Linearity Definition and 101 Threads

First, I need to check that I have the 3 notations correct for an inner product in finite vector spaces over a complex field; v* means: given the isomorphism V to V* then: (a) physicists and others: (u,v)=v*u ; linear in the second argument (b) some mathematicians: (u,v)=u*v; linear...

Reference textbook “The Physics of Waves” in MIT website: https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/ Chapter 2 - Problem 2.2 [Page 51] (see attached file) Question: In the content of Page 43 (see attached file), it also states that the amplitude of...

Here is a link to the book. This question is about the section between the end of page 6 to the start of page 9. That section discusses the "generic nature of linearity". Let me go through the reasoning. Suppose there is a particle moving along the ##x##-axis with potential energy ##V(x)##...

From 03:39 the presenter describes a dual-gate MOSFET follower stage. He states that he gets better linearity by applying the same RF input to both gates (with different DC biasing of course). Considering that many MOSFET mixer circuits have RF and LO applied separately to the two gates...

Hello friends. Excuse my ignorance. Why is addition linear and not multiplication?

Hi, I think this is a simple question, but I just wanted to ask how I could go about showing this in a scientific manner. I will try to use an analogy later on which is, I hope, a simple way to understand what I am doing. What I am trying to do: I am trying to investigate whether the...

How can we evaluate the linearity of a dependent voltage/current source? Components like resistors are easy to deal with because they always obey a linear equation. Dependent sources are more complicated because their voltage/current relationships depends on other parameters within the circuit.

please see below.

Hi everyone, I am a student studying color pigment fingerprints and how we can improve their identification. Especially in cases where these are mixed with each other to create variants. We are using the VIS and NIR regions (400-1000nm) to obtain our observations. I have a few questions...

I've included the problem statement and a bit about the function but my main issue is with the equation after "then" and the one with the red asterisk. I don't understand why the Laplace transform for a u(t)*e^(-t/4) isn't (1/s)*(1/(s+1/4)). The book I am reading says it's(1/(s+1/4)).

I have a question related to linearity of power spectral density calculation. Suppose I have a time series, divided into some epochs. If I compute PSD by Welch's method with a time window equal to the length of an epoch and without any overlap, I obtain this result: If I calculate the...

A system is linear if it satisfies the properties of superposition and homogeneity. Superposition: adding the inputs of two systems results in the addition of the two outputs. Ex) x1(t) + x2(t) = y1(t) + y2(t) Homogeneity: multiplying the input by some scalar value is equal to the output...

There are several ways to show that the Lorentz transformations must be linear. What's the best/more intuitive argument in your opinion?

dear yall with tranditional wave equation on the gre book it says by the linearity in function f which represents wave. it leads to the principle of superposition. I get an intuition about with a standing wave with cos(x)cos(t) you can break it down to pair of left and right moving waves. i...

The building of theoretical mechanics can be constructed using only the first and the second derivatives (those of coordinates in case of kinematics: velocity and acceleration and those of energy in case of dynamics: force and gradient thereof). It is obviously unavoidable if one wants to deal...

Hey all, I don't understand what makes a differential equation (DE) linear. I found this: "x y' = 1 is non-linear because y' is not multiplied by a constant" but then also this: "x' + (t^2)x = 0 is linear in x". t^2 also isn't a constant. So why is this equation linear?

Hey! :o Let $C[0,1]$ and $C^1[0,1]$ be the space of continuous and continuously differentiable (respectively) functions $x:[0,1]\rightarrow \mathbb{R}$ with the supremum norm $\displaystyle{\|x\|=\sup_{t\in [0,1]}|x(t)|}$ and $T_0, T_1, T_2: C^1[0,1]\rightarrow C[0,1]$ maps with...

Hello all, you may already know that Q.M. is a linear theory however there is something called nonlinear Sch. eq. for example Gross-Pitaevskii equation. How can such a thing exist considering that Q.M. is a strictly linear theory. Cheers.

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on Rn" I need some help with an aspect of Theorem 9.2.1 ... Theorem 9.2.1 reads as follows: Theorem 9.2.1 refers to and relies on Theorem 9.1.10 ... ... so I am...

