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. binbagsss

    Green's method- linear differential operator

    Homework Statement 2. Homework Equations 3. The Attempt at a Solution [/B]- So with the (from what i interpret of the notes this is needed) the same boundary conditions when time is fixed, we can relate the 'fundamental problem'- the initial condition ##t=0## given by a delta...
  2. M

    Linear Algebra: Verifying A^2-2A+7I=0

    Homework Statement Verify that A^2-2A+7I=0Homework Equations A is a squared matrix and I is the identity matrix. The Attempt at a Solution I squared a matrix, which I called A, by multiplying the two A matrices together, then I subtracting the new matrix with the third matrix 2A, then I added...
  3. lonelypancreas

    Are ALL parallel plate capacitors linear?

    Homework Statement Are ALL parallel plate capacitors linear? Homework Equations D = permittivity * E The Attempt at a Solution Capacitance is linear if permittivity is constant. Is there a case where a parallel plate capacitor is not linear?
  4. F

    A Differential of Multiple Linear Regression

    Say you have a log-level regression as follows: $$\log Y = \beta_0 + \beta_1 X_1 + \beta_1 X_2 + \ldots + \beta_n X_n$$ We're trying come up with a meaningful interpretation for changes Y due to a change in some Xk. If we take the partial derivative with respect to Xk. we end up with...
  5. 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...
  6. Another

    Question ODE non-homogeneous Linear

    Mod note: Member warned that the homework template is NOT optional find yp (particular integral) (D2 + 4D + 5) y = 2 e-2xcos(x) ((D+2)(D+2)+1) y = 2 e-2xcos(x) yp = [1/((D+2)2 + 1)] ⋅ 2e-2xcos(x) yp = e(-3)x ∫ ∫...
  7. W

    Designing a linear DC power supply

    Homework Statement Design, giving the necessary ratings and characteristics necessary for the transformer, bridge rectifier, reservoir capacitor and regulator, a 12 V, 2 A regulated power supply. Also identify any heat sink requirements. The available mains supply is 230 V ±10% @ 50 Hz...
  8. M

    A Centering variables, linear regression

    I am working with multiple regression with two independent variables, and interaction between them. the expression is: y = b1x1 + b2x2 and b3x1x2 The question is: does one center both independent variables at the same time, when checking for the significance of the effect of the independent...
  9. G

    Wondering if these two First Linear Order IVPs are correct

    Homework Statement I am having trouble proving if the equation i have found for number 1 is correct. I have posted my solution to get back to the main problem in the first photo below. For number 2 I am having trouble isolating for 1 y(x). Did i do the integration and setup properly?Homework...
  10. D

    Characteristic linear dimension (Reynolds' number)

    How can I choose the characteristic linear dimension? For example in pipe it is its diameter, but on a surface is the length, on a flat plane it can be measured as 4A/P. I was having problems determining the characteristic linear dimension for a diffusion problem in a "rectangular" pipe. I don't...
  11. D

    Help intagrating hertzmeter into analog linear power supply

    First off I'm not an engineer nor do I have a electronics background. I have an old Linear (HP 6284A Power Supply). I would like to be able to display Hz/CPS. I need to measure the speed of a tattoo machine. (basically two coils that make and break connection through out its use. From what...
  12. S

    Algebra Recommended books for linear algebra and multi-variable calculus

    hey everyone just started university and the jump i feel is huge from a level and was just wondering if you guys knew of any books that had linear algebra and/or several variable calculus in them but displayed and explained stuff in a clear simple way? or if anyone has any websites that do the...
  13. Eclair_de_XII

    Conceptual question with first-order linear PDEs.

    Homework Statement "Show that a solution of the homogeneous PDE ##au_x+bu_y+cu=0## cannot be zero at one, and only one point in the plane." My interpretation of this is that ##u(x,y)## is zero everywhere on the plane except on that point ##(x_0,y_0)##. Homework Equations ##w=bx-ay## ##z=y##...
  14. S

    Hoffman, Kunze Linear Algebra book: which topics to study QM?

