What is Linear: Definition and 1000 Discussions

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".

View More On Wikipedia.org
Recent contents Top users
  1. Whiteboard_Warrior

    I Parameterization of linear operators on the holomorphisms

    Let D represent the differentiation of a single-parameter holomorphism, with respect to its parameter x. It's clear that for any sequence of holomorphisms g on x, sigma[k=0,inf](g[k](x)*D^k) is a linear operator on the space of holomorphisms. Is this a complete parameterization of the linear...
  2. M

    MHB Integrating Factors for Solving Linear Differential Equations

    Can i have help with this linear differential equation ? First, i divided by (1-x^2) to be like dy/dx + p(x)y= q(x). But i could not obtain Q(x). Any help will be welcomed.
  3. L

    Conservation of Linear momentum

    Homework Statement A 500g putty ball moving horizontally at 6m/s collides with and sticks to a block lying on a friction-less horizontal surface. If 25% of the kinetic energy is lost, what is the mass of the block? Homework Equations initial (i) = final (f) m1v1+m2v2 = m1v1+m2v2...
  4. J

    Solving a fifth order non-homogeneous differential equation

    Homework Statement Find the general solution of y^{(5)}-y(1)=x The Attempt at a Solution I found the complementary function by substitution of the solution form y=e^{kx} giving k=0,1,-1,i,-i, so y_{cf}=a_0+a_1e^x+a_2e^{-x}+a_3e^{ix}+a_4e^{-ix} Now for the particular integral, the general...
  5. S

    MHB The parameter λ in linear integral equations

    Hi the parameter $\lambda$ in linear integral equations say Fredholm that appears in front of the integral containing the kernel i.e. $y(x)=f(x)+ \lambda \int_{a}^{b} \,k(x,t) y(t) dt $ can $\lambda$ be adjusted for the convergence of the method like in using Picard's successive...
  6. V

    Find the linear acceleration of points on a rolling cylinder

    Homework Statement [/B] A uniform cylinder with radius r is rolling down an incline without slipping as shown in diagram below. It's angular acceleration is α and acceleration of it's center of mass C is a. What will be the linear acceleration of point of contact P and another point Q on the...
  7. P

    Linear elasticity, strutural physics, pressure load

    I have been studying structural physics lately, in particular the linear elastic eqns, governed by the equation of motion. https://en.wikipedia.org/wiki/Linear_elasticity I am trying to solve these equations numerically. I was thinking of a scenario of a rod being compressed on one end by some...
  8. M

    Taking Calc 2 and Linear Algebra at the Same Time

    I'm in a bit of a dilemma. At my university, Calc 2 is a prerequisite to Linear Algebra. However, I have been told that it's totally doable to take both at the same time. What do you guys think? Do most universities wave this requirement if you take them concurrently? Thanks!
  9. mastermechanic

    Linear Approximation of F(x) at x=1.001

    <Moved from a technical section and thus a template variation> 1-) Question: Let f, g and h be differentiable everywhere functions with h(1) = 2 , h'(1) = - 3 , g(2) = -1 , g'(2) = 5 , f(-1) = 4 , f'(-1) = 7. Approximate the value of function F(x) = f(g(h(x))) at point x= 1.001 2-) My...
  10. Mr Davis 97

    B Certainty of being able to solve linear equation

    I have the linear equation ##a + (2a - b) \sqrt{2} = 4 + \sqrt{2}##. I commonly hear that for linear equations, if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions. However, how does that jive with this example, where there is one...
  11. R

    MHB Solving system of linear inequalities

    How do I solve this? Which of the following ordered pairs (x,y) satisfies the system of inequalities below? y>-3x+5 y≤ x-2 Choices: (2,1) (1,4) (4,1) (3,-6) Please. Thanks.
  12. R

    MHB Can You Balance Two Jobs and Meet Your Weekly Financial Goals?

