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. 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...
  2. 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...
  3. 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...
  4. 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##...
  5. S

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

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  6. M

    Linear Transformations if the design matrix

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  7. Schaus

    Linear Algebra - REF with another variable

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  8. J

    3D Eccentric Impact -- Final Linear and Angular Velocities

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  9. S

    I Ordered set which is not linear

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  10. 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...
  11. B

    A Hahn-Banach From Systems of Linear Equations

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  12. C

    Transmission over a linear barrier (QM)

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  13. U

    Conservation of linear momentum in this system

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  14. D

    Courses Linear Algebra or Computer Science?

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  15. C

    Conservation of Linear Momentum and Covariance

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  16. M

    A R2 Addition in OLS Regression: Unrelated Variables?

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  17. A

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

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  18. Eclair_de_XII

    More linear charge density troubles....

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  19. A

    Current versus dynamic force on linear actuators

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  20. 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...
  21. 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...
  22. J

    I Solving a system of linear equations using back substitution

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  23. T

    Understanding Linear Motors: Right Hand Rule & Current Flow

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  24. Eclair_de_XII

    What is wrong with my linear charge density calculations?

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  25. G

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

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  26. Pushoam

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  27. F

    I Question about second order linear differential equations

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  28. L

    B Generalized Linear equation of a cube

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  29. G

    A Selected solution of linear equations

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  30. M

    Charge moving with a constant linear velocity....

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  31. Pushoam

    Uniform polarization and linear dielectrics

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  32. B

    Linear Actuator Sizing for Angular Force (BMX starting gate)

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  33. 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?
  34. 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?
  35. R

    Courses Is proof based Linear Algebra be similar to Abstract Algebra

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  36. Mr Davis 97

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

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  37. M

    Estimating Eigenvalues from linear ODE

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  38. S

    Linear Least-Squares Regression: a_0 = ?

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  39. Eclair_de_XII

    Trouble understanding linear transformations in this context

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  40. 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...
  41. 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)##...
  42. O

    Courses Linear Algebra vs Deterministic Operations Research for CS

    Hey all, I'm currently working on my CS degree with a mathematics minor. After this Fall, I will only have one more course to take to finish my minor. I'm debating between Linear Algebra and Deterministic Operations Research. I do have other options, but these seem to be most applicable to CS...
  43. M

    I Linear regression and probability distribution

    I have some data that I want to do simple linear regression on. However I don't have a lot of datapoints, and would like to know the uncertainty in the parameters. I.e. the slope and the intercept of the linear regression model. I know it should be possible to get a prob. distribution of the...
  44. Drakkith

    Linear Independence of Two Functions

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  45. Elroy

    Linear Algebra Problem: Solving for Euler between two ordered bases

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  46. Ackbach

    MHB Linear Regression Gradient Descent: Feature Normalization/Scaling for Prediction

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  47. M

    MHB What is the Linear Equation for the Yield of an Orange Grove?

    Imagine that you own a grove of orange trees, and suppose that from past experiences 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...
  48. S

    MHB Geometry in Linear Systems

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  49. P

    Electric Field of Linear Charge along a Line Segment

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  50. SetepenSeth

    Linear Algebra - Kernel and range of T

    Homework Statement Let ##T:M_2 \to M_2## a linear transformation defined by ##T \begin{bmatrix} a&b\\ c&d \end{bmatrix} = \begin{bmatrix} a&0\\ 0&d \end{bmatrix}## Describe ##ker(T)## and ##range(T)##, and find their basis. Homework Equations For a linear transformation ##T:V\to W##...
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