Algebra Definition and 999 Threads

In math, we normally proceed by learning elementary arithmetic and then elementary algebra. For me algebra is all about assigning a symbol to an unknown value and manipulating it to find its value. Now I don't understand how concepts like groups, fields, rings etc. fall into the algebra...

Hey PF, as my thread a week or two ago said I am currently planning on taking a college Calc and Analytic Geometry class with formal education only up to Geometry. I am very proficient in Geometry, good in algebra 1, have some experience in Trig and a little in Algebra 2. For this reason I was...

In peskin p. 160 forth paragraph they say to verefy Ward identity in equation 5.74. I don't succeed, they say some algebra is needed. I conjecture that this some algebra is what i miss. Any help will be appreciated - thanks a lot.

Apologies if this should be in homework section but I thought it best suited here. Been revising past papers but with no solutions. the following questions all require just a true or false answer. Any help or confirmation of my answers would be appreciated. 1 - every N x N matrix has N...

I know De-Morgan's law that $$ -(p∧q) = -p∨-q $$ Also $$ -(p∨q) = -p∧-q $$ But for material implication and bi conditional operations there are also some transformation. What is the law or proof for it? Like $$ p⇒q = -p∨q $$ $$ p ↔q = (p∧q) ∨ (-p∧-q) $$ There may be other properties also that I...

Homework Statement ##J=r^{2}\dot{\phi}## [1] ##\dot{r^{2}}=E^{2}-1-\frac{J^{2}}{r^{2}}+\frac{2MJ^{2}}{r^{3}}+\frac{2M}{r}##. [2] (the context is geodesic equation GR, but I'm pretty sure this is irrelevant). where ##u=r^{-1}## Question: From these two equations to derive...

Homework Statement [/B] Matrix A = 1 1 0 1 Matrix B = a b c d Find coordinates on a, b, c, d such that AB = BA. Homework EquationsThe Attempt at a Solution I calculated AB and BA with simple matrix multiplication, but am not sure where to go from here. AB = a + c a + b...

Hello everybody, in Schwartz' QFT book it says (p. 483 - 484) In Problem 25.3 this is repeated asking the reader for a proof. I wonder though if this is really true. I know this can be proven for Lie algebras of compact Lie groups (or to be precise, every representation is equivalent to a...

I found out I can pick up a second major in math should I elect to take a two semester sequence in abstract algebra. My first major is in chemical engineering. Right now, I plan on taking a two semester sequence in either: 1) probability with measure theory, 2) abstract algebra (Dummit and...

Let $P$ be a function defined on $[0, 1]$ such that $P(0)=P(1)=1$ and $|P(a)-P(b)|<|a-b|$, for all $a\ne b$ in the interval $[0, 1]$. Prove that $|P(a)-P(b)|<\dfrac{1}{2}$.

Given series:1,2,5,12,25... How did they get :##T_n=a(n-1)(n-2)(n-3)+b(n-1)(n-2)+c(n-1)+d## And for series like 3,7,13,21,... they have given ##T_n=an^2+bn+c## How do you get these equations?

So I have the metric as ##ds^{2}=-(1-\frac{2m}{r})dt^{2}+(1-\frac{2m}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}##* I have transformed to coordinate system ##u,r,\phi, \theta ##, where ##u=t-r*##(2), where ##r*=r+2m In(\frac{r}{2m}-1)## and to the coordinate system ##v,r,\phi, \theta ##, where...

Homework Statement I have the expression [((1-x)/x) * x^(2/1-x)] / x^(2x/1-x) I want to simplify this expression.Homework Equations None The Attempt at a Solution I am not so good with algebra. But I tried to find a common factor for x^(2/1-x)] / (x^2x)/1-x and then add them. However, it...

https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/ Sorry for the long title but ST = string theory. Just thought it was interesting news personally since string theory has been elusively hard to prove or observe(at least the particles it claims to predict, notably...

[FONT=Times New Roman]I would like to know some general properties of the modulo (remainder) function that I can use to rewrite expressions. For example, say we wanted to prove the following by rewriting the right-hand-side: $$ \Big{\lfloor} \frac{n}{d} \Big{\rfloor} = \frac{n - n \pmod d}{d}...

