Algebra Definition and 999 Threads

I'm currently working through the book Algebra by I.M. Gelfand and A. Shen on my own. (As advised here: https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/#toggle-id-1) This isn't really homework so I wasn't quite sure where to post this. Anyway, I've got a question...

I am planning to self-teach myself Trigonometry and all the other required fields before jumping into calculus. I have compiled a list of books that I have researched and would like your opinions and recommendations. Books: Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry...

Homework Statement I have been going through a textbook trying to solve some of of these with somewhat formal proofs. This is a former Putnam exam question. (Seemingly the easiest one I have attempted, which worries me). Consider a polynomial function ƒ with real coefficients having the...

Hi guys, there's a sample problem in m textbook where they simplify an expression from x3-6x2+12x-8/x2-4x+4 to (x-2)3/(x-2)2. Can you explain how they solved this? For reference, I'm learning about limits

Homework Statement What is the next radius outwards of this Apollonian gasket? R = radius of outer circle = 5 r1 = radius of largest inner circle = 3 r2 = radius of second largest inner circle = 1 a = unknown radius Homework Equations C = 2πr A = πr2 d = 2r The Attempt at a Solution Make a...

I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some remarks of Bresar on the centre of an algebra ... Commencing a section on Central Algebras, Bresar...

I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some remarks of Bresar on the centre of an algebra ... Commencing a section on Central Algebras, Bresar writes the...

I am wanting to get the expression up to ##O(\epsilon^{2}) ## : To show that ##\frac{1}{2w_{k}} (\frac{1}{w_{k}-k_{0}-i\epsilon} + \frac{1}{w_{k}+k_{0}-i\epsilon})## ##=## ## \frac{1}{k_{v}k^{v} + m^{2} - i\epsilon}##, [2] where ##w_{k}^{2}=k^{2}+m^{2}##, ##k## the variable, and (this seemed...

Problem 1 Simplify/solve: 2*81/2-7*181/2+5*721/2-50 Attempt at solution: a1/2=√a ⇒ 2*√8 - 7*√18 + 5*√72 - 50 = 2√8 - 7√18 + 5√72 - 50 = ? Do not know how to proceed beyond this point. Have experimented with little luck. Problem 2 Simplify/solve: a-1(1+1/a2)-1/2 * (1+a2)1/2 Attempt at...

Homework Statement I am wanting to show that ##\Delta (t) = 1/q (\sum\limits^{\infty}_{n=0} p(n)q^{n})^{24} ## where ##\Delta (q) = q \Pi^{\infty}_{n=1} (1-q^{n})^{24} ## is the discriminant function and ##p(n)## is the partition function, Homework Equations Euler's result that : ##...

The question is : Is it true that two matrices with the same characteristic polynomials have the same trace? I know that similar matrices have the same trace because they share the same eigenvalues, and I know that if two matrices have the same eigenvalues, they have the same trace. But I am...

Homework Statement Prove that $$(\bar{a} + b)(b+c) + a\bar{b}$$ where ##a,b## can be from the set ##B\in\{0, 1\}## equals $$a+b+c$$ Homework Equations Rules of Boolean Algebra 3. The Attempt at a Solution [/B] My attempt: ##\bar{a}b + \bar{a}c + bb + bc + a\bar{b}## ##b(\bar{a} + 1+c) +...

I've been studying graduate level Linear Algebra from Steven Roman's Advanced Linear Algebra (Springer, GTM). It is a terrific book, but many of the concepts are extremely abstract so that I find it difficult to retain what I've learned. Can anyone point me to some books/reading on the...

Given that $x,\,y$ and $z$ are non-zero real numbers such that $x + y + z = 3$ and $xy + yz + zx = −1$. Evaluate $$\frac{(x + 1)(y + 1)}{x + y}+ \frac{(y + 1)(z + 1)}{y + z}+ \frac{(z + 1)(x + 1)}{z + x}$$.

I'm currently an applied math major. I'm creating a schedule for my next semester and I have the choice to take either complex variables or vector analysis with linear algebra and a college geometry course(elective of choice), but I don't know which pairing will be less stressful. I am currently...

Homework Statement Express the function Y= (abd + c)' + ((acd)'+(b)')' as the complete disjunctive normal form: 2.1 by applying Boole's theorerm, Homework EquationsThe Attempt at a Solution I separated the equations to two terms (T1,T2) T1= (abd + c)' T2=((acd)'+(b)')' T1= (abd+c)'...

