Linearity Definition and 104 Threads

Linearity is the property of a mathematical relationship (function) that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are nonlinear.
Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle.
The word linear comes from Latin linearis, "pertaining to or resembling a line".

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  1. D

    Linearity of Y=mx+b: Exploring Its Properties & Approximations

    I understand that a linear relation needs to satisfy both the property of superposition and homogeneity. Y(x)=mx+b does not satisfy both property at the same time yet any equations in this form are called a "linear function" and it is used in linear approximation. For example, sin(x), which is...
  2. G

    Exploring the Relationship Between Theorem 4.5 and p's and q's

    Homework Statement here is theorem 4.5 The Attempt at a Solution How can theorem 4.5 even relate to the question? The question deals with p's and q's and the theorem deals with u v w.
  3. G

    Is Theorem 4.5 Linear Despite Condition A Failing?

    Homework Statement this is theorem 4.5 The Attempt at a Solution The book says that condition A fails. How can you even know? Condition A has a w in it and there is no W in the above question.
  4. S

    Linearity of Maxwell's equations as a result of special relativity.

    Ok, some background: In the static case, the force on a charge is the multiplication of the charge into the electric field {\bf{E}}, defined by Gauss' law, the force on a moving charge with velocity {\bf{v}} is given by the multiplication of the charge (which is Lorentz-invariant) into the...
  5. J

    Statisics - linearity and best-fit in 3 dimensions

    Howdy folks I've gotten a number of answers to this in various fora, some contradictory. I need to do 3 things to a set of datapoints in 3space (X,y, and z real values). 1)Test for linearity (pearson's R?). 2)If passed, find line of best fit (SSE?) 3)See if line of best fir is...
  6. T

    Linear Algebra: Linearity of inner product

    Homework Statement An inner product is linear in both components. Homework Equations <x,y> = <conjugate(y),conjugate(x)> <x+y,z> = <x,z> +<y,z> The Attempt at a Solution I thought it was true . It is obvious that it is linear for the first component by definition Attempt to show...
  7. T

    Linearity in Differential Equations

    Homework Statement Is the following differential equation linear: yy' + 2 = 0 The Attempt at a Solution I have the definition of linear as being a_0 (t) y^{(n)} + a_1(t) y^{n-1} + a_2 (t) y^{n-2} ... = 0. Now, presumably y is a function of t. Thus, I could define y = a_0 (t) and...
  8. I

    Building a JFET Transistor Cascade for Gain, Linearity & Stability

    I need help on how to cascade two or three Jfet transistor and the calculation of the values to have a good amount of gain. The cascade also should help in providing good linearity, have less distortion and should stable good stability.Thanks, good help will really be appreciated by me.
  9. W

    Is Linearity of PDE Operator Lu = du/dx + u * du/dy Verifiable?

    Homework Statement Is the operator Lu = du/dx + u * du/dy linear? Homework Equations Linearity occurs for L[u+cv] = L[u] + cL[v] The Attempt at a Solution I know this isn't linear because of the second term, but I don't understand why I can't write the operator as L =...
  10. T

    How Do Stability and Linearity Determine System Behavior?

    confused!..system stability and linearity 1)-Is y(n)=cos{x(n)} a stable system?? and is the condition s=Ʃ|h(k)|<∞ for stability valid only for LTI systems? actually my book solves the given problem using the above method..but according to me the given system is not LTI SINCE ZERO I/P...
  11. T

    Is the Function f[x_] := x + 2 Truly Linear?

