Linear Definition and 1000 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. M

    MHB What is the difference between a linear spline and a linear B-spline?

    Hey! :o We want to construct a linear spline that at the points $x_0=-1$, $x_1=0$, $x_2=1$ has the values $b_0=0$, $b_1=1$ and $b_2=3$. The spline should get from $S_{x,1}$ with $x=\{x_0, x_1, x_2\}$ to $[-1,1]$. Do we want to find a function of the following form...
  2. P

    Linear Algebra, Find a matrix C st CA = B

    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...
  3. S313

    Linear Motion Formulas for an Inclined Plane

    Homework Statement The initial speed of a ball up an inclined plane is 3 m/s upwards. The ball reaches its highest point after 1.2 seconds. a) find the acceleration. b) how far has the ball gotten during this time? c) how far has the ball rolled after 0.4 s and after 2.4 seconds? Homework...
  4. Ty Ellison

    Linear Algebra: Parametric Solution Set

    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...
  5. B

    Linear Momentum to Angular Momentum

    Homework Statement A 10 g bullet traveling at 400 m/s strikes a 10 kg , 1.2-m-wide door at the edge opposite the hinge. The bullet embeds itself in the door, causing the door to swing open. What is the angular velocity of the door immediately after impact? Homework Equations p[/B]= mv L = Iω...
  6. mertcan

    I Linear programming -- Bland rule degeneracy

    hi, my question is related to a proof involves bland rule for avoiding the degeneracy. Initially I emphasized some sentences which have importance in attachment/file with yellow color. At the beginning, it says xs is entering variable and when it enters objective value does not change( because...
  7. Sorcerer

    I What are the differences between matrices and tensors?

    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...
  8. R

    Is the linear mapping T(x) one to one?

    Homework Statement This is for a linear algebra class, but it's taught my mathematicians, for mathematicians and not physicists or engineers so we write pseudo-proofs to explain things. In Exercises 37–40, let T be the linear transformation whose standard matrix is given. In Exercises 37 and...
  9. Decimal

    Rayleigh Scattering and linear polarization

    Hello, I am having a bit of trouble understanding the concept of Rayleigh scattering as it relates to light from the sun. My book states that unpolarized light will hit the atmosphere, and the scattered light will be mainly horizontally polarized, while the unscattered light remains...
  10. Ben Geoffrey

    I Free Vibrations of a linear triatomic molecule

    This question is regarding equation 6.59 in Classical Mechanics Goldstein 3rd edition. I don't understand how he reaches that step. He says he does by inverting 6.41. Is he taking the transpose matrix composed of the eigen vectors by meaning inversion ? I thought I could upload the screenshot...
  11. B

    What Static Load Should a Linear Actuator Handle for a 5kg Object?

    Hello, I measured an object using an electric scale. It is 5kg. I want to select a linear actuator such that when placed vertically, it can keep the object on hold at a fixed height. In this case, what static load (in lbs and Netwon) should the actuator be capable of handling? I am confused by...
  12. G

    Ways to Spin a Flywheel using linear motion

    I am to try to find simple way to spin a spinning flywheel. What linear motion device acting on the outside edge of a spinning flywheel can be used to increase the speed of the flywheel. Need to use a repeating pattern such as pushing your hand down over and over on the edge of the wheel to...
  13. SemM

    A Does Commutativity Affect Linearity?