    I've started self-studying quantum mechanics. It's clear from google searching and online Q.Mech lectures, I'll need to understand linear algebra first. I'm starting with finite-dimensional linear algebra and Hoffman, Kunze is one of the widely recommended textbooks for that. I need help...
  15. M

    Linear Transformations if the design matrix

    Homework Statement given that X is an n × p matrix with linearly independent columns. And $$X^∗ = XA$$ where A is an invertible p × p matrix. a) Show that: $$X^*{({X^*}^TX^*)^-}^1{X^*}^T = X{(X^TX)^-}^1X^T$$ b) Consider two alternative models $$M : Y = Xβ + ε$$ and $$M^∗ : Y = X^∗β ^∗ +...
  16. Schaus

    Linear Algebra - REF with another variable

    Homework Statement Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', 'a = ', or 'a ≠', then specify a value or comma-separated list of values. x1−2x2+2x3 = −1...
  17. J

    3D Eccentric Impact -- Final Linear and Angular Velocities

    Homework Statement I have been thinking about this problem for comparison of experimental data to theoretical for a project. A hammer with known mass mh and mass moment of inertia Ih is held stationary at 90 degrees. It is released as swings down. At 0 degrees it impacts a block, initially...
  18. S

    I What Is a Non-Linear Ordered Set?

    Hello! I was introduced in the Real Analysis class the notion of ordered set. I am not sure I understand the concept of linear and non-linear ordering. Can someone explain this to me? Thank you!
  19. 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...
  20. B

    A Hahn-Banach From Systems of Linear Equations

    In this paper on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables ##c_i = A_{ij} x_j##: \begin{align*} \begin{array}{ccccccccc} 1 & = & x_1 & + & x_2 & + & x_3 & + & \dots \\ 1 & = & &...
  21. C

    Transmission over a linear barrier (QM)

    Homework Statement Suppose we have a potential such that $$ V = \left\{ \! \begin{aligned} 0 & \text{ if } x<0\\ \mathcal{E}x & \text{ if } x>0, x<L\\ \mathcal{E}L & \text{ if } x>L \end{aligned} \right. $$ for some electric field ##\mathcal{E}##. I'm trying to find the transmission...
  22. U

    Conservation of linear momentum in this system

    A pet mouse sleeps near the eastern edge of a stationary, horizontal turntable that is supported by a frictionless, vertical axle through its center. The mouse wakes up and starts to walk north on the turntable. Is the momentum of the system constant? i understand that the initial momentum is...
  23. D

    Courses Linear Algebra or Computer Science?

    I am going to have two slots available this year for electives and I want to use one of them for Astronomy. For the other, I am struggling to decide between Linear Algebra or Computer Science (CIS 210 at my university) which focuses on Python programming. If I can only choose one, which is more...
  24. C

    Conservation of Linear Momentum and Covariance

    Homework Statement Assume two masses m1' and m2' are moving in the positive x-direction with velocities v1' and v2' as measured by an observer in S' before a collision. After the collision, the two masses stick together and move with velocity v' in S'. Show that if an observer in S' finds...
  25. M

    A R2 Addition in OLS Regression: Unrelated Variables?

    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...
  26. A

    I Finite universe, quantized linear momentum, and the HUP....

    Hi all, Yet another question: if the universe is finite, then linear momentum should be quantized (I assume in a similar manner to an infinite potential well since there are boundary conditions). My question is, then, if one computes a value for ##\Delta p## (momentum variance), is the variance...
  27. Eclair_de_XII

    More linear charge density troubles....

