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".

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  1. A

    MHB Are Mutually Perpendicular Vectors Always Linearly Independent?

    In a problem I am working on, it is given that $V_1, V_2, ... , V_n$ are mutually perpendicular vectors in a space defined with a certain scalar product. I need to prove or disprove that $V_i$ are linearly independence regardless of any definition of scalar product. I think the solution should...
  2. PhotonSSBM

    Potenital of a cylindrical, linear dialectric in a E-Field

    Homework Statement A very long cylinder of linear dialectric material is placed in an otherwise uniform electric field ##E_0##. Find the resulting field within the cylinder. (The radius is ##a##, the susceptibility ##X_e##, and the axis is perpendicular to ##E_0##) Homework Equations Boundary...
  3. A

    Two Coaxial Linear Magnetic Materials

    Homework Statement Consider a two-layered cylindrical wire with inner-layer permeability μ1 and outer-layer permeability μ2. A line current I runs through the center in the z direction. Calculate the bound currents and the magnetic field produced by the bound currents. Homework Equations [1]...
  4. StarWarsNerd

    B Calculating Linear and Angular Quantities from Applied Force

    In the image above, a centroid with radius 1 is depicted. F1 is pointing directly at point A (which is the center of the circle), and F2 is pointing directly at point B. The radius for finding the torque would be the perpendicular between the center of the object and the force vector, so r1...
  5. W

    Linear Differential Equations and Linear Operator Problem

    Homework Statement I'm not sure how to approach this. The question involves linear operators and a non-homogenous differential equation. Here is the question: https://s15.postimg.org/cdmw80157/Capture.png Homework Equations They are given in the question The Attempt at a Solution I really...
  6. Valour549

    I 2nd Order Linear ODE w/ nonconstant coefficient

    xy'' + 2xy' - y = 0 Honestly no clue where to start, Wolfram Alpha gives a rather complex answer lol (http://www.wolframalpha.com/input/?i=xy%27%27%2B2xy%27-y%3D0)
  7. M

    I Is the set {e^x, x^2} linearly independent?

    Hello all. I have a question about linear dependency. Suppose we have a set ##S## of functions defined on ##\mathbb{R}##. ##S = \{e^x, x^2\}##. It seems very intuitive that this set is linear independent. But, we did something in class I'm unsure about. Proof: Let ##\alpha, \beta \in...
  8. nysnacc

    Are the Wronskians set up and compared in this linear independent check?

    Homework Statement Homework Equations The Attempt at a Solution I set up the Wronskian?? and compare them?
  9. O

    2-D Linear Momentum, Collision Problem

    Good evening! I'm having trouble completing an assignment related to inelastic collisions and momentum. Homework Statement Here is the statement: You are in great shape on this hot July morning. You are working as an intern for the Montreal Police (SPVM) Collision Investigation unit. You...
  10. M

    MHB Finding B^-1 in 3x3 Matrices with Linear Algebra

    if A and B are 3x3 matrices such that: ABC = I, |3A|=81 and |C^T|= 2 , how to find |B^-1| I couldn't solve this because there is not much given.
  11. Kernul

    Find out if it's linear transformation

    Homework Statement Does a linear transformation ##g : \mathbb{R}^2 \rightarrow \mathbb{R}^2## so that ##g((2, -3)) = (5, -4)## and ##g((-\frac{1}{2}, \frac{3}{4})) = (0, 2)## exist? Homework EquationsThe Attempt at a Solution For a linear transformation to exist we need to know if those two...
  12. Rectifier

    Linear algebra - linear equation for a plane

    The problem I am trying to write the equation for the plane on the following form ## ax + by + cz + d = 0 ## $$ \begin{cases} x = 1 + s - t \\ y = 2 - s \\ z = -1 + 2s \end{cases} $$ The attempt ## s, t ## are the parameters for the two directional vectors which "support" the plane. $$...
  13. B

