Algebra Definition and 999 Threads

Please, I need some clarifications about second direction, in the file attached, $$ \text { Then ad } x \text { ad } y \text { maps } L \rightarrow L \rightarrow I \text {, and }(\text { ad } x \text { ad } y)^2 \text { maps } L \text { into }[I I]=0 \text {. } $$Thank you in advance,

I found 3 papers on The Exceptional Jordan Algebra in physics arXiv:2305.00668 (hep-ph) [Submitted on 1 May 2023] CKM matrix parameters from an algebra Aditya Ankur Patel, Tejinder P. Singh Download PDF We report a theoretical derivation of the Cabibbo-Kobayashi-Maskawa (CKM) matrix...

The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...

For two quantum oscillators, I have raising and lowering operators and , and the number operator . I need to check if operators below follow commutation relations. Now as far as I know, SU(2) algebra commutation relation is [T_1, T_2] = i ε^ijk T_3. So, should I just get T_1 and T_2 in...

Homework Statement: About semidirect product of Lie algebra Relevant Equations: ##\mathfrak{s l}_2=## ##\mathbb{K} F \oplus \mathbb{K} H \oplus \mathbb{K} E## Hi, Please, I have a question about the module of special lie algebra: Let ##\mathbb{K}## be a field. Let the Lie algebra...

Homework Statement: please, could you help me to know hoe I compute the Casimir element of lie algebra sl(2), I know the basis and their relations, but i could not find the book explain in details how we get the Casimir element.. I think it is related to killing form, but also I could not find...

I am currently learning some maths from “Precalculus by James Stewart”. I was wondering if that’s ok? Is it ok to just dive straight into it or go back and brush up my algebra 2 ? I was wondering what are some good textbooks on algebra 2 by the way? Thank you. (This is all for the love of physics).

This is the defining generator of the Lorentz group which is then divided into subgroups for rotations and boosts And I then want to find the commutation relation [J_m, J_n] (and [J_m, K_n] ). I'm following this derivation, but am having a hard time to understand all the steps: especially...

Let ##P(x,y)## be a multivariable polynomial equation given by $$P(x,y)=52+50x^{2}-20x(1+12y)+8y(31+61y)+(1+2y)(-120+124+488y)=0,$$ which is zero at ##q=\left(-1, -\frac{1}{2}\right)##. That is to say, $$ P(q)=P\left(-1, -\frac{1}{2}\right)=0.$$ My doubts relie on the multiplicity of this point...

Here is an interesting book a student could do after after Algebra 1, or even integrate into an Algebra 1 course: https://www.amazon.com/dp/B077VV95N3/?tag=pfamazon01-20 And a website: https://www.calculussolution.com/ Several topics become easier, such as logarithms, when you know a...

It is easy to see that a matrix of the given form is actually an unitary matrix i,e, satisfying AA^*=I with determinant 1. But, how to see that an unitary matrix can be represented in the given way?

Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...

I'm watching a nice video that tries to explain how linear algebra enters the picture in quantum physics. A quick summary: Classical physics requires that physical quantities are single-valued and vary smoothly as they evolve in time. So a natural way to model classical physical quantities is...

The toric code is a basic computational model as follows: There are 2 operations that can be performed, A and B, on this grid. To compute the value A at each point on the grid, we transform the raw values at each dot (located in between two vertices) according to some predefined operators...

When working with fractions and when we have a fraction or equation with fractions like this one for example ##\frac{x}{x-1}+\frac{x}{x+1}=\frac{9}{4}## do we always need to set boundaries? Like, do we always need to write that x can't be a number that would give the denominator 0? In this...

I came across the following problem somewhere on the web. The original site is long gone. The problem has me stumped. May be sopmeone can provide some insight. (The problem seems too simple to post in the "Linear/Abstract Algebra" forum.) The Cost of Beer It was nearing Easter, and a group...

Hello, how can one proof that the dual of ##\mathbb{Z}^n## is ##\mathbb{Z}^n##? My idea: The definition of a dual lattice says, that it is as set of all lattice vectors ##x \in span(\Lambda)## such that ##\langle x , y \rangle## is an integer. When we now consider ##\mathbb{Z}^n## we see that...

