Algebra Definition and 999 Threads

Homework Statement given two unit vectors a= cosθi + sinθi b=cosΦi+sinΦj prove that sin(θ-Φ)=sinθcosΦ-cosΦsinθ using vector algebra[/B]Homework Equations sin(θ-Φ)=sinθcosΦ-cosΦsinθ[/B]The Attempt at a Solution axb= (cosθsinΦ-cosΦsinθ)k and I'm guessing that the change in sign has...

How do I get D by itself? This one's got me baffled [(A - D)^P] / [(B - D)^R] = (C - D)^(P - R)

Homework Statement 1. Show that the Lorentz algebra generator ##J^{\mu \nu} = i(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu})## lead to the commutation relation ##[J^{\mu \nu}, J^{\rho \sigma}] = i(g^{\nu \rho}J^{\mu \sigma} - g^{\mu \rho}J^{\nu \sigma}-g^{\nu \sigma}J^{\mu \rho}+g^{\mu...

I am currently in year 9 (9 grade for those in US) and I have a really rusty and a weak math background. I have 2 months of summer holidays coming up. I should be done with pre algebra in mid December. During my summer holidays I have more than 50 hours a week avalible for study and I was just...

Homework Statement Let ε = { (-∞,a] : a∈ℝ } be the collection of all intervals of the form (-∞,a] = {x∈ℝ : x≤a} for some a∈ℝ. Is ε closed under countable unions? Homework Equations Potentially De Morgan's laws? The Attempt at a Solution Hi everyone, Thanks in advance for looking at my...

Most of the results on google happily prove A+(B.C) = (A+B).(A+C), which is that OR is distributive (over AND). But as part of their proof, they use the law that AND is distributive (over OR), namely that A.(B+C) = (A.B)+(A.C) which I can't seem to find any algebraic proof for. So are there...

This is such a basic question that I'm embarrassed to even ask it but I don't really know where else to turn. I was reading through a derivation and one of the steps goes from \frac{\sqrt{a}}{n} = \frac{\sqrt{A}}{n+m} to n = \frac{\sqrt{a}*m}{\sqrt{A}-\sqrt{a}} I'm getting really...

In Bruce Cooperstein's book: Advanced Linear Algebra, he gives the following example on page 12 in his chapter on vector spaces (Chapter 1) ... ...I am finding it difficult to fully understand this example ... ... Can someone give an example using Cooperstein's construction ... using, for...

I'm grading for a linear algebra class this semester. The class is comprised entirely of engineering majors of various flavors. The homework assigned by the professor is almost entirely "proofs" they are fairly specific proofs. Really the only thing that designates them as proofs is that the...

Homework Statement The 1.1m bar is supported by a ball and socket support at A and 2 smooth walls. The tension in the vertical cable CD is 1kN. Draw the Free Body Diagram of the Bar. Determine the Reactions at A and B. Homework Equations Equillibrium equations. Perhaps the vector...

Homework Statement Let (S,\cdot ) be an algebraic structure where the operation \cdot is associative and commutative and also the following axiom is satisfied: \forall x,y\in S,\exists z\in S: zx = y\ \ (1) Prove that if for every a,b,c\in S, ac = bc, then a=b. Homework EquationsThe Attempt...

Homework Statement Am I doing this right? The question is... (ab) + (a' + b') = 1 Homework Equations (a) Commutative a · b = b · a a + b = b + a (b) Associative (a · b) · c = a · (b · c)...

Homework Statement Consider a semi simple lie algebra. Show that if ##T_a## are the generators of a semi simple Lie algebra then ##\text{Tr}T_a=0##. 2. Homework Equations The commutation relations of the generators and the cyclic properties of the trace. The Attempt at a Solution $$[T_a...

Say I have the algebra \mathbb{Z}_5[x]/I where is the I is the principle ideal generated by x^2+4. How do I find the ideals in A? I can't seem to find an explanation that is clear anywhere. Thanks!

Homework Statement Let ##x,y,z## be arbitrary integers and the GCD: greatest common divider between any two is ##1##. Let ##(G,\cdot )## be abelian. Prove the implication: a,b\in G\left (a^x = b^y = (ab)^z = e\Longrightarrow a=b=e\right ) where ##e## is the unit element of ##G##. Homework...

Hi, I am looking for a book that explains tensors and builds a working knowledge of tensors, like the book Linear Algebra by Friedberg Insel and Spence, which I thought explained things very well (if you haven't heard of it, its an intro. book on linear algebra). Thanks!