    Homework Statement An infinitely long line of charge has linear charge density ##λ=4.00_{10^{−12}} \frac{C}{m}##. A proton (mass ##m_p=1.67_{10^{-27}}kg##, charge ##e=1.602_{10^{-19}}C##) is ##r_a=0.18m## from the line and moving directly toward the line at ##v=1000\frac{m}{s}##. Homework...
  28. A

    Current versus dynamic force on linear actuators

    G'day guys So I currently have a 2x 8" stoke 150lbs dynamic force actuators with a maximum current draw of 5A. The exact one is the FA-PO-150-12-XX in the link below: https://www.firgelliauto.com/products/feedback-rod-actuator The load I have on it causes the actuators to draw almost its'...
  29. Drakkith

    Solution Sets of Linear Systems

    Homework Statement Q. Let A be an m by n matrix. Assume that the equation Ax=b is consistent for some b≠0 in Rn. Moreover, assume that Ax=b has a unique solution. Does the equation Ax=0 have a non-trivial solution? Is it possible to find a vector V in Rm so that the system Ax=V has infinitely...
  30. M

    MHB How do I write a linear equation from given information?

    Imagine that you own a grove of orange trees, and suppose that from past experience you know that when 100 trees are planted, each tree will yield about 240 oranges per year. Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases...
  31. J

    I Solving a system of linear equations using back substitution

    Hello, its been a while since I have taken linear algebra and I am having trouble understanding what a target vector is. I need to solve a system of linear equations in matrix form using back substitution and with different target vectors. I don't have a problem with back substitution, but I...
  32. T

    Understanding Linear Motors: Right Hand Rule & Current Flow

    Hi there, I attached an image of a basic linear motor. I'm a little confused if I use the right hand rule or left hand rule. I did it both ways and I think the left hand rule is the only rule that allows for a force being applied to the right. I assume the current is going down rather than...
  33. Eclair_de_XII

    What is wrong with my linear charge density calculations?

    Homework Statement "A straight, nonconducting plastic wire ##x=9.50_{10^{-2}}m## long carries a charge density of ##λ=1.3_{10^{-7}} C/m## distributed uniformly along its length. It is lying on a horizontal tabletop. If the wire is now bent into a circle lying flat on the table, find the...
  34. G

    What Could Be Causing the Malfunction in My Linear DC Power Supply?

    Lavolta BPS 305 0-30V and 0-5A. I was showing a young lad how you can energise a car ignition coil and get a spark on the output, and that this effect can be improved by adding a cap in parallel which forms a crude LC oscillator etc when the unit died. Silly, I know, but we got a bit carried...
  35. Pushoam

    B-field inside a linear magnetic sphere (in a uniform external B-field)

    Homework Statement Homework EquationsThe Attempt at a Solution Because of the external magnetic field ##\vec B_0 ## , a uniform magnetization will be in the direction of external magnetic field. Because off this uniform magnetization, there will be a uniform magnetic fied in the direction of...
  36. F

    I Question about second order linear differential equations

    Hi everybody. I need to learn how to solve this kind of equation by decomposing y in a serie of functions. All the examples I have seen are of homogeneous functions. I would be extremely thankfull if someone pointed me to some text in which this is done-explained. Thanks for reading.
  37. L

    B Generalized Linear equation of a cube

    As per my understanding, a linear equation with two variables form a line segment (ax=by+c or ax+by=c) and linear equation with three variables form a plane (ax=by+cz+d or ax+by+cz=d). Am I right? And if I am right, does an equation with four variables form a cube?
  38. G

    A Selected solution of linear equations

    Hi All, In my work I would like to solve a thousands of moderate size systems of linear equations, parallelly. The uniquum in my problem is that, I am not interested in the whole solutions of the systems, but only several elements of the solution vectors are interesting. To be more strict: Let...
  39. M

    Charge moving with a constant linear velocity....