    Probably I'm just being stupid but: According to Wikipedia. * Additivity (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition. * Homogeneity of degree 1: f(αx) = αf(x) for all α. f[x_] := x + 2; f[3...
  12. S

    Linearity and Orthogonality of Inner Product in Vector Space H

    Homework Statement Denote the inner product of f,g \in H by <f,g> \in R where H is some(real-valued) vector space a) Explain linearity of the inner product with respect to f,g. Define orthogonality. b) Let f(x) and g(x) be 2 real-valued vector functions on [0,1]. Could the inner product be...
  13. J

    Differential operator + linearity proof

    Is anyone able to give me a simple example showing the use of the differential operator to prove linearity in an ordinary differential equation ? thanks
  14. M

    Testing Linearity of Two Given Systems

    Homework Statement Two system are given as follows :- (a) \frac{dy}{dt} + sin(t)y(t) = \frac{df}{dt} + 2f(t) (b) \frac{dy}{dt} + 2y(t) = f(t)*\frac{df}{dt} Test linearity of systems. Homework Equations The Attempt at a Solution
  15. N

    Determining Linearity in First-Order Differential Equations

    So I just started taking an intro diff eq course and here's one of my homework problems: "Determine whether the given first-order diff eq is linear in the indicated dependent variable." (y2-1)dx + xdy=0; in y; in x I got the whole bit about the general form for linearity but I was...
  16. L

    Understanding T(\eta,X,Y) and Proving Linearity in X

    Consider the attached question and solution, The answer defines T(\eta,X,Y) = (\hat{T}(X,Y))(\eta) However, given the information that we have, I don't see how we know to do this? When I did this question, I decided that since \hat{T}(X,Y) is a vector and since covectors map vectors to...
  17. B

    Linearity of Harmonic Oscillator and Laplace's Equation

    Homework Statement Ok so basically I have two differential equations: 1) x'' + wx' + w^2 x = sin5t 2) del squared u = 0 The first is obviously just the equation for the driven harmonic oscillator. The second is Laplace's equation. The question asks: In both cases, is the linearity...
  18. B

    Is Linearity Always Exact in Differential Equations?

    Homework Statement Ok so basically I have two differential equations: 1) x'' + wx' + w^2 x = sin5t 2) del squared u = 0 The first is obviously just the equation for the driven harmonic oscillator. The second is Laplace's equation. The question asks: In both cases, is the linearity...
  19. R

    Linearity of Lorentz transformations

    I asked my prof why the Lorentz transformations had to be linear (which my textbook assumed when deriving them), and he mentioned some stuff about homogeneity and ended with "it's advanced, just believe". Can anyone offer a simple explanation?
  20. A

    Determining logarithm's linearity

    Is the following sufficient to prove linearity of y(t) = log (x(t)) ? In order for the system to be linear, this must be true: ax1(t) + ax2(t) \stackrel{response}{\rightarrow} a log(x1(t)) + a log(x2(t) ) (1) but for input ax(t): ax(t) -> log(ax(t))=log(a) + log(x(t)) thus ax1(t) + bx2(t)...
  21. Q

    Linearity, time invariance, causality

    Homework Statement For each of the following systems, determine whether or not the system is linear, time-invariant, and causal. a) y[n] = x[n]cos(0.2*PI*n) b) y[n] = x[n] - x[n-1] c) y[n] = |x[n]| d) y[n] = Ax[n] + B, where A & B are constants. Homework Equations The Attempt...
  22. N

    Proving Linearity of Transformation T on U

    Homework Statement Suppose U is a finite dimensional vector space and A = {u1, u2, ... , un} is a basis of U. Define T : U -> R(nx1) by T(v) = [v]A. (In other words: U is an n-dimensional vector space, A is a basis for U, and T is the transformation that takes a vector in U and finds its...
  23. J

    Establishing linearity in laplace's equation with no closed-form solution

    I'm writing a paper, and as a motivation to the forthcoming finite element modeling, I want to state, with some sort of "proof" that Laplace's equation in a heterogeneous volume: \del (sigma \del V) = 0 exhibits linearity. By "linearity", I mean that if a set of initial conditions...
  24. T

    Proving linearity of a function

    I'm stumped on what seems to be a simple proof question, but I don't know what to do. Question: (c) Show that the LSE of the mean Y0 = B0 + B1x0 is a linear function of the data Yi, for i = 1,2,…,n where x0 is a known constant. Could someone help me to at least start this problem? So...
  25. S

    Is \rhoutt + EIuxxxx = 0 a Linear or Non-Linear Math Problem?