    B Linear energy transfer for electrons

    Hi! I have a question about Linear Energy Transfer for electrons. According to the definition the linear energy transfer is equal to the energy dE which a charged particle loses at a distance dl. But what does dl mean in case of electrons? Is LET calculated per unit of full path length or per...
  14. Mayan Fung

    I Questions about linear transformations

    We learned that the condition of a linear transformation is 1. T(v+w) = T(v)+T(w) 2. T(kv) = kT(v) I am wondering if there is any transformation which only fulfil either one and fails another condition. As obviously, 1 implies 2 for rational number k. Could anyone give an example of each...
  15. karush

    MHB 10) AP Calculus linear functions

    $\textbf{10)} \\ f(x)\text{ is continuous at all } \textit{x} \\ \displaystyle f(0)=2, \, f'(0)=-3,\, f''(0)=0 $ $\text{let} \textbf{ g } \text{be a function whose derivative is given by}\\ \displaystyle g'(x)=e^{-2 x} (3f(x))+2f'(x) \text{ for all x}\\$ $\text{a) write an equation of the...
  16. L

    I 2° order linear, homogeneous, variable coefficients

    sin(x) * y''(x) + 2cos(x) * y(x) = 0 y(0) = 0 y'(0) = 1 how do I solve it? (I know the solution because I have created the diff. equation starting from a simple function). -- lightarrow
  17. thecourtholio

    Energy for Linear 1D Systems - 1D potential

    Homework Statement A particle of mass m moves along the x–axis under the influence of force ##F_x=-ax+bx^3## , where a and b are known positive constants. (a) Find, and sketch, the particle's potential energy, taking U(0) = 0 (b) Identify and classify all equilibrium points (c) Find the...
  18. K

    Proving That Any Vector in a Vector Space V Can Be Written as a Linear Combination of a Basis Set

    Homework Statement Show that any vector in a vector space V can be written as a linear combination of a basis set for that same space V. Homework Equations http://linear.ups.edu/html/section-VS.html We are suppose to use the 10 rules in the above link, plus the fact that if you have a lineraly...
  19. S

    Angular Velocity in Linear Momentum Equation

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

    How Do You Calculate Change in Kinetic Energy and Momentum for a Turning Loader?

    Homework Statement A 15 000kg loader traveling east at 20km/h turns south and travels at 25km/h. Calculate the change in the loader's a) kinetic energy b) linear momentum Homework Equations ΔKE= 1/2mvf^2 –1/2mvi^2 Δp= mv2 –mv1 m= 15000 vi = 20km/h = 5.56m/s vf= 25km/h = 6.94m/s The Attempt at...
  21. M

    I Linear least-squares method and row multiplication of matrix

    Suppose that I have an overdetermined equation system in matrix form: Ax = b Where x and b are column vectors, and A has the same number of rows as b, and x has less rows than both. The least-squares method could be used here to obtain the best possible approximative solution. Let's call this...
  22. mmcsa

    Determining linear velocity of pendulum

    Hello, I'm trying to develop a pendulum to test protective equipment so I want to work out the length I'll need to generate a desired velocity and the necessary mass I'll need for a specific moment of inertia. I know there are multiple ways to solve for linear velocity with equating Ek and Ep...
  23. A

    Linear differential equation hwk check

    1. Homework Statement Posted Homework Equations Posted The Attempt at a Solution Posted I just need the work to be checked.
  24. kyphysics

    Linear Algebra Any Great Linear Algebra Books for First-Time Learners?