    Charge moving in constant linear velocity does not produce magnetic field... If not, please provide an explanation...
  40. Pushoam

    Uniform polarization and linear dielectrics

    Can a sphere with a frozen - in uniform polarization ## \vec P ## be considered a linear dielectrics? Following the definition of dielectrics given in Griffiths: The electric field inside the sphere, ## \vec E = \frac {-P}{3 \epsilon_0} ## So, ## \vec P ≠ ε_0 χ_e \vec E ## as ## ε_0 χ_e##...
  41. B

    Linear Actuator Sizing for Angular Force (BMX starting gate)

    I am sizing a linear actuator to build a BMX gate. Please see the drawing below. I have a metal gate that is 20" tall and 9.5' wide. It will be steel and weigh approximately 150lbs. I am mounting the rear of the linear actuator on a clevis 8" off the ground. The front will be mounted with...
  42. S

    I Does chaos exist in circuits with linear elements?

    I have heard that chaos exists in all dynamical systems.Does this mean that chaos exists in circuits with linear elements too?Which software is best for analyzing chaos in electric circuits?
  43. T

    How would rotary motion be converted to linear in robot?

    How would the rotary motion of a robot / exoskeleton arm be converted into linear motion so that the arm can move objects and lift weights in a conventional manner, such as when people are weight lifting in a gym?
  44. R

    Courses Is proof based Linear Algebra be similar to Abstract Algebra

    I know both are different courses, but what I mean is, will a proof based Linear Algebra course be similar to an Abstract Algebra course in terms of difficulty and proofs, or are the proofs similar? Someone told me that there isn't that much difference between the proofs in Linear or Abstract...
  45. Mr Davis 97

    Proving det(A) > 0 for A^3 = A + 1 over R using linear algebra

    Homework Statement If A is an n x n matrix over R such that A^3 = A + 1, prove that det(A) > 0 . Homework EquationsThe Attempt at a Solution So, what I've done is factor the expression to get A(A+1)(A-1) = 1, then taking the determinant of both sides, I get det(A)det(A+1)det(A-1) = 1. I...
  46. M

    Estimating Eigenvalues from linear ODE

    Homework Statement Given $$u''(x)+\lambda u = 0\\ u(-1)=u(1)=0.$$ If ##\lambda_0## is the lowest eigenvalue, show that ##4 \lambda_0 = \pi^2##. Homework Equations $$\lambda_0 = glb\frac{(L(u),u)}{(u,u)}$$ where ##glb## denotes greatest lower bound and ##L## is the Sturm-Louiville operator. I...
  47. S

    Linear Least-Squares Regression: a_0 = ?

    Homework Statement Hello to everyone that's reading this. :) For this linear least-squares regression problem (typed below and also), I correctly find the value of g (which is what the problem statement wants to have found), but I was curious about the value of ##a_0## (and that's what this...
  48. Eclair_de_XII

    Trouble understanding linear transformations in this context

    Homework Statement "Show that every subspace of ##ℝ^n## is the set of solutions to a homogeneous system of linear equations. (Hint: If a subspace ##W## consists of only the zero vector or is all of ##ℝ^n##, ##W## is the set of solutions to ##IX=0## or ##0_vX=0##, respectively. Assume ##W## is...
  49. Eclair_de_XII

    Finding a basis for the linear transformation S(A)=A^T?

    Homework Statement "Find ##S_\alpha## where ##S: M_{2×2}(ℝ)→M_{2×2}(ℝ)## is defined by ##S(A)=A^T##. Homework Equations ##A^T=\begin{pmatrix} a_{11} & a_{21} \\ a_{12} & a_{22} \end{pmatrix}## ##\alpha= \{ {\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0...
  50. Eclair_de_XII

    Proving that determinants aren't linear transformations?

    Homework Statement "Determine whether the function ##T:M_{2×2}(ℝ)→ℝ## defined by ##T(A)=det(A)## is a linear transformation. Homework Equations ##det(A)=\sum_{i=1}^n a_{ij}C_{ij}## The Attempt at a Solution I'm assuming that it isn't a linear transformation because ##det(A+B)≠det(A)+det(B)##...
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