    I'm trying to see if \rhoutt + EIuxxxx = 0 is linear or non-linear where \rho, E and I are constants. I got L(u+v) = \rho\delta2u2/\deltat2 + EI\delta4u2/\deltax4 + \rho\delta2uv/\deltat2 + EI\delta4uv/\deltax4 = Lu + Lv. Does this mean it's linear or is there more to do.
  26. B

    Linearity of Partial Differential Equations

    Is this linear homogeneous, linear inhomogeneous etc... u_{t}-u_{xx}+xu=0 From that first one I get this \frac{u_{t}-u_{xx}}{u}=-x which I'm not sure is linear. Edit: Similar questions involve the following equations: iu_{t}-u_{xx}+\frac{u}{x}=0 and u_{x}+e^{y}u_{y}=0 Another Edit: I...
  27. H

    Testing System Linearity and Shift Invariance

    please anyone can help me how make check the linearity and shift invarient for the system I want to determine whether the system is linear and shift invarientby steps g(m,n) = f(m,-1) + f(m,0) + f(m,1) g(x) = (integration from +infinety to - infinety) f(x,z) dz please help me...
  28. N

    Linearity of Functions on Different Graphs

    Am I correct to say that a function that is linear on a log-log graph is also linear when it is plotted on a graph using regular scales on both axes, takes a square root form when plotted with the x-axis scaled regularly and the y-axis with a log scale, and exponential in appearance whn the...
  29. M

    Mappings, domains, codomains, proof of linearity

    Homework Statement For the following mappings, state the domain and the codomain. Determine whether the mapping is linear by using the definition of linearity: either prove it is linear or give a counterexample to show why it cannot be linear. i.) f(x1,x2,x3)=(2x2, x1−x3) ii.) g(x1, x2) =...
  30. M

    Determining Inner Product for P2: Non-Negativity, Symmetry & Linearity

    Hi, I was wondering how would i determine if <p,q> = p(0)q(0)+ p(1)q(1) is an inner product for P2. I know, we have to check for non-negativity, symmetry and linearity. Just not sure how. thanks!
  31. T

    Can we achieve linearity in helical springs like in cantilever design?

    For the design of a cantilever, assuming a Hookean material is used below its proportionality limit, stress will be proportional to strain. From this we can conclude that its behavior will be linear. However, how can we apply this knowledge towards helical springs? Is it possible to design a...
  32. K

    Confusing about linearity and nonlinearity

    Hi there, in the course of linear algebra, we talk about many on matrix and related properities. I wonder if any equations written in terms of matrices are linear? Could nonlinear equations also written in matrices? Thanks
  33. facenian

    Question on linearity of Lorentz transformations

    Hello.The way the transformation of coordenates in Special Relativity are ussually derived presuposes linearity or try do demostrate such linearity using wrong arguments. For example some authors state that since linear and uniform motion remains linear and uniform after the transformation this...
  34. N

    GR Math: how does tensor linearity work?

    So I'm reading these notes about differential geometry as it relates to general relativity. It defines a tensor as being, among other things, a linear scalar function, and soon after it gives the following equation as an example of this property of linearity: T(aP + bQ, cR + dS) = acT(P, R) +...
  35. A

    Linearity for energy storage devices

    Homework Statement I am supposed to prove that linearity holds for energy storage elements. Homework Equations So far I have got to know that linearity can be proved by superposition. The Attempt at a Solution I attempted to make a circuit with a resistor, capacitor, a voltage...
  36. I

    Proving Continuity and Linearity of a Function

    Homework Statement Given that f(x + y) = f(x) + f(y), prove that (a) if this function is continuous at some point p, then it is continuous everywhere (b) this function is linear if f(1) is continuous. Homework Equations definition of continuity The Attempt at a Solution (a) I...
  37. S

    Linearity Rules for Proving Non-Linearity

    Does anyone have a straightforward link to linearity rules? My textbook is not very helpful and my prof never knows what he is talking about :(. I have a few "prove this is not linear" questions to do f(x)=(|x1|,|x2|) f(x)=(1,2)+3x f(x)=(0,1) I should know this, but i tend to forget the...
  38. F

    What is the true nature of linearity?

