Linear Definition and 1000 Threads

I think that the intensity of output of circularly polarised light falling on an analyser should not depend upon the axis of analyser. Because it is circularly polarized. It keeps rotating. So, the output should be linearly polarised. The amplitude of E field in the direction of analyser is a...

> Let ##C## be the disk with radius 1 with center at the origin in ##R^2##. > Consider the following linear transformation: ##T: (x,y) \to (\frac{5x+3y}{4},\frac{3x+5y}{4})## > > What is the lowest number such that ##T^{n}(C)## contains at lest ##2019## points ##(a,b)##, with a and b integers.So...

Hi, I just have a quick question regarding the linear phase property of filters. It might be easier to provide some context before getting to the question, but feel free to skip to the bottom. Consider a system input as a discrete sequence obtained by sampling at t = 0, T, . . . , kT from an...

$\tiny{311.1.3.12}$ Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$ $ a_1\left[\begin{array}{r} 1\\0\\1 \end{array}\right], a_2\left[\begin{array}{r} -2\\3\\-2 \end{array}\right], a_3\left[\begin{array}{r} -6\\7\\5 \end{array}\right], b=\left[\begin{array}{r} -7\\13\\4...

In linear regression, one estimates parameters that are supposed to be linear with respect to the dependent variable, for instance ##y=\theta_0 e^x+\epsilon \ ,## or ##y=\theta_0+\theta_1 x_1+\theta_2x_2+...+\theta_n x_n+\epsilon \ . ## Is it not true that neither ##y(\theta_0)## nor...

Summary:: x Question: Book's Answer: My attempt: The coordinate vectors of the matrices w.r.t to the standard basis of ## M_2(\mathbb{R}) ## are: ## \lbrack A \rbrack = \begin{bmatrix}1\\2\\-3\\4\\0\\1 \end{bmatrix} , \lbrack B \rbrack = \begin{bmatrix}1\\3\\-4\\6\\5\\4 \end{bmatrix}...

Hi PF community, recently i learned about Calculus in one variables and several, so now i'd like to study linear algebra by myself in a undergraduate level, in order to do that i need some textbooks recommendations. I'll be waiting for your recommendations :).

Hi all, I'm currently designing a linear to linear transform mechanism (wedge cam design). Would like to check how much force required to push/ slide the roller down. To ease calculation, i have include some assumption: F2 = 1kg @ maximum compressed angle = 20degree F1 required = ? and how...

I tried to solve for y' the algebric second grade equation but I don't know how to solve the diff. equation either. Any help? Thanks. -- lightarrow

The question is as follows: Solutions given only contain part a) to c), which is as follows: So I now try to attemp d), e) and f). d) The magnetic field of a uniformly magnetized sphere is: $$ \vec B =\frac{2}{3}\mu_{o} \vec M = \mu_{o}\vec H$$ $$\frac{2}{3}\vec M = \vec H$$ The perpendicular...

Hi, I'm reading the following paper (L. Chua) about the state-of-art of dynamic non linear circuit analysis -- Chua_Dynamic_Circuits I've a doubt about Theorem 2 on section 3.2 On the Existence of the Resistor Function that establishes sufficient conditions for the existence of network...

He told me I "need to show that the Hamiltonian matrix elements you get by using those states have nonzero elements only on the diagonal." I understand what and how a diagonal matrix works, but what I don't understand is what those states are. Are they states I put in my "quantum mechanical...

Solution 1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##. From...

I am comfortable solving these types of problems, however I am having trouble setting up this problem and am unsure if I am doing this correctly so would someone be able to assist me? I first wrote an equation for the people: 10x +8y +11z. Then I made an equation for the flour: 4x + 12y + 8z...

The solution from my book: From $$\frac{3}{x+2}<\frac{13}{12}<\frac{3}{x} \tag1$$ It follows that ##13x<36<13(x+2)## x<3, i.e. x = 1 or 2. By checking, x=1 is not the solution and x = 2 satisfies the equation. However, how does the author deduce (1)?

