Linear Definition and 59 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".

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  1. C

    Ohm's Law graphing inversed gradient value

    Hey all. This is about Ohm's Law (and specifically resistance). When you plot the change in current vs the change in voltage you should get a linear trend line (providing it is from an ohmic device). The gradient should be the resistance. My questions is why does the gradient value need to be...
  2. K

    I Find the practical resonance frequencies in a system of linear differential equations

    Hi all, I would like to know what is the equation upon which I can use to determine the practical resonance frequencies in a system of second order, linear differential equations. First some definitions: What I mean by practical resonance frequencies, is the frequencies that a second order...
  3. YoungPhysicist

    I Area divided by linear function

    The original problem for anyone that can read Chinese: The problem defines a convex polygon with multiple points located in the first quadrant and the required task is to find a linear function y = ax that can spilt the polygon into two parts each...
  4. A

    I Linearity of DE's

    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?
  5. O

    Find the distance from the point P to a line - linear algebra

    Homework Statement Find the distance from point P (1,7,3) to the line (x,y,z) = (-2,1,4) + s(1,-3,4), s is a free variable Homework Equations projnQP = ( QP⋅n/(lengthQP)(lengthn) )(n) The Attempt at a Solution I'm not quite sure about how to find the normal (n) here, but if I make s=0, I'm...
  6. O

    Angle between lines, with free variables in equations?

    Homework Statement Find the cosine of the angle determined by the intersecting lines m1: (x,y,z) = (-2,1,4)+s(1,-3,4) m2: (x,y,z) = (-2,1,4)+t(-2,3,2) s and t are free variables Homework Equations a⋅b = length(a)length(b) cosθ The Attempt at a Solution I just did this equation using no...
  7. O

    Show that eigenvalue of A + eigvalueof B ≠ eigvalue of A+B?

    Homework Statement Let A and B be nxn matrices with Eigen values λ and μ, respectively. a) Give an example to show that λ+μ doesn't have to be an Eigen value of A+B b) Give an example to show that λμ doesn't have to be an Eigen value of AB Homework Equations det(λI - A)=0 The Attempt at a...
  8. kvothe18

    I Linear fitting in physics experiments with errors

    Hello! I have a question that maybe has to do more with Mathematics, but if you do experimental physics you find it quite often. Let's assume that we want to measure two quantities x and y that we know that they relate to each other linearly. So we have a set of data points xi and yi...
  9. SemM

    A Nonlinear operators vs linear

    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...
  10. S

    B Linear combinations

    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...
  11. G

    Why is linear momentum not conserved or realistic here?

    - This is the situation provided where it has to be demonstrated that linear momentum is not conserved or unrealistic. My assumptions are: there's no friction due to how Godzilla is moving, the sand is acting as a lubricant. Velocity is constant. This is an elastic collision. I also guess that...
  12. S

    I Linear Program:Multiple Optima for multivariable Obj. Func.?

    I know there can be an infinite number of solutions when the objective function with 2 variables has an equal slope as a constraint's slope (assuming the constraint is affecting the feasible region and not a redundant constraint). How can you know there are multiple optimal solutions for...
  13. L

    Linear and Non-linear Equations (QM)

    1. Problem Recall that we defined linear equations as those whose solutions can be superposed to find more solutions. Which of the following differential/integral equations are linear equations for the function u(x,t)? Below, a and b are constants, c is the speed of light, and f(x,t) is an...
  14. M

    A R1^2 + R2^2 = R3^2?

    Hey. I am working with OLS regression. First I run 2 regression operations with each having just one independent variable. Then I run another regression using both the independent variables from the first two regressions. If the explanatory "power" (R^2) in the third regression was to be the sum...
  15. R

    How stretching a rubber band, affects the range of a shot?

    Homework Statement I did an experiment on how stretching a rubber band, affects the range or distance of a projectile shot. Variable list: Independent Variable Amount of stretching Dependent Variable Range or distance I created my catapults on my own and used two different rubbers bands...
  16. M

    I F-test longitudinal data

    Can i Use a standard F-test on longitudinal data for a linear multiple regression? Mons
  17. M

    I Panel study, multiple linear regression, assumptions

    Hey. I am doing a project where I am studying a set of companies over a 7-year period. I am doing a multiple linear regression analysis either with fixed or random effects (so, it's a panel study). What I am wondering is if the general assumptions/requirements apply when using the fixed/random...
  18. 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...
  19. M