    Hi, I have in a previous thread discussed the case where: \begin{equation} TT' = T'T \end{equation} and someone, said that this was a case of non-linear operators. Evidently, they commute, so their commutator is zero and therefore they can be measured at the same time. What makes them however...
  14. R

    Find values of k in this linear system

    Homework Statement Find the values of k so that each of the following systems in unknowns x, y, and z has (i) a unique solution, (ii) no solution, (iii) an infinite number of solutions. x + y = kz = 1 x + ky + z = 1 kx + y + z = 1 Homework EquationsThe Attempt at a Solution I really don't...
  15. R

    Solutions to systems of linear equations

    Homework Statement Determine whether of not each system has a nonzero solution. x + 3y - 2z = 0 2x - 3y + z = 0 3x - 2y + 2z = 0 Homework EquationsThe Attempt at a Solution Using a ti-84 I put the matrix in rref and I get a nonzero solution for x and y and zero for z but when I find the...
  16. N

    B Sine/Cosine behaving like a linear function

    Hello all. After completing a problem in which we derived the formulas for potential and spring force energy as functions of time, with simple harmonic motion I noticed the equations are EXACTLY the same, but with sine and cosine switched. The equations were: A sin^2(pi * t) A cos^2(pi * t) I...
  17. M

    A Solving linear 2nd order IVP non-constant coefficient

    Hi PF! Generally speaking, how would one solve $$f''-a(x) f = 0 : f(0)=0,f'(0)=1$$ Or if you could point me to a source that would be awesome too!
  18. Velo

    MHB Solving Linear Equations: $Ax=b$ and Rank of A

    So, my linear algebra book, if you can call it that, says the following: $Ax=b$ is a system of linear equations with $m$ equations and $n$ variables. ${v}_{1}, {v}_{2}, ..., {v}_{n}$ are the vectors in the columns of $A$. The following are equivalent: (1) The system $Ax=b$ is possible for...
  19. M

    Green's Function of Linear ODE

    Homework Statement Find Green's function of ##u''+u=f##. Homework Equations What we all know. The Attempt at a Solution Let Greens function be ##G##. Then ##G''+G=\delta(x-x_0)##. This admits solutions superimposed of sine and cosine. Let's split the function at ##x=x_0##. Then we require...
  20. A

    Linear attenuation coefficient for gamma ray attenuation

    Homework Statement I have data for the count rate (of gamma rays) against the thickness of various materials (Al, Pb, Fe etc...). I want to work out the linear attenuation coefficient. Can I do this by plotting the natural log of count rate against thickness? I don't have values for...
  21. A

    I Classification of First Order Linear Partial Differential Eq

    How can I classify a given first order partial differential equations? Are all first order linear PDEs hyperbolic? Quora Link:https://www.quora.com/How-do-I-classify-first-order-PDE-elliptic-hyperbolic-or-parabolic-using-method-of-characteristics
  22. N

    Understanding Linear Algebra Subspaces and Matrices: A Homework Guide

    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"...
  23. M

    Solving Linear Motion Problems: Initial Speed, Time, and Height

    Homework Statement "A ball thrown straight up with an initial speed of 30/m/s. (a) Show the time it takes to reach its trajectory will be 3m/s (b) Show that it will reach a height of 45m" Homework Equations h=ut+1/2g^2 1/2gt^2 The Attempt at a Solution Part a of the question I am fine with...
  24. W

    Linear Regression, etc : Standard vs ML Techniques

    Hi All, This is probably trivial. What is the difference between techniques such as Linear/Logistic Regression, others done in ML , when they are done in Standard software packages : Excel ( incl. add-ons ), SPSS, etc? Why use ML algorithms when the same can be accomplished with standard...
  25. T

    Simple linear regression model

    Homework Statement For the first question , why when the humidity increase by 1 percent , the moisture content will increase by 0.2727 percent ? Shouldn't it the moisture content will increase by 0.2727 + 0.4911 percent when the humidity increase by 1 percent ? Second question , it's clear that...
  26. hideelo

    A Extending a linear representation by an anti-linear operator

    When we consider representation of the O(1,3) we divide the representation up into the connected part of SO(1,3) which we then extend by adding in the discrete transformations P and T. However, while (as far as I know) we always demand that the SO(1,3) and P be represented as linear operators...
  27. S

    Non linear field of a long bar magnet

    How can the "end effect " be reduced on a long Bar magnet where the (for example) North pole is along the length of the top side. For example a 0.5 cm square bar say 5" long is stuck (South) onto a metal backing. The center length of 4" has a linear field effect but the ends have a bloom...
  28. binbagsss

