Algebra Definition and 999 Threads

In the same way one can show that \nabla^{2}\theta=0 has only one smooth solution, namely \theta=0, I would like to show that \gamma^{i}\partial_{i}\epsilon=0 has only one smooth solution, where \gamma^{i} is a Dirac gamma matrix (or an element of the Clifford algebra), and \epsilon is a...

Homework Statement I stated this as a small attached photo, since i still need to learn to write in latex (see photo) Homework Equations How do you transform the first expression into the next expression (on photo)? The Attempt at a Solution [/B]I tried various manipulations with potens...

Homework Statement Hello there everybody! I'm reading a Linear Algebra textbook, specifically on LTV systems solutions. I'm trying to redo this example from the book: Homework Equations But I couldn't understand the passage: The Attempt at a Solution I mean. x1(0) = 1 and x2(0) = 0? I...

Homework Statement goal: solve for t; all else are constants $$cos(\omega t)=1-e^{-(\frac{t}{RC})}$$Homework Equations noneThe Attempt at a Solution i turned the cos to complex notation & rearranged $$e^{i\omega t}+e^{-(\frac{t}{RC})}=1$$ $$ln(e^{i\omega t}+e^{-(\frac{t}{RC})})=0$$ and i...

Hello, I have some problems with understanding some concepts in Quaternions and Clifford Algebra. For example, where can I learn the basic construcion of Clifford Algebra? I'm listing the equalities I did not understand and I appreciate it if you can help me with understanding these : Homework...

Hi all, So I'm going to have my first exposure to linear algebra and I've completed Calc 1 and 2. I've seen Axler and Shilov numerous times and I'm having a hard time choosing it. Here's my syllabus for my Linear Algebra Course. Matrices, Gauss reduction, invertibility. Vector spaces, linear...

1. Calculate the force needed to move the wire in the figure (Figure 1) if it holds a soapy solution (surface tension is 0.025 N/m) and the wire is 16.0 cm long. Express your answer to two significant figures and include the appropriate units. 2. γ(surface tension) = F (force) / l (length)...

Homework Statement let A be the matrix corresponding to the linear transformation from R^3 to R^3 that is rotation of 90 degrees about the x-axis Homework Equations find the matrix A The Attempt at a Solution I got stuck on rotating z component. I tried T([e1,e2,e3])=[0 -1 0]...

Homework Statement This is one part of a wider question, I'm only posting the part I'm having trouble with. $$ \begin{align} \text{Given an impedance network } B &= \frac{Z_1 \parallel Z_3}{Z_2 + Z_1 \parallel Z_3} \\ \text{show that: } \frac{1}{B} &= 1 + \frac{R_2}{R_1} + j\frac{\omega CR_2}{1...

Homework Statement Determine whether the set S spans R3. If the set does not span R3, then give a geometric description of the subspace that it does span S = [ (2,0,3) , (2,0,-1) , (6,0,5) , (4,0,6) ] Homework EquationsThe Attempt at a Solution I know S does not span R3 because the system of...

Hello, Please excuse me about my ignorance. I would like to know how SO(2,1) Lie algebra, is derived from operators and commutators. I have some notes, that the Lie algebra of SO(2,1) is derived from: [D,H]=-iH [K,D]=-iK [H,K]=2iD where D, H, and K are the "generators". I have no clue what does...

For next term, I was wondering if I should take a course in non-commutative algebra. As of now, I'm focusing on mathematical physics, specifically on the mathematical side of string theory; mirror symmetry and field theories. I know that commutative algebra as well as algebraic geometry are...

Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let a and g be elements of a Lie group G, the left translation L_{a}: G \rightarrow G of g by a are defined by : L_{a}g=ag which induces a map L_{a*}...

Okay, long story short, my mom could not work because of an illness so me and and family all had to chip in so fulfill the mortgage payment on our home. I worked two jobs and am a full-time student. I've been studying as much as I possibly can and may have to temporarily restrain from spring...

Homework Statement A car of mass m accelerates from speed v1 to speed v2 while going up a slope that makes an angle θ with the horizontal. The coefficient of static friction is μs, and the acceleration due to gravity is g. Find the total work W done on the car by the external forces. Homework...

Hi, I am trying to work through a proof/argument to show that the adjoint representation of a semisimple Lie algebra is completely reducible. Suppose S denotes an invariant subspace of the Lie algebra, and we pick Y_i in the invariant subspace S. The rest of the generators X_r are such that...

