occurs by the assembly $A$. One of the criteria for forming relations is that the assembly so obtained is again a relation, which is denoted $(\ast )$ by the notation

$(\mathrm{A}\mid x)\mathrm{R}$

and is called the relation obtained by substituting $A$ for $x$ in $R$, or by giving $x$ the value $A$ in $R$. The mathematical object A is said to satisfy the relation $R$ if the relation $(\mathrm{A}\mid x)\mathrm{R}$ is true. It goes without saying that if the letter $x$ does not appear at all in the assembly $R$, then the relation $(\mathrm{A}\mid x)\mathrm{R}$ is just $R$, and in this case to say that $A$ satisfies $R$ means that $R$ is true."

However, I do appreciate the textbook that is self-contained and appreciate the author devoted to mathematical reasoning so rigorously at the beginning of the chapter. I tried to find some textbook about mathematical logic but they are either too abstract or not thorough enough that seems to start from the most fundamental (i.e. from axiom and the most basic rule).

I have read relevant posts on the subject I am asking but can't decide the material right for me. I am wondering if there are any materials or textbooks that introduce mathematical logic rigorously and serve as a supplementary text for me to understand the first chapter of the book? If there really isn't any textbook that is not too abstract but rigorous enough, I am wondering if there are any other textbooks on abstract algebra that start from mathematical logic and build the whole system from the scratch?

Upper level algebra
Answered

Raul Walker
2022-07-04

I find that I usually think about this step in the problem as a black box, into which I put a set of equations and out of which comes a set of solutions. This is particularly true if the algebra involved includes many steps.

This is good in a lot of ways and bad in some others, but putting that aside, I'm fascinated that this step seems to come up in all areas of math. Is it possible to separate math from algebra? Are there any branches of math where long chains of algebra are not found or uncommon?

Upper level algebra
Answered

Kaeden Hoffman
2022-07-04

Does anyone have any advice?

PS: If this means anything at all, as I was not a mathematics major, I did not take analysis or abstract algebra.

Upper level algebra
Answered

Jovany Clayton
2022-07-03

For example, I know that Stokes's theorem is a generalization of the divergence theorem, the fundamental theorem of calculus and Green's theorem, among I'm sure many other notions.

I've read that pure mathematics is concerned mostly with the concept of 'generalization' and I am wondering which theorems/ideas/concepts, like Stokes's theorem, are currently celebrated 'generalizations' by mathematicians.

Upper level algebra
Answered

sweetymoeyz
2022-07-02

But take, for example:

$f(x)=\frac{3}{2x+1},x>0$

and

$g(x)=\frac{1}{x}+2,x>0$

I am so confused with the whole process of finding the ranges of functions, including those above as samples, that I can't even quite explain what or why.

Can someone please, in a very step-by-step process, detail exactly what steps you would take to get the ranges of the above functions? I tried substituting x-values (such as 0), and came up with $f(x)>3$, but that was mostly guesswork -- also, $f(x)>3$ is incorrect.

Also, is there an outline I can follow -- even for the thinking process, like check if A, check if B -- that would work every time?

Upper level algebra
Answered

Lucille Cummings
2022-06-16

$Ax=b$

with conjugate gradient using computers, if A is a very sparse matrix, it can be difficult to utilize the hardware computational power maximally

Does there exist some way to rewrite

$Ax=b\text{or}{A}^{T}Ax={A}^{T}b$

to be able to utilize hardware better?

One idea I had is that often ${A}^{2},{A}^{3}$ et.c. are increasingly non-sparse. Maybe it would be possible to take multiple steps at once..?

One motivation why this should be possible is that the Krylov subspaces which the Conjugate Gradient investigates are precisely the powers of the matrix. A second motivation why this is possible is of course Caley Hamilton theorem

$P(A)=0\Rightarrow {A}^{-1}={P}_{2}(A)$

For some polynomial ${P}_{2}$ other than P.

Upper level algebra
Answered

pokoljitef2
2022-06-13

(I think it's a flaw in the way mathematics is conventionally taught that one first learns answers to questions like this only by taking a later course for which the one introducing the concept is a prerequisite, and one has no prior notice of which course that might be. And in the case of power series, there are many such courses, but still very few compared to the number of courses a student might later take, so the situation is worse.)

Upper level algebra
Answered

Gabriella Sellers
2022-06-13

Please explain without invoking other algebraic objects such as modules,rings etc and by using the concepts regarding vector spaces only (as the book assumes no such background either, it is unlikely that any reader of that book will benefit from such an exposition). Everywhere I searched, I found the explanation in terms of those concepts only and being unfamiliar to those I couldn't get them at all. Thanks in advance.

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