Homework Statement Let A be the algebra \mathbb{Z}_5[x]/I where I is the principle ideal generated by x^2+4 and \mathbb{Z}_5[x] is the ring of polynomials modulo 5. Find all the ideals of A Let G be the group of invertible elements in A. Find the subgroups of the prime decomposition.Homework...

Homework Statement It is a potential barrier with E of the incoming matter wave E>U (greater than the height of the barrier). I have already done a LOT of algebra to get to the point where ##\frac{F}{A}=\frac{2kk'e^{-ikL}}{2kk'cos(k'L)-i(k'^{2}+k^{2})sin(k'L)}## which I won't bother writing...

Hello folks! New to this forum, so hoping I'm not retreading old ground. The Pauli matrices are spin angular momentum operators in quantum mechanics and thus are axial vectors. But in Clifford algebra in three dimensions they are odd basis elements and thus polar vectors. Hestenes, Baylis, other...

Homework Statement There is an equation in my class notes that I do not know how to solve. It is basically transforming equation A to equation B. It is very important to me that I know how to transform it. Homework Equations Attached as a picture. The Attempt at a Solution I do not know how...

Hey guys, So I'm stuck on another question from the previous one that I posted and would absolutely love it if I can get some help regarding how to attempt this. I literally have no clue at how to go by solving it. I have a feeling for question one that the cosine laws might come in handy but...

Hi, I have been wanting to do this for a while but not too sure how to go about it. I have the following geometric algebra \lbrace\mathbf{e}_{i}\rbrace_{i=0}^{3} which satisfy the following relations: \mathbf{e}_{i}\mathbf{e}_{j}=-\mathbf{e}_{j}\mathbf{e}_{i} and...

Hi, say we have entities A,B,C in a Relational Database. Is there a sort-of operational algebra that allows us to determine that if, say, A and B are in a many-many relation and B and C are in a 1-many relation, then A,C are in a certain relationship, i.e., a composition algebra between...

Homework Statement If an investment increased by 40% over 3 years and rose by the same percentage rate each of those years, what was the rate? Homework Equations new_value = old_value(1+r)^p The Attempt at a Solution Let value of the investment be 1000, then we have 1000(1+0.4) = 1000(1+r)^3...

Homework Statement Good day, I need to show that S_n=\mathbb{Z}_2(semi direct product)Alt(n) Where S_n is the symmetric group and Alt(n) is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product Homework Equations none The Attempt at...

Hi, I'm currently studying to become a chemical engineer. After learning differential equation and linear algebra, I've realized how useful they are in my engineering courses since they make setting up equations and solving them so much easier. So I was wondering if there are other math that...

Homework Statement I solved the problem myself but i have a question about the algebra Homework Equations n(A ∪ B ∪ C) = n(A) + n(B) + n(C) -n(A ∩ B) -n(A ∩ C) -n(B ∩ C) + A ∩ (B ∩ C) = n(A ∪ B) -n(A ∩ C) -n(B ∩ C) + A ∩ (B ∩ C) The Attempt at a Solution I knew i needed...

Homework Statement Construct a 3x3 matrix A and vectors b and c in R^3 so that Ax=b has a solution but Ax=c Homework EquationsThe Attempt at a Solution So I don't know where to start. I am not sure if the problem is asking me to create a matrix with real numbers or variables. What I do know is...

Homework Statement a. Lucy's Lunch and Latte has found that customers are put off by the local tourism tax of 9% that is added to their bill. If Lucy decides to cover the tax herself, rather than adding it to the customer's bill, what percent will the customer see in savings? b. Lucy decides...

So, I'm trying to think of some way to formulate a "Conservation of Symbols" Law of Mathematics. Something like this: By "symbol" I mean any atomic variable, constant, digit, operator, bracket, etc., that is present in a syntactically correct expression, equation, or inequality. So the...

This is only a minor question. I watched an on-line video recently on su2 and how it applies to Physics. Now, one of the first things the instructor did was to change the base to sl2. Fine and all, but she called sl2 "su2" for the whole video. Since the two Lie algebras have different...

Homework Statement Prove that if ({A_1, A_2, ..., A_k}) is a linearly independent subset of M_nxn(F), then (A_1^T,A_2^T,...,A_k^T) is also linearly independent. Homework EquationsThe Attempt at a Solution Have: a_1A_1^T+a_2A_2^T+...+a_kA_k^T=0 implies a_1A_1+a_2A_2+...+a_kA_k=0 So...

So basicly our teacher taught us in high school how to find the product of some equations but I do not understand it very well and I need someone to teach me how to solve this basic problem. The Equation is : (3x^2-4x+1)(4x^2+x-2) I do not know how to find the product of that problem can...