    Proving basic linear ODE results

    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'...
  20. D

    Engineering Solving a linear circuit #2

    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...
  21. D

    Engineering Solving a linear circuit

    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...
  22. W

    Show: Elementary row operations don't affect solution sets

    Homework Statement Show that elementary row operations don't affect solutions sets in linear systems Homework Equations - The Attempt at a Solution It's pretty easy to come up with a random linear system and perform ERO on them and showing that solutions are not affected, but is there all...
  23. D

    Engineering Solving a linear circuit

    Homework Statement In the circuit above ##E_1=E_2=E_3##, ##Ig_1=Ig_2=Ig_3## and ##R_1=R_2=R_3=R_4##. In the case when the switched is closed the power of the ideal generator ##E_2## equals ##P_{E_2}=-30W##. Then the switch opens. Calculate the ##ΔP_{E_1}##. Homework Equations 3. The Attempt...
  24. Euler2718

    I Linearizing a System of ODE's

    I am given the equations of Lorenz with respect to deterministic non-periodic flow: \frac{dX}{dt} = Pr(Y-X), X(0)=X_{0} \frac{dY}{dt} = -XZ + rX - Y, Y(0) = Y_{0} \frac{dZ}{dt} = XY-bZ, Z(0) = Z_{0} where Pr is the Prandtl number, r = Ra/Rac is the ratio of the Rayleigh number to its...
  25. W

    2nd Order Linear ODE-Derivation of system-issue

    Homework Statement How exactly they combined equation1 and equation2 and got that system? I don't get that part. Homework Equations A*(dy/dt)= -k*y eq1 A*(dz/dt)=ky-kz eq2 The Attempt at a Solution I tried substituting the 1st ky in the 2nd equation and then differentiating but I don't...
  26. S

    Angular Velocity in Linear Momentum Equation

    m1v1 = m2v2 v = rω m1(rω)1 = m2(rω)2 m1ω1 = m2ω2 Does this make sense?
  27. RoboNerd

    Question about conservation of linear momentum

    Homework Statement [pardon my crude drawing] Say I have a ball that hits the a rod at rest pivoted around hinge A. The ball in the diagram is the solid black, and the rod is the thin line in the diagram and is rotating around the square-shaped hinge at the bottom. I am not an artist but...
  28. mpoli

    Linear Programming Case Study - Case Problem

    Homework Statement Linear Programming Case Study - Case Problem ( Page # 109 Decision making methods) “The Possibility” Restaurant? In the case problem, Angela and Zooey wanted to develop a linear programming model to help determine the number of beef and fish meals they should prepare each...
  29. T

    All possible planes, given two points

    Homework Statement Find the equation of all planes containing the points P(2, -1, 1) and Q(1, 0, 0) Homework Equations The Attempt at a Solution I use PQ to get a vector, (-1, -1, 1). I some how need to use another vector so I can use the cross product to find the planes. So i let another...
  30. Alanay

    -66 over -99 into a graph?

    Okay, so I'm down to the last equation. -12/11 x 10 + 54/11 I get -66/-99. Is this right? If so how do I put it into the graph. -12/11 x 10 = -120/110 + 54/11 = -66/99 (I think I've went wrong somewhere)
  31. Ryan Reed

    How calculate iris radius and cavity radius in accelerator?

    In linear accelerators that use a disk loaded structure (traveling wave), how would you calculate the iris(disk hole) radius, cavity radius, and disk thickness according to the wavelength
  32. T

    I Linear Transformation notation

    I'm confused about the notation T:R^n \implies R^m specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?
  33. T

    B What are the values in a vector?

    I'm trying to understand the concept of vectors. Vectors have magnitude and a direction. When I read vector with some values \textbf{x} = \left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right) = \left(\begin{array}{c}1\\2\\3\end{array}\right) I'm not sure what these values are. Are the values...
  34. G

    Does LIGO discovery rule out (quasi-)linear theories?

    As the discovery matches templates based on GR, and the regime is of very strong gravitational fields and very high speeds (relativistic speeds), and there is a 90% match between model and measured data, this does rule out linear or quasi linear alternative theories of gravity?
  35. B

    Linear momentum and Angular momentum

    Homework Statement Two objects with mass of m are connected with a rod with length 2l and with no mass. The center of the rod is pinned so that it can spin. Object with mass M comes with speed v and sticks to m. There is no friction. 1) What is the angular speed w after collision? 2) FInd the...
  36. kostoglotov

    Soln space basis for all constant coeff homo linear DE's?