I came to know that time evolution in classical mechanics is highly non linear. Is there any case that it become linear?

Why can we calculate factor of safety using Strain (fail)/ Strain (allow) only if the load applied is linearly related to the stress developed within the member? What happens if the two are not related? Also, my textbook says factor of safety can also be calculated by stress (fail)/ stress...

Homework Statement Let be T : ℙ2 → ℙ2 a polynomial transformation (degree 2) Defined as T(a+bx+cx²) = (a+1) + (b+1)x + (b+1)x² It is a linear transformation? Homework Equations A transformation is linear if T(p1 + p2) = T(p1) + T(p2) And T(cp1)= cT(p1) for any scalar c The Attempt at...

Bell inequality in page 171 of https://www.scientificamerican.com/media/pdf/197911_0158.pdf is ##n[A^+B^+] \le n[A^+C^+]+n[B^+C^+]## In page 174 we can see that this causes linear dependency according to angle. How to derive this? Let us suppose that angle between ##A^+## and ##B^+## is 30°...

Homework Statement Determine whether the system is linear Homework Equations Superposition The Attempt at a Solution I am comfortable solving the case where the bounds are from negative infinity to t. I have provided an example of that solution I found online. I attempt to solve that...

Homework Statement For the network of constant current shown in Figure 4 it is known that R1 = 50 Ω and , R = 10 Ω. When the switch P is in the 1-position , current I = 50 mA and Ip = 70 mA known i . When the switch P is in the 2-position , current I' = 40 mA and Ip' = 90 mA are known ...

Homework Statement Consider the linear system: dx/dt=x-y dy/dt=x+3y a. show that the function (x(t), y(t))=(te2t, -(t+1)e2t) is a solution to the differential equation (easy) b. Solve the initial value problem dx/dt=x-y dy/dt=x+3y y(0)=(0,2) need help with part b not a Homework Equations...

Hi, I have a question from my germanium lab, which is how to estimate the max deviation (non linearity ) of a system. I plot the relation between the deviation of the energy as a function of true energy and got this ( see pic), but didn't know what to do next. Thanks.

This is more of a general question than a specific homework question, because it popped up in more than 1 problem. If you have 'x' has the independent variable and 'y' as the dependent variable, you can determine the linearity in 'y' by seeing if any of the derivatives (dy/dx) are being raised...

This isn't a homework problem so hopefully this section is fine. I came across something that's bothering me while reviewing PDEs. Take something like: u_{x}(x,t) = 1. which has the general solution: u(x,t) = c_{1}(t) + x. Wolfram says this is linear but if I take a different solution: v(x,t) =...

Homework Statement Is the equation (x2sinx + 4y) dx + x dy=0 linear This problem also asks me to solve it, but I don't have a problem with that part. Homework Equations An equation is linear if the function or its derivative are only raised to the first power and not multiplied by each other...

Homework Statement Homework Equations Ohm's Law: v = iR KVL: ∑ V = 0 KCL: ∑ I = 0 Linearity: kiR = kvThe Attempt at a Solution Okay, so to apply superposition, I'm supposed to turn off all independent sources except one. I began by turning off the voltage source, and I then I used KCL to...

Dear all, I am trying to understand the vector triple product. ## x\times (y \times z) ## As the vector triple product of x,y and z lies in the plane ## (y \times z) ## the vector ## x\times (y \times z) ## can be written as a linear combination of the vectors ## \pm y ## & ## \pm z## In the...

Homework Statement My textbook (Advanced Engineering Mathematics, seventh edition, Kreyszig) indicates that if u1 and u2 are solutions to a second-order homogeneous partial differential equation, and c1 and c2 are constants, then u where u = c1u1 + c2u2 is also a solution, this is the...

Hey! :o Let $K \leq K(a)$ a field extension with $[K(a):K]=n$. $K(a)$ is a vector space over $K$. How can I show that the map $\varphi : K(a) \rightarrow K(a)$, with $\varphi(e)=ae$, is a $K-$linear map??

Hi , I have no problem to solve but just a bit of confusion on what determines the linearity of an ODE. Let's say the equation is (1 x^2) dy/dx + y = 0 Is it linear ? I would incline to say yes because the dependent variable and its derivatives are not in a product with each other but the...