    Hello, I'm kind a new to this forum. Been lurking here for a while and got some thought that I'd like to get some advice on. But since I'm new, I better present myself a bit. I'm a programmer since my youth (age 42 now), and never got a formal degree in anything. Nevertheless, I've always...
  39. N

    Linearity of momentum operator

    Is there a theory or any physical reasoning for the momentum operator being a linear, rather than an antilinear, operator?
  40. Y

    Linearity of Heat Conductance - Is Heat Transfer the Same in All Directions?

    Heat conductance - linear? Hello forum friends, I have stumbled upon the fallowing heat conduction problem: Consider a heat source of constant power embedded inside a solid with a constant heat capacity and conductance. Around the source is a box with a constant temperature, which cools the...
  41. Saladsamurai

    Is a nth order ODE considered linear if n=2?

    I am reading through my Diff Eqs Text and I follow most of the lingo. However I am just a tad confused by the statement: An nth order ODE is said to be linear if F is linear in y,y',...y^(n) Then it gives the example: a_n(x)\frac{d^ny}{dx^n}+a_{n-1}(x)..+a_0(x)y=g(x) It then says: 'On the...
  42. D

    Linearity, Time Invariance, Causality, ETC.

    Homework Statement Is the following input/output (x is input, y is output) system linear, time invariant, causal, and memoryless? Answer yes or no for each one.Homework Equations y(t)=2x(t)+3The Attempt at a Solution My instinct tells me it's linear, but for some reason I have trouble showing...
  43. D

    Calculating Power in Linearity Properties with Two Sources

    I've been presented with a question, but I am not sure how to approach it: A linear resistor containing two sources drives a 100ohm load resistor. Source #1 delivers 250mW to the load when source #2 is off. Source #2 delivers 4W to the load when source #1 is off. My question is, how would...
  44. B

    Linearity of A Hermitian Operator

    Can anybody give me a hint about how can i show that if an operator is linear then it's hermitian conjugate is linear. Thanks for your help from now.
  45. T

    Proving Linearity: x(t) -> y(t)

    how can i prove mathematically if a system is linear or not? i mean, i know the system must obey proportionally law and superpositon, but i don't know how apply into it. well, if anyone could help me, the systems i need to prove are: x(t) -> y(t)= Cx(t) + k x(t) -> y(t)= &int; (from minus...
  46. R

    A question on linearity of functionals

    Suppose we have a bounded linear functional f defined on L1 (the sequence space of all absolutely summable sequences) and we take the natural (Schauder) basis for L1, that is, the set of sequences (E1,E2,...,En,...) that have 1 in the n th position and everywere else zero. Pick x in L1. Then...
  47. J

    Proving Linearity of L:R(4)→R(4)

    I need some help solving this...not even sure how to start... Let L:R(4) goes to R(4) be the linear transformation defined by -matlab notation, the value is a 4x1 column L ( [ a b c d])=[ a-b 0 c-d 0 ] Show...
  48. B

    Linearity of the Loentz transformations

    Please give a plausibile justification for the linearity of the Lorentz transformations. Would you accept: The Lorentz transformations should be linear because to one event in I should correspond a single event in I'
  49. S

    Complex Inner Products: Skew Symmetry & Linearity

    Previously i learned from maths that the complex inner product satisfied skew symmetry and linearity in the first component, ie - <aA+bB,C> = a<A,C> +b<B,C> But after studying Shankar in quantum mechanics, he claims the linearity is in the ket vector, ie - <A,bB+cC> = b<A,B> +c<A,C> which...
  50. D

    Linearity of Schrodinger Equation

    Could someone please address this question? How do you algebraically demonstrate the superposition principle revealed by the Schrodinger equation (ie. If Psi1(x,t) and Psi2(x,t) are both solutions then Psi(x,t)= Psi1(x,t)+Psi2(x,t) is also a solution.)?
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