Algebra Definition and 999 Threads

Homework Statement If I have two eigenfunctions of some operator, that are linearly indepdendent e.g ##F(x) , G(x)+16F(x) ## and ##F(x)## has eigenvalue ##0##, does this mean that ## G(x) ## must itself be an eigenfunction? I thought for sure yes, but the way I particular question I just...

Homework Statement Hi I am looking at part a). Homework Equations below The Attempt at a Solution I can follow the solution once I agree that ## A^u U_u =0 ##. However I don't understand this. So in terms of the notation ( ) brackets denote the symmetrized summation and the [ ] the...

Homework Statement Suppose that we have a system consisting of two interconnected tanks, each containing a brine solution. Tank A contains x(t) pounds of salt in 200 gallons of brine, and tank B contains y(t) pounds of salt in 300 gallons of brine. The mixture in each tank is kept uniform by...

Homework Statement Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K. Homework Equations f1, ..., fn are field morphisms => Ker (fi) = 0 (injective) The Attempt at a Solution I...

Homework Statement |x-2| < a is a necessary condition for x^2 -3x -10 < 10 . What is the range value of a? |x-2| < a is a sufficient condition for x^2 -3x -10 < 10. What is the range value of a?[/B] The options are a>= 4 a>=3 0<a<=2 0<a<=3 0<a<=5 The Attempt at a Solution Range of x for x^2...

This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...

Hey guys, I have never finished algebra 1 in high school which was around 10 years ago. I want to start myself on a good foundation so I can learn more advanced math. My ultimate goal in the end is to learn meteorology; so I need to start somewhere. For algebra 1 are you able to list things...

Homework Statement Suppose that A is a 3 x n matrix whose columns span R3. Explain how to construct an n x 3 matrix D such that AD = I3. "Theorem 4" For a matrix A of size m x n, the following statements are equivalent, that is either all true or all false: a. For each b in Rm, Ax = b has a...

I am following a book and can't arrive at the same answer. Not sure what to try next. (1-x)(1-0.03)^2 the book then says = 0.667 x= 0.291 my attempt (1-x)(1 - .03)(1 - .03) then i get confused (1-x)(0.9409) not sure ><

Why do students say, as a typical excuse, "I don't need this algebra stuff because it is not required for success after high school"? What do you say? I say we use algebra everyday and don't even realize it.

Homework Statement i'm viewing an example written in class. it looks like this: f(x1, x2, x3, x4) = [(not x1) * x2 * x4] ∨ [x2 * x3 * x4] what should be function after applying absorption law? Homework Equations i know how another option called "gluing" works: [x1 * x2 * x3] ∨ [(not x1) *...

Hello! I am a bit confused by some definitions. We have that a Lie algebra is abelian if ##[a,b]=0## for all ##a,b \in L## and ##L'## is an invariant subalgebra of ##L## if ##[a,b]=0## for all ##a \in L'## and ##b \in L##. From here I understand that ##L'## is abelian. Then they define a...

I have this word problem, and was wondering how I would go about creating a system of equations. Here is the question: Problem: You are a small forest landowner, and decide you want to sustainably harvest some of timber on your property. There are costs related to the infrastructure needed to...

Post moved by moderator, so missing the homework template. series ##{a_n}## is define by ##a_1=1 ## , ##a_2=5 ## , ##a_3=1 ##, ##a_{n+3}=a_{n+2}+4a_{n+1}-4a_n ## ( ##n \geq 1 ##). $$\begin{pmatrix}a_{n+3} \\ a_{n+2} \\ a_{n+1} \\ \end{pmatrix}=B\begin{pmatrix}a_{n+2} \\ a_{n+1} \\ a_{n} \\...

Please, can someone help me with this? Let $M$ be a left $R$-module over a ring $R$. Let $J$ be a left ideal in $R$ generated by $r$: $J=Rr=<r>$. Now $JM=\{am \ | \ a \in J \ and \ m \in M\}$ Prove that $JM$ is a submodule of $M$. This is an example in Rotman's book "Advanced Modern Algebra"...

Hello, I've been trying to improve my algebra since I've never been particularly good with math. I'm going through Serge Lang's Basic Mathematics textbook and while I have been learning a lot his proof-based exercises are a pain to get through and the back of the book only provides answers for...

