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. P

    Electric Field of Linear Charge along a Line Segment

    Homework Statement Consider a straight line segment of 3L and with a linear charge density λ. Determine the electric field, E, of at point P, which is a point within the segment and along the axis. (figure attached) Homework Equations dE=kdQ/r^2 The Attempt at a Solution I attempted solving...
  2. 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##...
  3. N

    Analysis Good books on linear algebra and real/complex analysis?

    Hey everyone! (new to the forum) I am currently trying to self study more advanced mathematics. I have taken up to multivariable calculus and have taken a class for an introduction to mathematical proofs/logic (sets, relations, functions, cardinality). I want to get a head start on the...
  4. SetepenSeth

    Linear Algebra - Standard Matrix of T

    Homework Statement Let T: ℝ² → P² a linear transformation with usual operations such as T [1 1] = 1 - 2x and T [3 -1]= x+2x² Find T [-7 9] and T [a b] **Though I'm writing them here as 1x 2 row vectors , all T's are actually 2x1 column vectors (I didn't see a way to give them proper...
  5. SetepenSeth

    Linear Algebra - Linearity of a transformation

    Homework Statement Let be T : ℙ2 → ℙ2 a polynomial transformation (degree 2) Defined as T(a+bx+cx²) = (a+1) + (b+1)x + (b+1)x² It is a linear transformation? Homework Equations A transformation is linear if T(p1 + p2) = T(p1) + T(p2) And T(cp1)= cT(p1) for any scalar c The Attempt at...
  6. M

    B What Does 'Simultaneous' Mean in Linear Equations?

    The term simultaneous in simultaneous linear equations does not make sense to me? Would you explain the what simultaneous mean there? Example: "We have all solved simultaneous linear equations - for example, 2x + y = 4 x - 2y = -3 " Source: Linear Algebra by Fraleigh/Beauregard. Thank you.
  7. B

    B What do "linear" and "abstract" stand for?

    What does "linear" in linear algebra and "abstract" in abstract algebra stands for ? Since I am learning linear algebra, I can guess why linear algebra is called so. In linear algebra, the introductory stuff is all related to solving systems of linear equations of form ##A\bf{X} = \bf{Y}##...
  8. B

    Linear ordinary differential equation.

    Homework Statement ##\dfrac{dy}{dx} + y = f(x)## ##f(x) = \begin{cases} 2 \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}## ##y(0) = 0## Homework EquationsThe Attempt at a Solution Integrating factor is ##e^x## ##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)## ##\displaystyle ye^x = \int e^x...
  9. J

    Electromagnetism - Linear charge

    Homework Statement q=1.602*10^-19 point 1 L=1mm=r1 v=1.1*10^6 at point 2 F=1.44*10^-12 at point 1 Homework Equations E=(1/4πε)*(q/r) ΔV=∫E*dr=(1/4πε)*q∫(1/r)=(1/4πε)*q*ln (r2/r1) ΔU=ΔK=mv^2/2 ΔK=mv^2/2=ΔV*q=q*(1/4πε)*Q*(ln(r2/r1))...
  10. JulienB

    3D quantum harmonic oscillator: linear combination of states

    Homework Statement Hi everybody! In my quantum mechanics introductory course we were given an exercise about the 3D quantum harmonic oscillator. We are supposed to write the state ##l=2##, ##m=2## with energy ##E=\frac{7}{2}\hbar \omega## as a linear combination of Cartesian states...
  11. E

    Linear expansion by heat for an object

    Homework Statement http://i.imgur.com/swYr8aw.jpg Homework Equations delta L=LalphadeltaT The Attempt at a Solution sorry for s**ty handwriting look at #8[/B] http://i.imgur.com/59Ew8vJ.jpg edit:click the links please. The forum software cropped the quality of the photo.
  12. SetepenSeth

    Linear Algebra - Linear (in)dependence of a set

    Homework Statement Let { u, v, w} be a set of vectors linearly independent on a vector space V - Is { u-v, v-w, u-w} linearly dependent or independent? Homework Equations [/B] A set of vectors u, v, w are linearly independent if for the equation au + bv + cw= 0 (where a, b, c are real...
  13. J

    I Tangential force on a linear solenoid plunger

    Hello, I know how a plunger in a pull solenoid is pulled to the center of the solenoid. What I am wondering is the following. Does the plunger experience a centering force towards the center of the coil? If the plunger is moved slightly off-axis from the center of a circular coil, will the...
  14. L