    You can work a total of no more than 41 hours each week at your two jobs. Housecleaning pays \$6 per hour and your sales job pays \$9 per hour. You need to earn at least \$252 each week to pay your bills. Are the answers: x+y\le 41 6x+9y\ge252 How would you solve this? Thanks
  13. A

    Automotive Calculate the position for a linear acutator

    Hi everyone! So I'm working on project where I have to design a hinge system for a hard lid that is fitted onto the back of a pickup truck. In this system, I will be using 2x linear actuator(s) that will need to lift a total weight of roughly 95kg. At the moment, I've got a selection of 600N...
  14. P

    I Factors that determine the coefficient of linear expansion

    ΔL=LαΔT The concept is interesting and applying its formula isn't even tedious, but what are the real factors that determine α here? I understand that thermal expansion is a direct consequence of the average separation between atoms. And that the coefficient can be found through different...
  15. V

    Linear perturbation to harmonic oscillator

    Homework Statement Find the first-order corrections to energy and the wavefunction, for a 1D harmonic oscillator which is linearly perturbed by ##H'=ax##. Homework Equations First-order correction to the energy is given by, ##E^{(1)}=\langle n|H'|n\rangle##, while first-order correction to the...
  16. B

    I'm stuck on solving a linear quadrupole problem

    Homework Statement [ Homework Equations I found this is about linear quadrupole and these are potential of it. [/B] The Attempt at a Solution So, i solve the problem (a) by using potential. This is what I've done, I'm not sure if it is right though.[/B] And i have no idea how can i...
  17. A

    Question about linear independence

    Homework Statement Homework EquationsThe Attempt at a Solution if there exists a set with 3 vectors, and all of them are linear independent, then by definition no linear combination of the 3 vectors can equal to 0. I believe that's an accurate definition right? So in this case, the answer...
  18. M

    Max Comparisons in Binary Search: Understanding Efficiency and Implementation

    I have two questions over linear and binary serch. 1. Homework Statement Q1: In binary search, after three comparisons have been made, only ___ of the array will be left to search. Q2: The maximum number of comparisons that a binary search function will make when searching for a value in a...
  19. KennethK

    Calculate the spectrum of a linear operator

    <mod note: moved to homework> Calculate the spectrum of the linear operator ##T## on ##B(\ell^1)##. $$T(x_1,x_2,x_3,\dots)=(\sum_{n=2}^\infty x_n, x_1, x_2, x_3, \dots)$$I think the way to do it is to find the point spectrums of ##T## and its adjoint ##T^*##. But I don't know how to calculate...
  20. Adgorn

    I Regarding the linear dependence of eigenvectors

    Let's say we have a set of eigenvectors of a certain n-square matrix. I understand why the vectors are linearly independent if each vector belongs to a distinct eigenvalue. However the set is comprised of subsets of vectors, where the vectors of each subset belong to the same eigenvalue. For...
  21. M

    How Do Matrix ODEs Relate to Determinants and Traces?

    Homework Statement Please bear with the length of this post, I'm taking it one step at a time starting with i) Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices. (i) If F : I → gl(n, R) satisfies the matrix ODE F'...
  22. W

    I Use of R^2 adjusted in Simple Linear Regression-Excel?

    Hi All, I am kind of confused at the fact that Excel uses the measurement of adjusted ## R^2 ## for simple linear regressions ; please see below. What does that even mean, what is the adjustment for?
  23. C

    Linear Algebra - Finding coordinates of a set

    Homework Statement Find the coordinates of each member of set S relative to B. B = {1, cos(x), cos2(x), cos3(x), cos4(x), cos5(x)} S = {1, cos(x), cos(2x), cos(3x), cos(4x), cos(5x)} I am to do this using Mathematica software. Each spanning equation will need to be sampled at six separate...
  24. N

    I Non linear interpretations of QM and string theory?

    Are interpretations that are non linear with respect to the wave function (eg. GRW, transactional) compatible with string theory?
  25. J

    MHB How Do You Find the MLE of Lambda in a Sum of Two Poisson Distributions?