    I Function of 2 variables, max/min test, D=0 and linear dependence

    ##f(x,y)## a critical point is given by ##f_x=0## and ##f_y=0## simultaneously. the test is: ##D=f_{xx}f_{yy}-(f_{xy})^2 ## if ##D >0 ## and ##f_{xx} <0 ## it is a max if ##D >0 ## and ##f_{xx} >0 ## it is a min ##D >0 ## is is a saddle if ##D =0 ## it is inconclusive, and ##f_x## and...
  29. S

    B Exploring Linear Combinations in Quantum Mechanics

    Hi, I read that linear combinations of a state, Psi, can be as: \begin{equation} \Psi = \alpha \psi + \beta \psi \end{equation} where ##\alpha## and ##\beta## are arbitrary constants. Can however this be a valid linear combination?\begin{equation} \Psi = \alpha \psi \times \beta \psi...
  30. LarryS

    I Measures of Linear Independence?

    My formal education in Linear Algebra was lacking, so I have been studying that subject lately, especially the subject of Linear Independence. I'm looking for functions that would qualify as measures of linear independence. Specifically, given a real-valued vector space V of finite dimension...
  31. F

    Ball rolling down an incline angular and linear acceleration

    Homework Statement A ball with moment of intertia bmr^2 rolls without slipping down a plane inclined at angle O. What is it’s linear acceleration? Homework Equations Fnet= ma T = F x r T = IO’’ a = rO’’ The Attempt at a Solution Fnet = mgcos(O) - FfT = FfrT = IO’’ Ffr = bmr^2O’’ Ff =...
  32. D

    Solving Systems of Linear Equations in Two Variables- Graphs

    Homework Statement Solve the system of equations: { (1/2)x-y=3 and x=6+2y Homework Equations NA The Attempt at a Solution The solution is 3=3, which is an identity, which means that there is an infinite amount of solutions to the system. Here's where my question lies (asked my teacher but...
  33. H

    I Equality of two particular solutions of 2nd order linear ODE

    I got the following two integral for the a particular solution of a 2nd order linear ODE $$(D-a)(D-b)y = g(x)$$ by using inverse operators ##\frac{1}{D-a}## and ##\frac{1}{D-b}##. The two different integrals are obtained by operating these operators in different order on y to get a particular...
  34. W

    I Interesting Article Ref: Sudoku and Linear Algebraic aspects

    Hi, This looked interesting. Determinant and Linear-Algebraic aspects of Sudoku puzzles: http://www.student.montefiore.ulg.ac.be/~merciadri/docs/papers/sudoku.pdf
  35. N

    Linear polarizers: angle between the electric & magnetic fields

    From wikipedia: "An electromagnetic wave such as light consists of a coupled oscillating electric field and magnetic field which are always perpendicular; by convention, the "polarization" of electromagnetic waves refers to the direction of the electric field." A polarizer is an object that...
  36. valovato

    Momentum transfer from linear to angular motion

    I am trying to create a momentum trap to calculate the velocity of a projectile when it hits the trap. It essentially consists of a plate if known mass at the bottom of a pendulum of known radius. When the projectile hits the plate, the degrees that the pendulum rotates are recorded. I believe...
  37. Drakkith

    Finding the Standard Matrix of a Linear Transformation

    Homework Statement Let ##T:ℝ^3→ℝ^2## be the linear transformation defined by ##\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\mapsto \begin{bmatrix} x_1 + x_2 + x_3\\ 0 \end{bmatrix}##. i. Find the standard matrix for ##T##. Homework EquationsThe Attempt at a Solution For this problem I was...
  38. C

    MHB Find accelerations- Angular and linear

    Dear Every Body, I need some help. First question: Is the tension of the string is the same or different? A pulley of moment of inertia 2.7 kg · m2 is mounted on a wall as shown in the following figure. Light strings are wrapped around two circumferences of the pulley and weights are attached...
  39. M