I am working on related rates problems involving figuring out how area of a square increases per second based on how much one side increases per second (or how the area of a circle increases based on increase of the radius, etc.). I was wondering about the practical significance of problems like...

Hello, I hope the equation formatting comes out right but I'll correct it if not. So far, I have attempted to write ##\ddot{a}_k(t) = \sum_{n}(u^{k}_n)^*\ddot{q}_n(t) ##. Then I expand the right hand side with the original equation of motion, and I rewrite each coordinate according to its own...

In the link below, there is a force tangent to a wheel that travels distance Ds. There is also a torque that moved thru an angle. Why do they consider them equal? There are two works, shouldn't they be added together...

b) c and d): In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct? e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find two.Otherwise does b-d above look correct? Thanks in advance!

Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...

Hi everyone hope you are well, I would like to express what I have done for this question: Proving and employing caratheodory theorem we can say that any point in polyhedron can be expressed as a convex combination of at most n+1 points (where n is the space dimension) in same polyhedron that...

Hey! 😊 I am looking at the following exercise but I think that I miss something. The statement is the following: We are given the following system of equations: \begin{align*}2a-2c+d-2e=&-2 \\ -2c-2d+2e=&\ \ \ \ \ 3 \\ d+2e=&-2\end{align*} 1) Is the system in echelon form? Justify. 2)...

I tried it but I don't know how to evaluate the integral on the last equation. Help.

Could you check if what I did is right and help me at e) and f)? Thank you!

Does circular freefall of an object involve cancellation of gravitational and inertial forces, as it does in linear freefall?

I understand the Frechet derivative of a bounded linear operator is a bounded linear operator if the Frechet derivative exists, but is the result always the same exact linear operator you started with? Or, is it just "a" bounded linear operator that may or may not be known in the most general case?

a) Total weight ##W=W_1+W_2+...+W_{25}##.$$E[W]=E[W_1]+E[W_2]+...+E[W_{25}]=25\times76=1900\,kg$$$$\sigma_W=\sqrt{V(W_1)+V(W_2)+...+V(W_{25})}=\sqrt{25\times(16)^2}=80\,kg$$ b) Since ##W## is a linear combination of normal distributions, the reproductive property tells us that ##W## is also...

As the brief summary explains, I'm looking for a linear actuator which moves quickly and can move a moderately low load (say, 10kg) but which also features an adjusting hard-stop which can withstand significantly heavier. So effectively, the actuator can move in and out, preferably quickly, but...

I write a Linear Search code,then ı decided ask to user "r" and "int arr".I mean,User decide their r and arr numbers.I tried scanf("%d",r); command but doesn't work This code my first code before the decide Ask user : #include <stdio.h> int search(int arr[], int n, int r) { int i; for...

Hi everyone, sorry we are talking about pulsed power accelerators again. In this context, it is about the HERMES III again. According to https://apps.dtic.mil/dtic/tr/fulltext/u2/a351472.pdf HERMES III, a 16 terawatt pulsed power accelerator at Sandia National Labs, which fired electron beams...

##A^{x'} = T(A^{x})##, where T is a linear transformation, in such way maybe i could express the transformation as a changing of basis from x to x' matrix: ##A^{x} = T_{mn}(A^{x'})##, in such conditions, i could say det ##T_{mn} \neq 0##. But how to deal with, for example, ##(x,y) -> (e^x,e^y)## ?

Be f a orthogonal transformation and g being in the canonical form. $$[f^{-1}]^{t}[g][f^{-1}] = [g']$$ So this equation of isometries implies that the diagonal of g is +- 1, but, apparently, if it is minus one it can't be a group, and i don't know why. To make the things clear, if det = +1, in...

