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. H

    A Identical Fermions in identical linear combinations

    According to Pauli principle the two fermions can not occupy one state of a Hamiltonian. Can the two fermions occupy a state which is linear recombination of two states of the Hamiltonian?
  2. H

    B Linear amplitude decay function

    I collected data on a periodic mechanical motion. The amplitude is damped linearly. What mathematical function models linear amplitude decay. All I can find is exponential decay of a sine wave.
  3. icesalmon

    Independent study for linear algebra

    Hello, I just completed a first course in linear algebra and really enjoyed my studies. So much so that I want to pursue it more in the fall as an independent study, i am a EE major in college and was curious what directions might be useful for applications in that field.
  4. Z

    How to derive linear velocity from position and angular vel.ocity

    Hello! I'm trying to derive the linear velocity vector from the position vector and the angular momentum vector. I've seen on the internet that V = W x R (V,W and R are all vectors and x is the cross product) but I cannot for the life of me derive it! I've tried doing it by writing out the...
  5. T

    Partial Derivatives and the Linear Wave Equation

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  6. J

    I Video that show how a linear accelerator was built in 1967

    Hi folks, I have just found a video on youtube that shows how a 2 mile linear accelerator was built in 1967, I share it with you, hopefully you will enjoy it as much as I did:
  7. B

    I A Question about Notation and Continuous Linear Functionals

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  8. marsupial

    Calculate linear acceleration of a pivoted stick

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  9. E

    I If you were to perform a linear regression of log10(B) vs log10(x)....

    If you were to perform a linear regression of log10(B) vs log10(x) what would you expect the slope to be? The expected relationship between B and x is B(x) = μoI(2πx)-1
  10. Auburn2017

    Nonuniform Linear Acceleration

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  11. R

    I Dividing quadratic with linear

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  12. D

    Linear Transformation R4 to R4: KerT + ImT = R4

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  13. Mrq

    Admissions Solving Quadratic Equations with a Linear Polynomial Relation

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  14. J

    Linear quadratic state regulator

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  15. J

    A Linear Regression with Non Linear Basis Functions

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  16. F

    System of linear inequalities to find a vector

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  17. P

    Rotary to linear motion using a crank

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  18. M

    Linear Algebra: 2x2 matrix raised to the power of n

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  19. M

    Linear Algebra: Matrix Transformation

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  20. M

    Courses Should I take linear algebra over the summer?

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  21. srinaath

    I Why do we linearise a non linear system

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  22. goonking

    Finding a linear differential equation

    Homework Statement Homework EquationsThe Attempt at a Solution So the general solution is the sum of the null solution (Yn) and particular solution (Yp) I believe I just need to write: y = e2t + 5e8t - 5 + C and then find the derivative of both sides y' = 2e2t + 40e8t is this correct?
  23. qnach

    A Why does Jackson consider small antennas on page 412?

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  24. I

    I Is this Linear Accelerator Design Valid?

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  25. F

    MHB Help to find asymptotic solution of linear ode

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  26. D

    Show that the T is a linear transformation

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  27. Q

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  28. arpon

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  29. vtechno1729

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  30. L

    Word problem equation linear growth and slope

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  31. ibkev

    Linear (in)dependence of vector pairs and triples

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  32. S

    MHB Impact of variables in the an in non linear equation?

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  33. P

    Finding a matrix for a linear transformation

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  34. Dusty912

    What does it mean for a linear approximation to be reliable?

    Homework Statement In regards to linearization of a nonlinear system in differential equations. What does it mean for a linear approximation to be reliable to describe the long term behavior of the non-linear system around the equilibrium point? Homework Equations jacobian matrix The Attempt...
  35. B

    MHB Linear Algebra Expert Requested For Discussion

    I was wondering if there was anyone comfortable enough about basic to advanced linear algebra concepts who would be willing to participate in a chat/whiteboard based platform to discuss or answer questions concerning such concepts. I need clarification on a few things
  36. D

    I Linear dependency of Vectors above R and C and the det

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  37. Evangeline101

    Use discriminant for linear and quad. function.

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  38. S

    I Proving the Linearity of the Curl Operator in Electromagnetic Theory

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  39. N

    MHB Word Problem: Application Of Linear Equations (Typo In Textbook?)

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  40. D

    Linear dependence of functions

    Homework Statement check for linear dependecy[/B] f(x) = cosx and g(x) = xcosx 2 functions from R to R Homework EquationsThe Attempt at a Solution Why this is wrong: if i take the scalar a1 = 3, a2 = 1 i can do that since 3 is real, and a1 is in R. so 3f(3) + -1g(3) = 0 there for we have none...
  41. D

    I Linear dependence of functions

    Functions f,g from R to R. f(x) = xcosx, g(x) = cosx for x = 0, we get f(x) = 0, g(x) = 1. so for scalar t in R t(f(x)) + 0 * g(x) = 0 . ==> f(x) and g(x) are linearly idepenent. Is that right? if so in functions we search for an x that makes the function dependent?
  42. P

    Linear transformation representation with a matrix

    Homework Statement For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x1 + x2, 2x1 - x2), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}.Homework Equations T(v) is given, (x1+x2, 2x1-x2) The Attempt at a Solution Okay, I see...
  43. B

    Best way of turning rotational motion into linear motion?

    Hello everyone, I am not an engineer, so I apologize if this is a relatively simple question. What's the best way to turn rotational motion into linear motion under the following circumstances? 1. Rotational motion is driven by a vertical bolt. 2. When turned, the bolt will extend a bar...
  44. C

    MHB Linear Equation Help: Plot, Find Eq., Approx Y for X=5

    Hello, i have been studying for finals and i am stuck on a question on my study guide. the question is to make a scatter plot of a set of data, find the equation of the best fit line, and approximate the value of y for x = 5. the data is like this: x: 0 2 4 6 7 and the y is like : y: 2 7 14...
  45. i_hate_math

    Linear Transformation and Inner Product Problem

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  46. Danielm

    Proving Linear Independence and Spanning in Vector Spaces

    Homework Statement Prove the following: Let V be a vector space and assume there is an integer n such that if (v1, . . . , vk) is a linearly independent sequence from V then k ≤ n. Prove is (v1, . . . , vk) is a maximal linearly independent sequence from V then (v1, . . . , vk) spans V and is...
  47. Danielm

    Proof: Linear Dependence of Vectors in a Vector Space

    Homework Statement Prove the following theorem: Let (v1, . . . , vk) be a sequence of vectors from a vector space V . Prove that the sequence if linearly dependent if and only if for some j, 1 ≤ j ≤ k, vj is a linear combination of (v1, . . . , vk) − (vj ). Homework EquationsThe Attempt at a...
  48. L

    I Can Linear Programming Handle Combined Item and Shipping Costs Optimization?

    Hi everyone. I am tackling a problem related to the purchase of items that are available from different vendors, and from what I've seen so far it seems to me that it may be solvable by linear programming methods (I'm using R). However I'm not sure how to set up the problem to achieve the exact...
  49. S

    MHB Exploring Linear Algebra as a Computer Science Major

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  50. Josh Terrill

    B Linear regression with two data sets?

    I want to try to predict the USA summer highs using a linear regression. I know I can probably take data from the last 10 summers and plug that in, and use that to predict, but I'd like to use two data sources. 1 data source from the historical highs from past summers in the USA, and the 2nd...
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