    What are the best ones and why for a first-timer like myself (doing self-study)? Thanks very much everyone.
  25. R

    MHB Proving disjoint of Kernel and Image of a linear mapping

    I am working on a problem that goes like this: Show that $Ker (F) \cap I am (F) = \{0\}$ if $F: W \rightarrow W$ is linear and if $F^4 = F.$ I have the solution but there is one step which I need help: (the delineation is mine) (1) Suppose that there exists $x$, such that $x \in Ker(F) \cap...
  26. T

    Linear polarized microstrip patch antenna - design help

    hi there,im pretty new to antenna design, still have a lot of theory to go through, so please be kind. ;)I have to design a linearly polarized microstrip patch antenna (rectangular) that is fed with 2 coaxial probe feeds. The patch should be a square (L=W), so i can place 2 feeds orthogonally...
  27. P

    Linear Optimization Conversion (nested free variable)

    Homework Statement (This is a mock-up) Given maximize(x,y) Ax + | x - y | s.t. x + By - C >= 0 y >= 0 A, B, and C are some coefficients. Homework Equations Linearizing absolute expressions: z = | x |; where z >= x, z >= -x Linearizing free variables: let x = x+ - x-; where x+, x- >= 0...
  28. M

    I Expanding linear independent vectors

    Hi PF! The other day in class my professor mentioned something about expanding linear independent vectors, but he did not elaborate. From what I understand, if ##x_1,...,x_k## are linearly independent vectors in ##V##, where ##dimV=n>k##, how would you extend ##x_1,...x_k## to a basis ##\{...
  29. M

    Linear transformations, images for continuous functions

    Homework Statement Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$ a) Show ##T## is a linear transformation from ##C## to ##C##. b)What...
  30. P

    Is the system linear or nonlinear

    Homework Statement 3y(t)+2=x(t) Homework Equations k1y1(t) + k2y2(t) + 2(k1+k2) = k1x1(t)+k2x2(t) The Attempt at a Solution I know the system is non linear but I cannot explain why. It has something to do with 2(k1+k2) but I am unsure.
  31. R

    Quick question on intro to linear algebra book

    I'm looking at purchasing Algebra (2nd Edition) by Michael Artin, is this a good book to purchase as my first intro to linear algebra book for self learning?
  32. J

    MHB Two problems: linear distance and directed graph

    1.) Two volumes of an encyclopedia stand side by side, in order, on a shelf. A bookworm starts at page i of Volume I and bores its way in a straight line to the last page of Volume II. Each cover is 1 mm thick, and the first volume is 2/5 as thick as the second volume. The first volume is 40 mm...
  33. J

    Help finding general solution for 2nd order linear DE

    Homework Statement Find the general solution for the DE: t2y''-2y=0 Homework Equations These were given for other parts of the problem so I'm not sure if they're relevant. y1(t)=t2, y2(t)=t-1, y(1)=-2, y'(1)=-7 The Attempt at a Solution The t2 at the front was really stumping me and I'm not...
  34. Christopher117C

    Centripetal Force of a linear spring

    I was given a question relating centripetal force to solve A linear spring of negligible mass requires a force of 18.0 N to cause its length to increase by 1.0 cm. A sphere of mass 75.0 g is attached to one end of the spring. The distance between the centre of the sphere M and the other end P of...
  35. almarpa

    Algebra Similar book to Kleppner's Quick Caculus for linear algebra

    So anyone of you know a book that provides a gentle and quick refresher for linear algera, in the spirit of the book "Quick Calculus" by Kleppner and Ramsey? Now that I am studying quantum mechanics, I feel I need to review the linear algebra I studied during my engineering degree. Thanks.
  36. ShaiNaveh

    What is the formula for calculating force in a crank and slider mechanism?

    Hi all, I have a question regarding a simple crank and slider mechanism, but I'm not a mathematician. This means that long formulas scare me... (: I've been searching around and haven't found anything I can really understand. I'm trying to get a "ball park" formula to calculate the amount of...
  37. binbagsss

    QM Bra & Ket Linear Algebra Hermitian operator proof -- quick question

    Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...
  38. RJLiberator

    Semi-circle linear charge Electric Field

    Homework Statement A semi-circle of radius R has a charge Q uniformly distributed over its length, which provides a line charge density λ. Determine E at the origin. Homework EquationsThe Attempt at a Solution https://www.physicsforums.com/attachments/105239 I can tell by argument of...
  39. B

    MHB Finding the Measures of Angles in a Linear Pair

    <1 and <2 form a linear pair. If m<1 = 5x + 9 and m<2 = 3x + 11, find the measures of both angles.
  40. a_c_e

    Gas Spring and Linear Actuator calculation....