If each vector in basis B1 is scalar multiple of some vector in basis B2 then transition matrix PB1→B2 is diagonal.The column space of matrix A is the set of solutions of Ax = b.If A is n × n invertible matrix and AB is defined then row space of AB coincides with row space of B.Column space of...

Homework Statement Let ab=a and ba=b, show that a^2 = a and that b^2 = b Homework Equations none The Attempt at a Solution Not sure if I did this correct.. but here is what I did. Given: ab = a. Multiply both by left hand multiplication by a^-1 a^-1*a*b = 1. where a^-1*a is obviously...

Find the matrix for the transformation that projects each point in R3 (3-D) perpendicularly onto the plane 7x + y + 3z = 0 . The attempt at a solution is attached for question 1 (actually instructor's solution) I kind of understand it but ... why is n <dot> v = equation of the plane? Does v...

Hello, This is kind of a weird thought but if I obtained a bunch of different polarized cellophane squares as my set and considered my binary operation to be holding the squares over one another and observing the new color made, would that be considered math?? I think they could form a group...

I did some linear algebra studies and learned how to change between foreign bases and the standard basis: Change of basis matrix multiplied by the vector in coordinates with respect to the foreign basis equals the vector in coordinates with respect to the standard basis. Of course, this is...

Assuming the axiom of choice, show that a group $G$ has trivial automorphism group if and only if $G$ has less than three elements.

Here is a picture of the problem. Can anyone give me some hints on the problem? I've looked in my textbook, but I don't know what "s" means. I found stuff on the parametric vector form, and it gives me the equation x = su + tv, but I don't see any "t"'s in this problem. I first tried...

Hello guys ! I need your help with the next problem: --------------------------------------------------------------------- "Show that the equation: $\dfrac{mv_0}{k}-\dfrac{m^2g}{k^2}sin\alpha \cdot ln\left[ 1+\dfrac{kv_0}{mgsin\alpha } \right]$ is equivalent to: $\dfrac{{v_0}^2}{2gsin\alpha...

Dirac matrices satisfy the relations: \gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2g^{\mu\nu} I would like to understand why the dimension of this algebra in 3+1 dimensions is 4. If we're looking for all possible sets {\gamma^0,\gamma^1,\gamma^2,\gamma^3} of 4x4 matrices that satisfy this, how...

the first row 1 0 0 2 the 2nd row 0 1 2 0 the 3rd row 0 2 1 0 the 4th row 2 0 0 1 I would like to ask which is the most efficient way of solving this ques.Though i can solve but is long method, I know there must have some quick 1, appreciate if u can share it. thank you

Homework Statement G is a finite group. K is normal to G. If G/K has an element of order n, show that G has an element of order n. Homework Equations none. The Attempt at a Solution (Kg)^n = K for some Kg in G/K. (Kg)^n = (Kg^n) = K, hence g^n = 1 where g is an element of G. Is this...

Homework Statement The velocity of an object as a function of time is given by: Vx(t) = 12t2 - 5t + 40 m/s Vy(t) = 5t - 30 m/s What is position at 2 sec if the object has an initial position of x = 5 m and y = 8 m? What is the instantaneous acceleration at 10 s? Homework EquationsThe...

Please help me to understand the solution :( (I have made the sketch, is it correct?, where has the 1/4 comes from? is it like 1:3 (1 ratio 3) => numerator divided by numerator + denominator = 1/1+3 =1/4 (as given on page 6 at http://www.lowndes.k12.ga.us/view/14167.pdf )? or am I wrong? For...

Homework Statement If K is normal in G and |g| = n for some g in G, show that the order of Kg in G/K divides n. Homework Equations None The Attempt at a Solution Okay so I feel like I have a solution but I don't use all the information given so I'm trying to find holes in it... g^n = 1...

Homework Statement If H and K are subgroups of G, show HUK is a subgroup of G if and only if H < K or K < H ( the < meaning that all the elements of H are in K or all the elements of K are in H). Homework Equations None The Attempt at a Solution I believe the problem here is HUK might...