Well, my guess is that there is something wrong with the factors chosen, because ##\left\Vert \left(0,1,0\right)\right\Vert =1## and \begin{align} \left\Vert F\left(0,1,0\right)\right\Vert &=\left\Vert...

Hello, I am looking for one or more books in combination for self-study of abstract algebra. Desirable would be a good structure of the book with good examples of sentences and definitions. Of course, exercise problems should not be missing. I am now almost tending to buy the Algebra 0 book by...

I had learned everything in College Algebra in my Algebra II course in high school, and indeed (at least at my alma mater) in engineering, physics or math, no credit is even given for College Algebra. Perhaps what is going on here is that colleges can't trust that someone who has passed (even...

The following properties of big-O notation follow from the definition: (i) if ##f(x)=O(u(x))## as ##x\rightarrow{a}##, then ##Cf(x)=O(u(x))## as ##x\rightarrow{a}## for any value of the constant ##C##. (ii) If ##f(x)=O(u(x))## as ##x\rightarrow{a}## and ##g(x)=O(u(x))## as ##x\rightarrow{a}##...

This equation takes a present value (PV) to find mortgage payments, PMT: Alternatively, switching V for PV and T for PMT: V/T = r(1-r^n)/(1-r) What is an algebraic method to solve for "r"? Can it not be solved for? I realize I can just find out "r" by trial by error in Excel using the PMT...

Summary: What would be a good book for learning Linear Algebra by myself in my situation (which is explained in my post below)? I did an undergraduate Linear Algebra course about 18 years ago. The textbook we used was Howard Anton’s “Elementary Linear Algebra”. The problem is that I never...

I am looking for good textbooks in physics, algebra, and trigonometry textbooks that are up to date and a good read. I heard that Feynman’s Lectures was really good. Is it still up to date enough? Any opinions?

Hi, For an engineer who graduated and finished typical Cal A,B,C + Linear Algebra + ODE, what book do you recommend to start reading to be a transition to advanced pure math subjects like abstract algebra and number theory? I did deep google search & concluded that that book supposed to include...

[FONT=times new roman]Problem statement : [FONT=times new roman]Let me copy and paste the problem as it appears in the text. Attempt : From the "Relevant Equations" given above, we can compare to see that ##a-1 = -1## and ##a^2+2=3##. These lead (after some algebra) to the three values of...

I am trying to convert the attached picture into dirac notation. I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ> The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS. *Was going to type in LaTex but I...

My HS senior son is taking AP Calc this year and struggled with the first quiz, which was a review of some more difficult algebra - factoring higher degree polynomials, simplifying complicated fractions etc. Ordered him Schaum’s Int algebra for practice problems, but curious about any advice...

So this expression is apparently in Sz basis? How can you see that? How would it look in Sy basis for example? The solution is following. They are putting Sz as a basis, bur how do you know that Sz is the basis here? Thanks

Dear Everybody, I am about to teach my first course, College Algebra at my university as an instructor of record. Most of the students take this course is just for liberal arts requirement for critical thinking. I feel like I have too high expectation of my students when I should not have too...

Some textbooks I found online ( open source ) College Trigonometry 3rd Corrected Edition - STITZ ZEAGER OPEN SOURCE MATHEMATICS Precalculus 3rd, Corrected Edition - Lakeland Community College, Lorain County Community College A First Course in Linear Algebra - Robert A. Beezer Cheers.

Hello, I have been looking for textbooks for self-studying linear algebra, which seems to be quite an important course. I have read that in order to study quantum mechanics well, one must have a very good command of linear algebra. Some textbooks in my country are quite bad and only teach...

Summary: Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or calculus? Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or...

[FONT=times new roman]Problem Statement : I copy and paste the problem from the text to the right. Attempt (mine) : Given the inequality ##\dfrac{x}{x+2}\le \dfrac{1}{|x|}##. We see immediately that ##x\ne 0, -2##. At the same time, since ##|x|\ge 0\Rightarrow \frac{x}{x+2}\ge 0##. Now if...