    First of all I will use L to denote lambda the parameter of the distribution. X~Poission(nL), n$\in\Bbb{N}$, Y~Poisson(mL),m$\in\Bbb{N}$ with m$\ne$n S= aX+bY a,b real constants. Given observations x and y find the maximum likelihood estimator of L. The problem is I don't know what the pmf...
  26. J

    I Maximum tangential velocity from maximum linear velocities

    I have a particle that moves along a circular arc centered at origin in 2D plane. I have the following angular displacement function for time T0 through T3 and the following acceleration and velocity constraints in Cartesian coordinates. At t = T0, theta = 0, velocity = 0. At t = T3, velocity =...
  27. kuruman

    Frames of Reference: Linear Acceleration View - Comments

    kuruman submitted a new PF Insights post Frames of Reference: Linear Acceleration View Continue reading the Original PF Insights Post.
  28. woof123

    MHB Finding the Linear Equation Relating Price and Number of Widgets Sold

    Okay here's the question: If you sell widgets for 2 dollars, you sell 300 per day. If you sell them for 4 dollars, you sell 200 per day. assume linear relat., between price (x) and # widgets sold per day (y), write linear equation relating x and y I'm thinking I plot the x,y pairs, then...
  29. T

    MHB Linear Algebra Conditions: Solving for ab≠1

    The answers is b) ab≠1, but I have no clue how to get to that answer... Can someone help me? :D
  30. T

    I Linear Algebra Conditions: Solving for ab ≠ 1

    http://imgur.com/a/xIydC The answers is b) ab≠1, but I have no clue how to get to that answer... Can someone help me? :D
  31. M

    Linear Algebra: Matlab Question

    I am taking a linear algebra class, and it has a required lab associated with it. Here is the following problem that I must solve using Matlab 1. Homework Statement Write a function using row reduction to find the inverse for any given 2x2 matrix. Name your function your initial + inv(M), the...
  32. M

    MHB Linear Optimization: Possibilities

    Hey! :o To product a product $3$ square plates $A$ with measures $300\times 500$mm and $2$ square plates $B$ with measures $400\times 600$mm are needed. These plates must be cut out by two different workpieces with measres $1000\times 1000$mm (Type I) and $600\times 1500$mm (Type II). The...
  33. S

    Calculating Force and Impulse of a Bullet Hitting a Man

    Homework Statement A bullet of mass 0.011 kg is fired at a speed of 850 m/s. It embeds itself in a man of mass 90 kg. a) Find the momentum of the bullet before it strikes the man. b) Assuming it takes 0.1 seconds for the bullet to come to rest after hitting the man, calculate the average...
  34. Mr Davis 97

    I Verifying a linear transformation

    I am told that the trace function tr(A) is a linear transformation. But this function maps from the space of matrices to the real numbers. How can this be a linear transformation if the set of real numbers isn't a vector space? Or is it? Can a field also be considered a vector space?
  35. S

    A Charge-current interaction: linear terms in a given quark

    I am trying to figure out the interaction terms in the Lagrangian which are linear in the top quark.I find that these terms can only be found in the charged-current interaction since the neutral-current interaction and the gluon-fermion couplings are quadratic in the top quark. Now, I find two...
  36. V

    Should I consider Linear Kinetic Energy in this Equation

    Sorry If the thread name confuse you. I want to know if I want to find the Torque from deriving Kinetic Energy and the system has Object the Rotate and the rotating cause linear motion(v). Let's say it a Rolling Disc on the non-slope plane which has angular velocity ω and that ω cause it to move...
  37. T

    Linear acceleration, Angular acceleration, tension.

    Homework Statement A mass of 0.5 kg is suspended from a flywheel as shown in FIGURE 2. If the mass is released from rest and falls a distance of 0.5 m in 1.5 s. Mass of wheel: 3kg Outside rad. of gyration of wheel: 300mm Radius of gyration: 212mm calculate: (a) The linear acceleration of the...
  38. davidge

    Matrices and linear transformations. Where did I go wrong?