    From what I've seen so far, the basis of the solution space for all the constant coefficient homo linear DE's have been linear combinations of the exponential function e or of some polynomial multiplied by the exponential function. Is this always true that these DE's always result in solutions...
  37. S

    Matrix of a Linear Transformation Example

    Homework Statement Hi this isn't really a question but moreso understanding an example that was given to me that I not know how it came to it's conclusion. This is a question pertaining linear transformation for coordinate isomorphism between basis. Homework...
  38. D

    Definable relationships between 4 variables with 2 equations

    Hi, I saw a derivation in a book and I don't see the logical connection. Suppose 1.\ \ a=b \text{ and } 2.\ \ x=y Then 3.\ \ a-b= \lambda (x-y) makes sense to me since 0=\text{Anything}⋅0 However they said "similarly" 4.\ \ a+b= \mu (x+y) , and this I don't understand. To me a+b= 2a...
  39. G

    Quantum Mechanics, Cart filling, + friction as it moves

    Homework Statement A cart roles down the track with an initial velocity vo. Because of falling rain, water starts filling the cart such that its mass increases linearly with time. The rain that has fallen on the track cause the wagon to experience a frictional force characterized with a...
  40. N

    Find the approximate linear ODE system

    dx/dt = x-y^2 dy/dt= x^2 -xy -2x For each critical point, find the approximate linear OD system that is valid in a small neighborhood of it. I found the critical points which are (0,0),(4,2),(4,-2) but have no idea how to do the above question! please help!
  41. Dominic Luciano

    Constructing Equations from Points

    Hi guys, new here. Thank you for reading my post. I'm posting today because although I'm an engineer, I have some brain tumors that have impeded my ability to process most math above basic algebra (despite the fact that prior I'd gone all the way through stats in college). It didn't really start...
  42. M

    Conrod equal more torque?

    Hi, Sorry if this is in the wrong thread I'm new to the forums. I was recently talking my mate about how engines produce power and torque and how there's so many factors in it. I mentioned the length of conrod would be a factor but my mate disagreed that it has no effect on torque. To my...
  43. azeriAggie

    How can I control two outputs with a single PID-controller?

    Overview I'm trying to levitate a constrained permanent magnet with 2 electromagnets. I'm having trouble conceptualizing the control system for such an operation. Setup The permanent magnet is fixed onto a horizontal pendulum and is repelled by an electromagnet above and repelled by an...
  44. S

    Can someone explain to me what a matrix is in simple words?

    Ok so officially a matrix is a rectangular array of numbers, symbols, etc arranged in rows and columns that is treated in certain prescribed ways. But that doesn't help me understand a darn thing. From what I understand, a matrix is a math tool that can help you solve linear systems, represent...
  45. Prof. 27

    First Order Linear Differential Equation

    Homework Statement dv/dt = 9.8 - 0.196v Set in correct form: dv/dt + 0.196v = 9.8 Since p(t) = 0.196, u(t) the integration factor is given by: u(t) = e∫0.196 dt Multiply each term by u(t) and rearrange: (e∫0.196 dt)(dv/dt) + (0.196)(e∫0.196 dt)(v) = (9.8)(e∫0.196 dt) From now on we will set...
  46. B

    Abaqus 6.14 CAE - Imperfection Study

    The title of the project is as follows: A study of the effect of imperfections on the buckling capability of a soda can under axial loading. My group and I are aiming to carry out the following: Linear buckling analysis of soda can to gain first 5 eigenvalues Riks buckling analysis with...
  47. Prof. 27

    Linear Ordinary Differential Equation: Definition

    Homework Statement The website says this: "It is Linear when the variable (and its derivatives) has no exponent or other function put on it. So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is). More formally a Linear Differential Equation is in the form: dy/dx +...
  48. B

    Wronskian to determine L.D

    Homework Statement Hello, I was just looking for a quick tip: If I have three distinct solutions to a second order linear homogeneous d.e, how would I show that the wronskian of (y1,y2,y3)(x)=0? I know how to show the wronskian is not zero for a linearly independent set, but I'm confused...
  49. F

    Finding E Field at Point from non uniform charge density

    1. Portion of z-axis for which |z| < 2 carries a non uniform charge density of 10|z| (nC/m). Using cylindrical coordinates, determine E in free space at P(0,0,4). Explicitly show your integration. Homework Equations E = (1/4πε0) ∫ dQ*aR/(R2) The Attempt at a Solution...
  50. W

    Linear operator and linear vector space?

    hi, please tell me what do we mean when we say in quantum mechanics operators are linear and also vector space is also linear ?