I am trying to decompose some exponential operators in quantum optics. The interesting thing is that the operators includes operators from Su(1,1) algebra $$ [K_+,K_-]=-2K_z \quad,\quad [K_z,K_\pm]=\pm K_\pm.$$ For example this one: $$ (K_++K_-)^2.$$ But as you can see they are squares of it. I...

A problem that I have to solve for my Linear Algebra course is the following We are supposed to use Mathematica. What I have done is that I first checked that A is symmetric, i.e. that ##A = A^T##. Which is obvious. Next I computed the eigenvalues for A. The characteristic polynomial is...

Homework Statement Let A be an arbitrary m× n matrix. Find a matrix C such that CA = B for each of the following matrices B. a. B is the matrix that results from multiplying row i of A by a nonzero number c. b. B is the matrix that results from swapping rows i and j of A. c. B is the matrix...

Homework Statement [/B] Suppose the solution set of some system Ax = b , Where A is a 4x3 matrix, is *Bold characters are vectors* x_1= 1 + 3t x_2 = 2 - t x_3 = 3 + 2t where t is a parameter and can be any number. a) How many pivots are in the row echelon form of A? b) Let u, v, w be the...

Homework Statement My solution, is this correct? This is what I came up with. Y=A+((A*B)+B+C'+(B+C'*D)+D) Is it safe to say that it is correct or did I make a mistake?

Hello, Below are two results with their proof. Of course, there may be several ways to prove these results, but I just need some checking. Can someone check carefully if the math is OK ? (but very carefully, because if there is a failure, I will be murdered :-) ) ? thx. Claim 1: Let ##L/K## be...

I have not really finished studying linear algebra, I have to admit. The furthest I have gotten to is manipulating matrices a little bit (although I have used this in differential equations to calculate a Wronskian to see if two equations are linear independent, but again, a determinant is...

Homework Statement Prove that ##\vec {a} \cdot (\vec {b} \wedge \vec {C_r}) = \vec {a} \cdot \vec {b} \vec {C_r} - \vec {b} \wedge (\vec {a} \cdot \vec {C_r})##. Note that ##\vec {a}## is a vector, ##\vec {b}## is a vector, and ##\vec {C_r}## is an r-blade with ##r > 0##. Also, the dot...

Let L1 be the line passing through the point P1=(−2,−11,9) with direction vector d2=[0,2,−2]T, and let L2 be the line passing through the point P2=(−2,−1,11) with direction vector d2=[−1,0,−1]T Find the shortest distance d between these two lines, and find a point Q1 on L1 and a point Q2 on L2...

I am trying to create a formula for a spreadsheet to calculate values and it has been thirty years since high school algebra. Here are the formulas I need to rearrange. w=(d/r)/t v=d/t w is an unknown constant and I have values for v1 and r1 I need the formula to spit out v2 if I input r2 in...

Homework Statement I have an assignment for my linear algebra class, that I simply cannot figure out. Its going to be hard to follow the template of the forum, as its a rather simply problem. It is as follows: Given the following subspace (F = reals and complex) and the "linear image"...

Let me give some context. Let X be a compact metric space and ##C(X)## be the set of all continuous functions ##X \to \mathbb{R}##, equipped with the uniform norm, i.e. the norm defined by ##\Vert f \Vert = \sup_{x \in X} |f(x)|## Note that this is well defined by compactness. Then, for a...

I have a question regarding the Dirac notation arithmetic. Below is a measurement of a general 2 qubit state with the measurement operator M=|0><0| ⊗ I , where I is the identity operator. To go from equation (2) to equation (3), I've been converting all the Dirac notation to matrices and column...

Hi thanks to George, I found the following criteria for boundedness: \begin{equation} \frac{||Bf(x)||}{||f(x)||} < ||Bf(x)|| \end{equation} If one takes f(x) = x, and consider B = (h/id/dx - g), where g is some constant, then B is bounded in the interval 0-##\pi##. However, given that I...

<Moderator's note: Moved from a technical forum and thus no template.> The title isn't complete this is what I meant to say: Determine the values of aa and bb where the function has inflection points (x2+ax+b)(ex) I made the second derivative $$f''(x) = 2 e^x + 2 a e^x + b e^x + 4 e^x x + a...

Homework Statement Given for one-dimensional Galilean symmetry the generators ##K, P,## and ##H##, with the following commutation relations: $$[K, H] = iP$$ $$[H,P] = 0$$ $$[P,K] = 0$$ Homework Equations Show that the Lie algebra for the generators ##K, P,## and ##H## is isomorphic to the...