    May I start by saying hello to all and that I have visited the site before, but have yet to create an account until now. With that said, I am having difficulty calculating proper numbers for a project I am designing. Mind you, it has been about seven years since I was in any physics or...
  41. T

    Linear Algebra, subset of R2 not closed under scalar multipl

    Homework Statement Construct a subset of the x-y plane R2 that is (a) closed under vector addition and subtraction, but not scalar multiplication. Hint: Starting with u and v, add and subtract for (a). Try cu and cv Homework Equations vector addition, subtraction and multiplication The...
  42. tomwilliam2

    I Solving linear differential equations

    I have the following linear differential equations: ##A\dot{x} + By = 0## ##C\dot{y} + Dx = 0## Where x and y are functions of t, and A through D are constants. I can solve this fairly easily by differentiating the first equation, rearranging, and removing one of the variables, which gives me a...
  43. Keccogrin

    A 2d SWE - Linear Gravity Wave

    Hi, I have some questions about the 2D SWE (Shallow Water Equations). I would like to know if an exact (analytical) solution is available in the case of the (linear) surface "gravity" wave. Indeed, my example of interest is given by a square [0,L]2 with initial velocity field u = v = 0 and an...
  44. T

    How can I modify a LM337 circuit to increase current output?

    So many moons ago I made myself a pretty bog standard dual linear bench power supply using an LM317 and LM337 to give me -15 - 0 - +15. In an attempt to get more current out of it (for no real reason other than because I can) I got a larger transformer and a beefier rectifier, then using the...
  45. D

    Linear Algebra with Proof by Contradiction

    This is a linear algebra question which I am confused. 1. Homework Statement Prove that "if the union of two subspaces of ##V## is a subspace of ##V##, then one of the subspaces is contained in the other". The Attempt at a Solution Suppose ##U##, ##W## are subspaces of ##V##. ##U \cup W##...
  46. M

    Wrong answer on Linear Algebra and Its Applications 4th Ed.

    Homework Statement The Attempt at a Solution \left[ \begin{array}{cccc} 1 & 0 & 5 & 2 \\ -2 & 1 & -6 & -1 \\ 0 & 2 & 8 & 6 \end{array} \right] \sim \left[ \begin{array}{cccc} 1 & 0 & 5 & 2 \\ 0 & 1 & 4 & 3 \\ 0 & 0 & 0 & 0 \end{array} \right] From the RREF it is...
  47. F

    Expectation values linear harmonic oscillator

    hello :-) here is my problem...: 1. Homework Statement For a linear harmonic oscillator, \hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2x^2 a) show that the expectation values for position, \bar{x}, and momentum \bar{p} oscillate around zero with angular frequency \omega. Hint...
  48. T

    Linear Algebra, forcing a row exchange.

    the answer key said d is supposed to be 10. but there's a way to evade that row exchange. 1st picture is the question and the 2nd picture is the elimination steps.
  49. shintashi

    For gear teeth to linear shaft, what's the center point?

    I used to know this back when i worked with Flash animation, but I'm working on a 3D printer now and I've forgotten, when you have a set of teeth, on a linear shaft and a gear, do you base the unrolling of the gear, for the spacing, on the inside of the teeth, the outside of the teeth, or the...
  50. D

    I Linear equations and superposition of wavefunctions

    Hi. I have read many times that the Schrodinger equation is a linear equation and so if Ψ1 and Ψ2 are both solutions to the equation then so is Ψ1 + Ψ2. Is this use of the word linear the same as generally used for differential equations ? As the Schrodinger equation is also an eigenvalue...
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