The problem is to verify ##(g^n)^{-1} = g^{-n}## is true ##\forall n \in \mathbb{Z}##. Here is my proof: ## (g^n)^{-1} = (\underbrace{g \star g~ \star ~...~ \star g}_{n~ \mbox{copies}})^{-1} \iff## ##(g^n)^{-1} = [(g \star ~...~ \star g) \star g]^{-1}## Using ##(a \star b)^{-1} = b^{-1}...

Homework Statement If H is a subgroup of G and Ha = bH for elements a and b in G, show that aH = Hb. Homework Equations None needed The Attempt at a Solution I've basically just been fiddling around by right and left side multiplication of inverses and what not and can't seem to get it...

As per another thread I have mentioned that I am looking for information on Lie groups (and related Algebras). My personal library has come up dry in my Math texts, and the Physics texts where this comes up are so brief in their Mathematics that I can't really glean any useful (new) information...

The information I am given is : a door has two steel layers both are .47 mm thick, the door itself is 725 mm by 1800mm. The question asks, how thick of a layer of wood (oak) would have to be put in the door to limit the heat loss to 740kJ per hour? Temp inside is 18C and outside is -20C All...

How to find out the position vector of the centroid of tetrahedron , the position vectors of whose vertices are a,b,c,d respectively. I am familiar with the result, namely a+b+c+d/4 but want to know how to derive it without using the 3:1 ratio property. Any help would be appreciated. Thank you.

After having covered single variable calculus to a rather thorough degree, I would now like to move forward to linear algebra. As such I would like to enquire as to any recommendations for a text appropriate for what is essentially a beginner to the subject (I have received very basic...

Homework Statement Hello all, Can someone help me figure this out? "It takes cosmic stitching 2 hrs of cutting and 4 hrs of sewing to make a knit suit. To make a worsted suit, it takes 4 hrs of cutting and 2 hrs of sewing. At most 20 hrs per day are available for cutting and at most 16 hrs...

Homework Statement [/B] a1b1/x1= a2b2/x2solve for x2 Homework Equations no relevant questions apply, this is a stand alone review question The Attempt at a Solution [/B] multiply times reciprocal to get rid of denominator on left (x1) (a1b1/x1)= (a2b2/x2)(x1) a1b1 = (a2b2x1/x2) multiply...

Homework Statement I = Identity matrix Suppose that A^2 = A. Prove that I - 2A = (I - 2A)^-1 Homework Equations ahh don't know what to put here The Attempt at a Solution So I have to prove this thing is it's own identity... interesting.. I - 2A = I - 2A^2 (I - 2A^2)*(I - 2A)^-1...

Thanks.

I am a math major currently in a community college reputed for having an outstanding math department, lucky me :D. I am taking Calculus 2 this semester. Next semester I'll be taking Calculus 3 with linear algebra or discrete math. Can I take all three at the same time or would it be an overkill...

Let $$A(2,-1,1)$$, $$B$$ and $$C$$ be the vertices of a triangle where $$\overrightarrow{AB}$$ is parallel to $$\vec{v}=(2,0,-1), $$$$\overrightarrow{BC}$$ is parallel to $$\vec{w}=(1,-1,1)$$ and $$\angle(BAC)=90°$$. Find the equation of the line through $$\(A\)$$ and $$\(C\)$$ in vector and...

So, I'm interested in using my knowledge of elementary linear algebra (I can do projections, rotations, diagonalization, find eigenvalues/states/vectors, and a couple of other things) to learn other things based off of it. Is there an 'advanced linear algebra' sort of class? My institution...

So I had a quiz on Wednesday and got the problem wrong but don't know why. The question is: Use elementary row operations to reduce the matrix A= 3 1 -1 2 3 1 -4 0 2 to upper-triangular form. Each of these elementary row operations should have the form...

Homework Statement I've gotten to the point where I have ln(e^(-cx) - e^(-cy)) = ln(g) how do i solve this for x ?

Question is in this way

I find, in Kolmogorov-Fomin's Элементы теории функций и функционального анализа, at the end of § 5 of chapter IV, several statement on the spectral radius and the non-emptyness of the spectrum of a linear operator ina Banach space, which are left without proof. Nevertheless, in Tikhomirov's...

I've been skimming my String Theory text and I've been having a hard time understanding a couple of things. For now I'm simply going to ask...What is the Lie Algebra E8? (Not to be confused with E(8), 8 dimensional Euclidian space.) I've read the Wiki article and another on a different site...