Homework Statement Determine if the following sets are bases for P_2(R) b) (1+2x+x^2, 3+x^2,x+x^2) d) (-1+2x+4x^2, 3-4x-10x^2,-2-5x-6x^2) Homework Equations Bases IF Linear Independence AND span(Set)=P_2(R) RREF = Reduced Row Echelon Form The Attempt at a Solution My first question here...

Dear all, Can someone please recommend a english version of a linear algebra book. "Lineare algebra, Siegfried Bosch" I don't speak German so I cannot understand the chapters. If someone can help to identify a book with similar chapters that will be very helpful. you can find the full book...

Let $s,\,t,\,u,\,v$ be real numbers such that $s+t+u+v=0$. Prove that $(s^3+t^3+u^3+v^3)^2=9(st-uv)(tu-sv)(us-tv)$.

It's rather strange, isn't it? I did just fine in Algebra I, and even better in Algebra II, but now that I'm in Calculus I I'm finding that my algebra skills aren't as sharp as I thought they were. Often times i'll get the calculus portions of problems correct, but when it comes down to...

I have a question about Automorphisms. Please check the following statement for validity... An automorphism of a group should map generators to generators. Suppose it didn't, well then the group structure wouldn't be preserved and since automorphisms are homomorphisms this would be a...

In quantum mechanics, a physical quantity is expressed as an operator G, then the unitary transformation coresponding to the physical quantity is expressed as exp(-iG/ħt), being also an operator, where t is the tranformation parameter. G is actually the conservative quantity corresponding to the...

Homework Statement True or false? If T: ℙ8(ℝ) → ℙ8(ℝ) is defined by T(p) = p', so exists a basis of ℙ8(ℝ) such that the matrix of T in relation to this basis is inversible. Homework EquationsThe Attempt at a Solution So i think that my equations is of the form: A.x = x' hence A is...

What are some books that I can use master DiffyQ and linear algebra ? I want to improve significantly in E&M and quantum mechanics. I understand that I have to practice, practice, and practice , but can you provided me with with some advice ? Thank you.

Hello there, I was wondering if someone might be able to help me out with some intermediate steps please. I can't see how $$f_{tt} =-\int_{r_0}^0 \left(\frac{2Gm_0}{r} - \frac{2Gm_0}{r_0}\right)^{-\frac{1}{2}} dr$$ becomes $$f_{tt} =-...

I don't even know what is the general name for the thing whose specific name I am trying to find out, but in an algebra question, say 3x = 6y, there is the "assumption" that by dividing both side by 3, we will get x = 2y and this equation is as correct as the original one. What is thing that...

So, I have an equation: ~A * B * C * ~D + ~A * B * C * D + A * ~B * ~C * D + A * ~B * C * D + A * B * ~C * D + A * B * C * ~D where * represents "AND" and + represents "OR", ~ being NOT. Part of the reason I'm having trouble is due to the length of the equation. So far, I've managed to use...

Hello all, I am about a week into a quantum mechanics course in which the instructor is mostly going to follow along with Sakurai's Modern Quantum Mechanics. However, my linear algebra is pretty rusty and I have never taken an actual course in linear algebra, I have just learned what I have...

Homework Statement Suppose A is non-singular (nxn) matrix. Given that AB=(zero matrix), show that B=(zero matrix). Hint: Express A as [A1, A2, ..., An] and Express AB as [AB1, AB2, ..., ABn] Homework EquationsThe Attempt at a Solution [/B] I argued that because A is non-singular, A=[A1...

Hi all, Son came home with these questions and is unable to solve them. I am also struggling with these questions. I need to explain to my son how to get the solutions and not just the answers. Any help would be greatly appreciated! 1. Two cars are traveling along two straight roads which...

My professor was proving that the Dodecahedron is isomorphic to ##A_5## and in the process utilized the stabilizer (which one can intuit ) of an edge, vertex or face to determine the conjugacy class (which is hard to intuit) of elements of the same order. This seems like a valuable skill but I...

So I stumbled upon this problem: Solve: And I have attempted to solve it, however my solution doesn't match that of the book. The solution should be x+4, but I don't have a clue how to get to that. I'm sorry if this seems pretty rudimentary, but I just want some help because it's the first...

Homework Statement The Tensions in four cables are equal: |T1| = |T2| = |T3| = |T4| = T Determine the value of T so that the four cables exert a totl force of 12,500-lb magnitude on the support. (Inser a picture of 4 connected cables angled 9, 29, 40, 51 degrees respectively) My question...