Homework Statement I have two different experimental curves, and I would like to measure how closely a straight line fits each data, and which curve is more crooked. In statistics how can I measure this "linearity"? By the way this is about stepper motor step linearity (ideally it has to be a...

Hi, I was just wondering if : (y^2 - 1)*dx/dy + x = 0 In this case, x is the dependent variable. Is linear? I know it is but I want to understand why. My question is with the coefficients. The first coefficient has a y to the power of 2 to it and a constant. It is also a function of...

Homework Statement Prove or disprove the linearity of the following function y(x)=(z^2)x(z) Homework Equations I know how to determine linearity of functions in a 2-d plane but not in 3 dimensions. The Attempt at a Solution How can one attempt to plot this function by making a...

Hello, Given the complex linear mapping: T(z) = Az + B where A is real and B is complex. However trying to show that T(a * z1 + z2) = a * T(z1) + T(z2) does not work which implies the mapping is not linear? Why does not this rule apply here? Thanks.

Homework Statement Find Io in the network in the figure below using linearity and the assumption that Io = 1 mA. Figure:http://i.imgur.com/Xtu0VmG.jpgHomework Equations KCL, KVL, basic analysis techniques.The Attempt at a Solution The following values I have calculated correctly: VR1=9V...

Friends I have one doubt Below given equation is linear or non linear :)

Suppose I have this operator: ##D^2+2D+1##. Is the ##1## there, when applied to a function, considered as identity operator? Say: ##f(x)=x##. Applying the operator results in: ##D^2(x)+2D(x)+(x)## or ##D^2(x)+2D(x)+1##? If ##1## here is considered as an identity operator then the...

given an sop form of a multi variable boolean expression, how to judge if it is linear or not? is (x or y) linear? more generally, can a function be linear with an and in sop form?

Any comments on the following description from Kip Thorne, BLACK HOLES AND TIME WARPS, 1994, Box 10.1 would be appreciated. It seems odd to me that at some given curvature, gravity would become self sustaining...if that is what he is saying. We have previously discussed in these forums that...

I am trying to follow examples solved by the publisher of my book in order to understand the problem. However, I can't understand why he is solving it like this. What is confusing me, is why v1=(12+8)*1/8 why is v1 not 12*(1/8). Why is he adding the 8ohm resistor in there? Any help would be...

Homework Statement Show that the expectation operator E() is a linear operator, or, implying: E(a\bar{x}+b\bar{y})=aE(\bar{x})+bE(\bar{y}) Homework Equations E(\bar{x})=\int_{-\infty}^{+\infty}xf_{\bar{x}}(x)dx With f_{\bar{x}} the probability density function of random variable x...

Linearity vs. "takes straight lines to straight lines" Homework Statement Prove that if ##\Lambda:\mathbb R^n\to\mathbb R^n## is a bijection that takes straight lines to straight lines, and is such that ##\Lambda(0)=0##, then ##\Lambda## is linear. Homework Equations Fock's theorem implies...

Hi all, I found the following equations for the electric circuit shown Vc(t) = - 1/(R2*C) * Vc(t) – 1/C * i(t) --------------------1 Vt(t) = Vc(t) – R1*i(t) ------------------------------------ 2 I have the following questions 1.The first equation should be dVc(t)/dt = -1/(R2*C) *...

Homework Statement Using the definition of linearity to determine whether or not ech case is a linear homegeneous boundary condition: i.) Uxx(0,y)=Ux(0,y)U(0,y) ii.)Uy(x,0)=Ux(5,y) Homework Equations The Attempt at a Solution I know Uxx(0,y)=Ux(0,y)U(0,y) is not linear...

Homework Statement Check du/dt + d^2u/dx^2 + 1 = 0 Homework Equations L is a linear operator if: cL(u)=L(cu) and L(u+v)=L(u)+L(v) The Attempt at a Solution L = d/dt + d^2/dx^2 + 1 L(cu) = d(cu)/dt + d^2(cu)/dx^2 + 1 = c du/dt + c d^2(u)/dx^2 + 1 ≠ cL(u) = c du/dt + c...