    Hi everyone. Excuse me for my poor English skills. I did an exam today and my exam result was 13 of 40. I don't understand why it was my result, because while doing the exam I though I was doing it well, then the result was a surprise for me. I will write down the questions and after show my...
  39. davidge

    I Matrices and linear transformations. Where did I go wrong?

    Hi everyone. Excuse me for my poor English skills. I did an exam today and my exam result was 13 of 40. I don't understand why it was my result, because while doing the exam I though I was doing it well, then the result was a surprise for me. I will write down the questions and after write my...
  40. M

    MHB Which is the linear system of equations?

    Hey! :o A national economy has $3$ sectors: fishing, forestry, boat building. One fishing boat is needed, to catch two tonnes of fish. To produce four tonnes of wood, one tonne of fish is needed to feed the forestry workers. Two tonnes of wood are needed, to build one fishing boat. There...
  41. Eclair_de_XII

    Linear Algebra Can anyone recommend a good linear algebra/ODE book?

    In particular, I would like a book that explains the theory of function spaces, linear transformations, to explain the ways one can interpret ODEs as linear transformations on function spaces. I need to read up on these for a project in my Intro to Differential Equations class. Any suggestions...
  42. M

    Linear actuator use in all electric mini-digger

    Hi all, I've got this idea in my head of building a mini-loader like a http://www.multione.com/mini-articulated-loaders/ . I'm interested in replacing the hydraulics with linear actuators and wondered if anyone here could shed some thoughts/advice. Given the driveline would also be fully...
  43. Mr Davis 97

    Linear independence of polynomials of different degree

    Homework Statement Let S be a set of nonzero polynomials. Prove that if no two have the same degree, then S is linearly independent. Homework EquationsThe Attempt at a Solution We will proceed by contraposition. Assume that S is a linearly dependent set. Thus there exists a linear dependence...
  44. Mr Davis 97

    Linear Independence of a Set of Vectors

    Homework Statement Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. Homework EquationsThe Attempt at a Solution I think that that it would be easier to prove the logically equivalent statement: Prove that a set S of vectors...
  45. Dethrone

    MHB Optimizing Linear Regression Cost Function

    I'm trying to optimize the function below, but I'm not sure where I made a mistake. (This is an application in machine learning.)$$J(\theta)=\sum_{i=1}^n \left(\sum_{j=1}^{k}(\theta^Tx^{(i)}-y^{(i)})_j^2\right)$$ where $\theta$ is a $n$ by $k$ matrix and $x$ is a $n$ by 1 matrix...
  46. michaelgtozer

    I 1st year linear algebra question

    given P(-1,1,2), Q(-3,0,4), R(3,2,1), find an equation of the line through P that is parallel to the line through Q and R. All the words after the given three points really confuse me and I just need some help on where to start to tackle this problem. Thanks
  47. D

    Engineering What Voltage Should Be Applied to Achieve a Change in Capacitor Energy of 4uJ?

    Homework Statement This is the second problem from our test preparation cycle and i partially solved and understood it. The circuit in the picture is in the state of switch being open (STATE 1: SWITCH OPEN). Calculate ##E1## so that the change in the electrical energy of the capacitor is...
  48. D

    Engineering Understanding thevenin's circuit in linear circuits

    Homework Statement This is one of the problem i had had some help in solving but some things remain unclear to me and i will highlight them by posting the exact question in the brackets next to the line of solution so that you can see what bothers me. Hopefully you can help me better understand...
  49. M

    MHB Injective linear mapping - image

    Hey! :o Let $F$ be a field and $V,W$ finite-dimensional vector spaces over $F$. Let $f:V\rightarrow W$ a $F$-linear mapping. We have to show that $f$ is injective if and only if for each linearly independent subset $S$ of $V$ the Image $f(S)$ is linearly independent in $W$. I have done the...
  50. Eclair_de_XII

    How would you explain to a dummy how a diff. eq. is linear?

    Homework Statement This is the definition I got from my book: "A crucial classification of differential equations is whether they are linear or nonlinear. The ordinary differential equation ##F(t,y,y',...,y^{(n)})=0## is said to be linear if F is a linear function of the variables ##y...
Back
Top