[FONT=georgia](I could solve the problem but could not make sense of the solution given in the text. Let me put my own solutions below first). [FONT=times new roman]1. Problem Statement : I copy and paste the problem to the right as it appears in the text. 2. My attempt : There are three...

Summary: Calculating a "fixed profit" on inventory [Mentor Note -- after an initial move to the schoolwork forums, this thread turns out to be a General Math question after all (see posts #5-6), so it is moved back to the General Math forum] I have 1000 products for sell in a store. Some make...

[FONT=times new roman]Problem statement :[FONT=times new roman] Let me copy and paste the problem as it appears in the text on the right.Attempt (myself) : By looking at ##\large{\sqrt{x+2}\ge x}##, from my Relevant Equations above, we have the following : 1. Outcome ##\mathbf{x \ge 0}##, since...

In [this post][1] user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle". [Here][2] I proposed three shapes that could work. The common principle behind them being that if the unit curve is...

Hi all, I have a formula which i might have expected to have brackets, but it hasn't, so i need to correctly interpret it. Could anyone please offer any pointers as to how it should read? The formula is '1.2 . h - 0.2 . x' Can i assume that it should be '(1.2 . h) - (0.2 . x)'? Thanks Zamb

The variable is already isolated on one side. I dont' know how to solve for R though. Any help? Thank you.

*Kindly note that i created this question (owned by me). My Approach, ##\dfrac {(x+y)(4x+6y)}{(5x-5y)}##=##-1## ##(x+y)(4x+6y)=-5x+5y## ##\dfrac {4x+6y}{-5x+5y}##=##\dfrac {1}{x+y}## to get the simultaneous equation, ##4x+6y=1## ##-5x+5y=x+y## ... ##4x+6y=1## ##-6x+4y=0## giving us...

The first image is the question and the second is the answer. My thinking is let's say North is positive, and South is negative. Fixed point O is the starting point. Then the question becomes +(2a-b)-(3a+2b). The answer should be -a-3b. I cannot fathom why the book gives the answer as a+b. Any...

I'm self-studying tensors from a book that doesn't have exercises. The book is Semi-Riemannian Geometry by Newman. To get a better feel for index manipulation, tensor results and calculations, I'm looking for a book that has many exercises in these topics. I'd be grateful if those knowledgeable...

Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks! My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the...

My intuition about the Lie algebra is that it tries to capture how infinitestimal group generators fails to commute. This means ##[a, a] = 0## makes sense naturally. However the Jacobi identity ##[a,[b,c]]+[b,[c,a]]+[c,[a,b]] = 0## makes less sense. After some search, I found this article...

I think the answer is an even function as the function ##x^2## is an even function and thus, is symmetrical w.r.t. Y axis. The question I have is how to do this problem algebraically. I tried to graph some functions on GeoGebra to verify my answer. a) ##y = ln(x^2)## b) ##y = sin(x^2)##...

Computer languages handle the scope of variables in a precise way so that if one symbol, such as "k" is used in different contexts, the program keeps these separate. When sophisticated human beings re-use symbols in writing mathematics, they can keep things straight, but I don't think they...

Is Advanced Linear and Matrix Algebra by Nathaniel Johnston a good book on linear algebra? Will it teach me all I need to know? Is there any calculus in it despite the name? I never took a course on linear algebra so I'm looking for something that teaches everything and includes calculus with...

Problem statement : Let me copy and paste the problem statement from the text : Attempt at solution : I could not solve the problem reducing the L.H.S into the R.H.S. However, I could solve the problem by expanding the R.H.S. into the L.H.S., though it is less than satisfactory. Below is my...

Problem : Let me copy and paste the problem statement as it appears in the text, as shown above. Attempt : I can sense there is an "elegant" way of doing this, but I don't know how. I show below my attempt using ##\text{Autodesk Sketchbook}##. I hope am not violating anything. Ok so I have...