Hi I got stuck at the following circuit problem which involves linear algebra, since I am not a physics major, I don't even have the basics to get started. Please shed some light, really appreciate it! Thanks a lot. 1. Homework Statement Consider a long chain of resistors wired up like this...

Hi It's just that last step I'm not getting, so you have: [1 / Kz] - [1 / (2K)z] = [ (2K)z - Kz ] / [(2K)z * Kz] = [ (2)z - 1 ] / [(2K)z*] Then what? Thanks

Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f. I'm having trouble defining a function to prove this. Could anyone give me a start on this?

My book says in the slow motion approx, so ## v << c ##, ##v=\frac{dx^{i}}{dt}=O(\epsilon) ## It then states: i) ##\frac{dx^{i}}{ds}=\frac{dt}{ds}\frac{dx^{i}}{dt}=O(\epsilon) ## ii) ## \frac{dx^{0}}{ds}=\frac{dt}{ds}=1+O(\epsilon) ## The geodesic equation reduces from...

This problem is so simple that I'm not exactly sure what they want you to do: Let A and B be n x n matrices such that AB = BA. Show that (A + B)^2 = A^2 + 2AB + B^2. Conclude that (I + A)^2 = I + 2A + A^2. We don't need to list properties or anything, just manipulate. This all seems...

I am trying to learn the formalism of qm, so i am following the book linear algebra done right but is it worth it to study every proof? I mean what is the attitude to follow with such a proof oriented book to eventually have a solid basis in the libear algebra of qm?

Homework Statement Homework Equations A=LU, U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2, The Attempt at a Solution I used MATLAB and the relations: U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2, to find a solution I found U^-1*L^-1 , let =B...

I decide to self-study linear algebra. I have heard words about some good books on this subject such as Sheldon Axler's. Unfortunately his book is only loanable for 4 days in my university library. There is this book from S. Lang that I can borrow for one month, so what do you think about this...

Homework Statement Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression: R.RT= I with respect to the parameter t...

Homework Statement Show that the given matrices are row equivalent and find a sequence of elementary row ops that will convert A into B. a = 2 0 -1 1 1 0 -1 1 1 b = 3 1 -1 3 5 1 2 2 0 Homework EquationsThe...

Homework Statement "Let L1 be the line having parametric equations : x = 2 - s, y = -1 + 2s, z = 1+s and L2 be the line: x = 1 +t, y = 2+ t, z =2t . a. Do the lines intersect? If so, find the point of intersection. b. Find the point P on the graph of L1 that is closest to the graph of L2...

The question word for word : "Write the equation satisfied by all of the points P(x, y, z) that are at the same distance from the point F(0, 0, 4) and the plane z = 0." I figured I could maybe start by finding the distance between point F and the plane z = 0 but I can't figure out how to...

Homework Statement How do you know if say [(x_1,y_1),(x_2,y_2)] = x_1x_2 + 7y_1y_2 ? or any other equation? Homework EquationsThe Attempt at a Solution

I was given the following as a proof that the inertial tensor was symmetric. I won't write the tensor itself but I will write the form of it below in the proof. I am confused about the steps taken in the proof. It involves grade projections. A \cdot (x \wedge (x \cdot B)) = \langle Ax(x...

Recently I have known an algebraic theorem on polynomial while learning the method of partial fraction. The theorem is : Any polynomial can be written as the product of linear factors and irreducible quadratic factors. I did not find an intuitive proof of this theorem.I asked a question in this...

Warning: This is going to be a bit long. (Apparently my post was too long so it wouldn't render at all. I've split this into two threads.) I worked out some basic Algebraic properties of a Lie Algebra. This is similar to my previous thread about SU(2) but as I don't know this example I'm...

First, let me say that I am a senior physics undergrad. I have failed Linear Algebra once before. Otherwise I am a straight A student. I'm also taking Ordinary Differential Equations right now, and I breeze through that class without a care in the world. I'm not sure if I've developed some sort...

ab'c + a'b + bc' + abc = ac + a'b + bc' (How to further reduce this?) Kmap gives B + AC

I was taught that when you have a polynomial fraction where the denominator is of a higher degree than the numerator, it can't be reduced any further. This seems wrong to me for a couple of reasons. 1. If the denominator can be factored some of the terms may cancel out 2. Say you have the...