I came across something that is completely counter-intuitive, and I'm wondering if I'm correct or not. If a square has a side that is .8m someone would do .8 time .8 which is .64. How can an area be smaller than a side I thought and so I looked it up and found only one site that said something...

Hi there, I have been searching multiple websites and forums but have not found a cohesive answer to my queries. I am currently going through this textbook to learn pre-algebra: https://www.amazon.com/dp/0618250034/?tag=pfamazon01-20 Also, is this a good book to be studying as an autodidact...

I seem to remember this Algebra identity being covered in one of my classes years ago, but it has cropped back up in studying the relativistic doppler effect for light. Can anyone please show me the intermediate steps to show that: (1+x)/(sqrt(1-x^2) = sqrt((1+x)/(1-x)) or similarly...

I am trying to learn about free groups(as part of my Bachelor's thesis), and was assigned with Hungerford's Algebra book. Unfortunately, the book uses some aspects from category theory(which I have not learned). If someone has an access to the book and can help me, I would be grateful. First...

<Moved from a homework forum. Template removed.> I can't find any documentation on how to do this. I remember in linear algebra how to find the incidence matrix of an electrical network of purely resistors. Put how do I find it of a RLC circuit with resistors, inductors, and capacitors? I can't...

Homework Statement Verify that A^2-2A+7I=0Homework Equations A is a squared matrix and I is the identity matrix. The Attempt at a Solution I squared a matrix, which I called A, by multiplying the two A matrices together, then I subtracting the new matrix with the third matrix 2A, then I added...

(A1−A2,B1−B2,C1−C2)∧(A1,B1,C1)(A1−A2,B1−B2,C1−C2)∧(A1,B1,C1) ##=((A1−A2)∗B1−(B1−B2)∗A1)∗(\hat x \wedge \hat y)+((C1−C2)∗A1−(A1−A2)∗C1)∗(\hat z \wedge \hat x)+((B1−B2)∗C1−(C1−C2)∗B1)∗(\hat y \wedge \hat z)## Is this the correct exterior product?

I'm trying to expand the following using Newton's Generalized Binomial Theorem. $$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$ where $$0<\delta<<1$$ But the condition for this formula is that...

How do I: Find Y when AxY=B, BxY=C, and B is unknown? (A and C are known) Example: If A=100 and C=169 then Y=1.3 and B=130 I assume I use log or pow but cannot figure it out. Thanks.

Sorry, I posted too soon. I was able to figure it out.

hey everyone just started university and the jump i feel is huge from a level and was just wondering if you guys knew of any books that had linear algebra and/or several variable calculus in them but displayed and explained stuff in a clear simple way? or if anyone has any websites that do the...

I've started self-studying quantum mechanics. It's clear from google searching and online Q.Mech lectures, I'll need to understand linear algebra first. I'm starting with finite-dimensional linear algebra and Hoffman, Kunze is one of the widely recommended textbooks for that. I need help...

Homework Statement Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', 'a = ', or 'a ≠', then specify a value or comma-separated list of values. x1−2x2+2x3 = −1...

So I am in calculus 3 this year and have passed both calc 1 and 2 with a B and C+ respectively. I could have gotten a better grade but was lazy. I was lazy by using calculators and not actually learning the arithmetic and algebra. Now one serious issue I have is Trig. I can never remember trig...

I may be doing Algebra I tutoring for high school students soon. What are some good resources for exercises and intuitive/novel explanations for topics some students find sticky, etc.? One resource I'm sure I'll be using is the Schaum's Outline of Elementary Algebra, 3ed. What I'd also like is...

I am going to have two slots available this year for electives and I want to use one of them for Astronomy. For the other, I am struggling to decide between Linear Algebra or Computer Science (CIS 210 at my university) which focuses on Python programming. If I can only choose one, which is more...

Hi, I'm stuck on a problem from my quantum homework. I have to show <p1|p2> = ∫(from -1 to 1) dx (p1*)(p2) is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on...

Homework Statement I am trying to solve for ##P## in the equation: $$Q=\frac{2RP}{\sqrt{\sigma_{T}^{2}+\left(2RPr\right)^{2}}+\sigma_{T}} \tag{1}$$ The correct answer must be: $$\boxed{P=\frac{Q\sigma_{T}}{R(1-r^{2}Q^{2})}} \tag{2}$$ I am unable to get this expression. Homework...