I'm using the actuator that says its good for 270lb https://www.princessauto.com/en/detail/12v-dc-12-in-stroke-270-lb-linear-actuator/A-p8507840e I'm using the 4rth pully system from here Pulley diagram I'm using it to lift 40 lbs So from the diagram I figured that to lift 40 lbs I need an...

Electric field for the semi-circle $$E = - \frac {πKλ} {2R} $$ In this case E is equals to 10 N/C Electric field for the straighten wire $$E = 2Kλ * ( 1 - \frac {2y} {\sqrt{4y^2 + L^2}})$$ In this case E is equals to 8 N/C What I'm searching is R, λ, and the length of the wire, so I think...

I want to take some courses that involve heavy math, so I have been learning maths on the khan academy site: precalculus, calculus, statistics etc. But one fundamental area of maths the khan academy site doesn't have is a course on linear algebra. I really need to learn and use linear algebra in...

A new development race car of mass 400 kg accelerates from 20 km/h to 200 km/h over a distance of 120 metres. Frictional and wind resistance forces can be assumed to be 1000N during the acceleration. Determine the following: (i)The average acceleration from 20 km/h to 200 km/h (ii)The time...

I tried using equations listed above to get the answer, but I get stuck with a and Vi. What am I doing wrong? The answer is 6.7 seconds.

Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...

Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...

I'm not a statistician, but this has been bothering me for a bit. Suppose we have the simple model Y= aX + b + U where Y,X and U are taken to be random variables representing the explanatory variable, the independent variable and the error term respectively. In the case of a stochastic...

Hi, I'm trying to solve this integral and then isolate V, but I can't get the right answer. I don't know where is my errors. I probably muffed the integral. ##-bv -cv² = m\frac {dv}{dt}## ## \int_0^t dt = - m \int_{Vo}^v \frac {dv}{bv+cv^2} ## I get this after the integration ##t =...

The options are $$rank(B)+null(B)=n$$ $$tr(ABA^{−1})=tr(B)$$ $$det(AB)=det(A)det(B)$$ I'm thinking that since it's invertible, I would focus on the determinant =/= 0. I believe the first option is out, because null (B) would be 0 which won't be helpful. The second option makes the point that...

Hello everyone, I looked for the best physics forum to ask this question because the subject interests me a lot. The authors of this preprint (https://arxiv.org/abs/1804.10053) seem to claim that their approach (using linear canonical transformations) is a new alternative way for the...

First, a little context. It's been a while since I last posted here. I am a chemical engineer who is currently preparing for grad school, and I've been reviewing linear algebra and multivariable calculus for the last couple of months. I have always been successful at math (at least in the...

I know that a tensor can be seen as a linear function. I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.

Hello I am looking for an introductory linear algebra book. I attend university next year so I want to prepare and I want to become an engineer. I have a good background in the prerequisites, except I don't know anything about matrices or determinants. I am looking for the more application side...

The way I want to solve it is the way that I always want to solve separable linear diffEqs: after some trivial algebra and an easy integral I end up with t = (-1/2) ln (20-2x) +C Easy enough, solve for x(t) yields x(t) = 10 - (1/2)e^(-2t) + C Solve for C when x(0) = 3 yields C = -13/2 But...

I attached a drawing of the problem for a better understanding and my attempted solutions. The first point is fairly simple but there's something that I can't figure out. dq=λdx=kxdx Q=∫ k x dx from 0 to L -> Q=k[x^2/2]0-L -> Q=(L^2/2)k -> k=2Q/L^2 This is what I came up with. I integrated...

I am working on developing a micro reflex sight and finally got the optical science of it sorted out, but now I reached a new impasse (hopefully the last really difficult one!). The device has to have an adjustment mechanism so the user can align the reticle with the bullet impact point, which...

I am currently enrolled in Multivariate Calculus and am looking to get build up a solid base of mathematics for undergraduate physics curriculum. I am looking for a Linear algebra book that will aid me in my quest. I currently own Axler's Linear Algebra Done Right, but I fear it is too...