Homework Statement Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B−1 = CA. Homework EquationsThe Attempt at a Solution I think I have this figured out, just checking it. Heres what I got: ABC=I (ABC)B-1=IB-1 (B*B-1)AC=IB-1 I*AC=IB-1 Cancel I using left...

I'm looking at: http://arxiv.org/pdf/gr-qc/9712019.pdf, deriving the FRW metric, and I don't fully understand how the Ricci Vectors eq 8.5 can be attained from 7.16, by setting ##\partial_{0} \beta ## and ##\alpha=0## I see that any christoffel symbol with a ##0## vanish and so so do any...

Homework Statement (a) Show acceleration is perpendicular to velocity (b)Show the following relations (c) Show the continuity equation (d) Show if P = 0 geodesics obey: Homework EquationsThe Attempt at a SolutionPart (a) U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} +...

Homework Statement I am confused a little bit. I have been asked to simplify this function: A/B + /BC + /AB I seriously can't seem to know how to simplify this as for me its already simplified. Homework Equations N/A The Attempt at a Solution The best I could do is : (/A+/B) * (A+B+C) I did...

Homework Statement Two pumps of different sizes can empty an entire fuel tank in 4.8 hours. Used alone, the larger pump would empty the tank 4 hours less than would the smaller pump. If using only the smallest pump How long will it take to empty the tank? Homework Equations X = small pump...

Homework Statement Prove that set of all onto mappings of A->A is closed under composition of mappings: Homework Equations Definition of onto and closure on sets. The Attempt at a Solution Say, ##f## and ##g## are onto mappings from A to A. Now, say I have a set S(A) = {all onto mappings of A...

Homework Statement Prove that if A is an n x n matrix with the property A3=A, then det(A)=-1, det(A)=0, or det(A)=1 Homework EquationsThe Attempt at a Solution At first I started with the property A3=A I then applied the determinant to both sides. From this point I don't really see any...

<< Mentor Note -- OP has been advised to type their questions into the forum next time, instead of inserting images >>[/color] 1. Homework Statement Dear Mentors and PF helpers, I have a question from today's lesson. Homework EquationsThe Attempt at a Solution Is 1) and 2) both...

Recommend a self study book for linear algebra with complex numbers

I need some help in understanding the reasoning and analysis in the solution to Exercise 4.5 in Robert C. Wrede and Murray Spiegel's (W&S) book: "Advanced Calculus" (Schaum's Outlines Series). Exercise 4.5 in W&S reads as follows: https://www.physicsforums.com/attachments/3918 The...

Now ℝxℝ≅ℂ, seen by the map that sends (a,b) to a + bi. ℂ is a field, so the product of any two non-zero elements is non-zero. However, this doesn't seem to hold in ℝxℝ, since (1,0) * (0,1) = (0,0) even though (1,0) and (0,1) are non-zero. What am I missing? Also, the zero ideal is maximal in ℂ...

Homework Statement The vectors that are perpendicular to (1,1,1) and (1,2,3) lie on a ____. Homework Equations The Attempt at a Solution This is really straight forward, but I cannot validate the answer to myself. The textbook says that they should lie on a line, but why is this? Obviously if...

Is it a must to know clifford algebra in order to derive the dirac equation? I recently watch drphysics video on deriving dirac equation and he use two waves moving in opposite directions to derive it, without touching clifford algebra. If this possible, what is the intuition behind it?

<<Mentor note: Please always use descriptive thread titles.>> First of all, the title is such that it attracts most views.You see, in class our professor did some goofing around numbers and variables in the relativistic energy momentum relation: E2=(pc)2+m02c4 Since the energy required to...

1. Are uncountable unions of sigma algebras on a set X still a sigma algebra on X? 2. Are uncountable intersections of sigma algebras on a set X still a sigma algebra on X? (I think this statement is required to show the existence of sigma algebra generated by a set) 3. If 2 is true, can we...

Hello loves, I know there are post very very similar to this, but I just needed some help understanding it. any help is truly appreciated. 1. Homework Statement A model rocket is launched straight upward with an initial speed of 50 m/s. It accelerates with a constant upward acceleration of 30...