Bert and Ernie are running around a regular polygon with x sides, all of length 12m. They start from the same point and run in opposite directions. If Bert is twice as fast as Ernie, how far will Ernie have traveled when they meet?

Homework Statement 23. In a ABCD quadrilateral let P,Q,R,S be midpoints of sides AB,BC,CD and DA. Let X be the intersection of BR and DQ, and let Y be the intersection of BS and DP. If ##\vec{BX}=\vec{YD} ## show that ABCD is a parallelogram . Homework Equations ## (\vec{a}\cdot\vec{b})=0##...

I know that I can use guess and check, but I was wondering if there was an easier way? I got 34

Hello! Is there any rule to do sums and products like the one in the attached picture (Lie.png) without going through all the math theory behind? I understand the first (product) and last (sum) terms, but I am not sure I understand how you go from one to another. Thank you!

Homework Statement I am trying to solve for x in this equation: Homework EquationsThe Attempt at a Solution I tried solving it like a quadratic equation. This is the most simplified I could get it. Can someone please help me find a simpler expression

Homework Statement I want to show that ## \sum\limits_{n=1}^{\infty} log (1-q^n) = -\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty} \frac{q^{n.m}}{m} ##, where ##q^{n}=e^{2\pi i n t} ## , ##t## [1] a complex number in the upper plane.Homework Equations Only that ## e^{x} =...

Hello PF, I have just finished my first semester in college and did Calc. 3. Now for the spring semester i have to take differential equations and i have been given the advice that linear algebra comes in handy when dealing with DEs. So can anyone recommend a good introduction for linear algebra...

Homework Statement $$\lim_{x \to -∞}{\sqrt{x^2 + bx + c} - x}.$$Homework EquationsThe Attempt at a Solution So in problem 1, once I got to a point where I am to divide by the highest power in the denominator(x) I get something like: $$\lim_{x \to -∞}\frac{bx+c}{\sqrt{x^2+bx+c}+x}$$ Now what I...

Hi! First off, I am actually a math / econ major. I hope I'm still welcome here I am trying to figure out if it's worth it to take both of these courses or just one of them. I have not taken LA before. Course 1: Addition, subtraction and scalar multiplication of vectors, length of vector...

Homework Statement Given an operator ##D(\alpha)=\exp\ (\alpha a^{\dagger}-\alpha^{*}a)## and a function ##g(a,a^{\dagger})##, where ##a## and ##a^{\dagger}## are operators and ##\alpha## and ##\alpha^{*}## are complex numbers, show that ##D^{-1}(\alpha)g(a,a^{\dagger})D(\alpha)=g(a+\alpha...

Homework Statement Find critical numbers of the function: F(x)=t^3/4 - 2t^1/4 Derivative I got: F'(x)=3/4 t^-1/4 - 1/2 t^-3/4 Homework EquationsThe Attempt at a Solution I have found the derivative and I understand I must pull out a t in order to find critical numbers, and run across this...

Homework Statement Let ##g_k = 2cos(k/2)## and ##z=e^{ip(N+1)}## where N is an integer. There are two simultaneous equations: ##E^2 = (g_k + e^{ip})(g_k + e^{-ip}) = 1 + g_k^2 + 2g_k cos(p) ## [1] ##(1+z^2)E^2 = (g_k + e^{-ip})^2 z^2 + (g_k + e^{ip})^2##[2]...

Homework Statement X,Y,Z(coordinates) What function corresponds to the best tunnel shape? g = 9.8 m/s^2(earth gravity) Homework Equations F(x)=Y G(x)=X^2 in the xy plane G(z)= sin(X) in the xz plane H(x)= parabolic sinusoid(X^2 and sin(X) both in the xy plane) The Attempt at a Solution I have...

if A and B are 3x3 matrices such that: ABC = I, |3A|=81 and |C^T|= 2 , how to find |B^-1| I couldn't solve this because there is not much given.

I am following some lecture notes looking at the invariance of Poincare transformation acting on flat space-time with the minkowski metric: ##x'^{u} = \Lambda ^{u}## ##_{a} x^{a} + a^{u} ## [1], where ##a^{u}## is a constant vector and ##\Lambda^{uv}## is such that it leaves the minkowski...