    Constructing a 3x3 Linear system question

    Homework Statement Construct a 3 × 3 example of a linear system that has 9 different coefficients on the left hand side but rows 2 and 3 become zero in elimination. If the right hand sude of your system is <b1,b2,b3> (Imagine this is a column vector) then how many solutions does your system...
  40. M

    Finding values to make a linear system consistent

    Homework Statement Given the following matrix: I need to determine the conditions for b1, b2, and b3 to make the system consistent. In addition, I need to check if the system is consistent when: a) b1 = 1, b2 = 1, b3 = 3 b) b1 = 1, b2 = 0., b3 = -1 c) b1 = 1, b2 = 2, b3 = 3 Homework...
  41. G

    Energy Flow in Coaxial Cable with Linear Free Charge Density

    Homework Statement An infinitely long cylindrical capacitor with inner radius a and outer radius b carries a free charge per unit length of ##\lambda_{free}##. The region between the plates is filled with a nonmagnetic dielectric of conductivity ##\sigma##. Show that at every point inside the...
  42. Javier Mont

    How to understand linear momentum?

    I understand for linear momentum the measure of the dificulty to take a body to the rest. But when I do the next exercises, I find that it requires less force for one body compared to the other with same linear momentum. Can someone explain me what to understand in an intuitive way what to...
  43. Math Amateur

    MHB Why Does Lemma 8.4 Equal This Expression?

    The "Operator Norm" for Linear Transformations ... Browder, Lemma 8.4, Section 8.1, Ch. 8 ... ... I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra...
  44. Math Amateur

    MHB Operator Norm for Linear Transformations: Browder Ch. 8, Section 8.1, Page 179

    The "Operator Norm" for Linear Transfomations ... Browder, page 179, Section 8.1, Ch. 8 ... ... I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra...
  45. Erenjaeger

    I How Do You Formulate Decision Variables in a Linear Programming Problem?

    I was given the problem attached in the photo below and the first question is to define the decision variables and formulate the problem as a linear program. There are no solutions online, so it would be helpful if someone on the mighty PF could check them to see if they are correct, thanks...
  46. S

    What is the Maximum Speed Limit for a Linear Shaft in Reciprocating Motion?

    Hi all. I am not a mechanical engineer but I've been working on an idea which consists of a linear shaft with Reciprocating motion. Since the device only works with rather a high speed shaft, I wanted to know what are the speed restrictions of a linear shaft? So let's give it some numbers. The...
  47. B

    Tangential acceleration, linear acceleration, and torque

    What is the difference between tangential and linear acceleration of a circular object (let's say a ball)? Also, how does the torque contribute to linear acceleration?
  48. H

    How Much Energy Is Needed to Move and Rotate a Weighted Tray Using Rubber Tires?

    Hi, The goal here is to increase/decrease the angle x by 5 degrees. the black circles represent a rubber tire (same as car tires, only smaller), 5cm in diameter, rotating on it's center, powered by a motor. each wheel has 40 psi (2.76 bar) of air in it. the yellow tray on both diagrams are...
  49. U

    MHB Simple question regarding linear regression model poisson

    The question: Suppose $Y$ is discrete and only takes on non-negative integers and that the conditional distribution of $Y$ given $X=x$ is Poisson, that is, $$P(Y=y|X=x) = \frac{\exp(-x'\beta) (x'\beta)^y}{y!}$$ where $y = 0, 1, 2, \cdots$. First compute $E(Y|X=x)$ and $Var(Y|X=x)$, does this...
  50. D

    B What is a Linear Relationship?

    So, my pre-algebra class is learning about linear relationships, but can anyone help explain it to me a little more? I tried google, but honestly, I want to hear other people before jumping to conclusions. So, my question is, what is a linear relationship?
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