The problem I am trying to write the equation for the plane on the following form ## ax + by + cz + d = 0 ## $$ \begin{cases} x = 1 + s - t \\ y = 2 - s \\ z = -1 + 2s \end{cases} $$ The attempt ## s, t ## are the parameters for the two directional vectors which "support" the plane. $$...

Hi, I've somehow gone the past year without paying attention to the order of the indicies when one is upper and one is lower i.e. that in general ##g^{\mu}## ##_{\nu}## ##\neq g_{\nu}## ## ^{\mu}##. A have a couple of questions : 1) ##g^{u}## ##_{v} x^{v}=x^{u}## [1] ##g _{v} ## ##^{u} x^{v}...

Homework Statement [/B] example problem: 5 = [(x)(4+x)] / (4-x) answer: 5 Homework Equations Unsure what to use. 3. The Attempt at a Solution Not sure what my professor did, but I thought that if i multiply by the reciprocal of something, I have to balance by multiplying the other side as...

In the representation theory of Lorentz transformations, the words Clifford algebra and Dirac algebra are used interchangeably. However, there is a distinction between the two. Indeed, the Dirac algebra is the particular Clifford algebra ##Cl_{4}({\bf{C}})\equiv Cl_{1,3}({\bf{C}})## with a basis...

Let the generators of the SU(2) algebra be ##\tau_{1}##, ##\tau_{2}## and ##\tau_{3}##. Consider an ##N## dimensional representation, which means that the ##\tau_{i}## are ##N \times N## matrices which act on some ##N##-dimensional vector space. Consider the ladder operators...

The Poincare algebra is given by isl(2, R) ~ sl(2,R) + R^3. What exactly does the i stand for? Thanks a lot in advance!

The question I'm stuck on is: P = NKBT/(V-Nb) - aN2/(V2) -----> (1) Re-arrange variables in the Van Der Waals equation of state, Eq. (1), so that V always appears in the equation as V/(3Nb) and P appears as 27b2P/a. Then T should appear in the combination 27b kBT/(8a). Call these...

The problem I have been trying to solve a long problem but my answer differs from my books answer with just a few peculiar terms. My answer: ##x_1' \vee x_0'x_1x_2 \vee x_0x_1'x_2' \vee x_2## Book: ##x_1' \vee x_2 ## My question is: Is it possible to simplify ##x_0'x_1x_2 \vee...

Homework Statement Use the definition of exclusive or (XOR), the facts that XOR commutes and associates (if you need this) and all the non-XOR axioms and theorems you know from Boolean algebra to prove this distributive rule: A*(B (XOR) C) = (A*B) (XOR) (A*C) Homework Equations All the...

The problem This is not a complete homework problem. I am at the last step of the solution to a long problem and only interested to know whether these following expressions are equivalent. My answer: ## a \oplus ab \oplus ac ## Answer in my book: ## a \oplus b \oplus c ## The attempt I...

What are the best ones and why for a first-timer like myself (doing self-study)? Thanks very much everyone.

Homework Statement This question has two parts. There is a quarter or radius R that is charged with a net force of Q. A point like charge of net charge q,is at a distance z from the center of the quarter. Q1: Under what condition could we use Coulomb's law to find the magnitude the force...

The problem I am trying to show that ##a'c' \vee c'd \vee ab'd ## is equivalent to ## (a \vee c')(b' \vee c')(a' \vee d) ## The attempt ## (a \vee c')(b' \vee c')(a' \vee d) \\ (c' \vee (ab'))(a' \vee d)## The following step is the step I am unsure about. I am distributing the left...

So I want to read Euler's : Elements of Algebra. However, I see many editions on Amazon. What is the best version of this book?

I'm looking at purchasing Algebra (2nd Edition) by Michael Artin, is this a good book to purchase as my first intro to linear algebra book for self learning?

So anyone of you know a book that provides a gentle and quick refresher for linear algera, in the spirit of the book "Quick Calculus" by Kleppner and Ramsey? Now that I am studying quantum mechanics, I feel I need to review the linear algebra I studied during my engineering degree. Thanks.

0.0162 * 0.000000000000002 divided by 0.000054 * 0.02 I keep getting 0 but when I try to see if its correct, it tells me its not. What the heck is the answer?

#1. How many classrooms would be necessary to hold 1,000,000 inflated balloons? (Assume one balloon is about 1 ft3 and a typical classroom is about 30 ft × 45 ft × 15 ft. Round your answer to the nearest number of classrooms.) #2. Approximately how high would a stack of 1 million \$1 bills be...

Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...