    A A question about linear response and conductivity

    I am trying to derive the DC electrical conductivity using the pertubation theory in Interaction picture and linear response theory. If working in a energy eigen basis and using the density matrix, the Fourier transform of the susceptibility can be written as ##\chi {(\omega )_{ij}} =...
  15. vktsn0303

    A Understand Convolution, Singularity, Kernel, etc: Math Reading Guide

    I'm reading a book on vortex methods and I came across the above mentioned terms, however, I don't understand what they mean in mathematical terms. The book seems to be quite valuable with its content and therefore I would like to understand what the author is trying to say using the above...
  16. U

    I Error in declination of linear regression

    During a lab exercise we measured different masses of a magnetic material on a scale while changing the strength of the magnetic field it was in. Afterwards we plotted the masses and the fieldstrength hoping to find a linear slope. Then we drew a linear slope by using linear regression and found...
  17. D

    Stimulated Raman in linear regime

    In the course of another thread I was lend to think about the Raman effect. I also read about the stimulated Raman effect and found that it is usually described as a third order nonlinear effect where a power of two of E is assumed to drive the nuclear vibration. I don't quite see why this is...
  18. J

    I Comoving mass density leads to linear cosmology?

    The Friedmann equation expressed in natural units (##\hbar=c=1##) is given by $$\left(\frac{\dot a}{a}\right)^2 = \frac{l_P^2}{3}\rho - \frac{k}{R^2}$$ where ##t## is the proper time measured by a comoving observer, ##a(t)## is the dimensionless scale factor, ##l_P=\sqrt{8\pi G\hbar/c^3}## is...
  19. arnbobo

    Schools Will a B in Linear Algebra hurt my grad school chances?

    I had a pretty tough schedule this semester, so I'm getting my first B. I otherwise wouldn't be too sad, but I hear Linear is pretty important in upper-level physics and astronomy. So, will this hurt my chances of getting into grad school? I am (was? rising sophomore) only a first-year, and I do...
  20. Adgorn

    Proof regarding linear functionals

    Homework Statement Let V be a vector space over R. let Φ1, Φ2 ∈ V* (the duel space) and suppose σ:V→R, defined by σ(v)=Φ1(v)Φ2(v), also belongs to V*. Show that either Φ1 = 0 or Φ2 = 0. Homework Equations N/A The Attempt at a Solution Since σ is also an element of the duel space, it is...
  21. U

    Reynolds transport theorem derivation - linear momentum

    I've managed to derive the form of Reynolds transport theorem as a bilance of linear momentum of the system: \left (\frac{\vec{\mathrm{d} p}}{\mathrm{d} \tau} \right )_{system}=\frac{\mathrm{d} }{\mathrm{d} x}(\int_{V}^{ }\vec{v}\cdot \rho dV)+\int_{A}^{ }\vec{a}dm+\int_{A}^{ }\vec{v}\cdot \rho...
  22. T

    Linear dependency matrix problem

    Homework Statement Hi guys, I am having an issue understanding what to do with this question. The question is displayed below: I have hand wirtten my working, as I don't now how to do matrices fully on latext. I used the definition to get this far for part a, but not sure about the second...
  23. S

    I Γγ Coincidence and unsuccessful linear gate method results

    I had this setup (see attached photo) for the linear gate method in a γγ coincidence experiment. Using a Na22 source. The pulse from the movable detector enables the gate of the MCB, and any corresponding pulse from the fixed detector that arrives within the gate interval will be considered...
  24. T

    Linear, non linear and homeogenous and non homeogenous

    Homework Statement Hi guys, I am having a bit of trouble with this question: S2. It the linear non linear and homogeneous parts. I think it is a linear equation, as I always think dy/dx (y)=H(x), but is there a way to show this, also for non linear cases. I believe the second part to this...
  25. M

    MHB Linear conqruence and relations problem

    Suppose that the relation R is defined on the set Z where aRb means a = ±b. Establish whether R is an equivalence relation giving your justifications. Find the set of solutions of each of the linear congruence: a) x ≡ 3 (mod 5). b) 2x ≡ 5 (mod 9).(please write the full solutions thanks)
  26. P

    Linear force vs Torque vs Axial force

    Is it true that torque and axial force are stronger than linear force? If I hit something with an acute angle and progressively turn it into a lower angle while moving forward,is it applying more force?( for example needle insertion)
  27. T

    How to solve for the general solution in a vector linear relationship?

    Homework Statement Find the general solution to the equation: ##{u}\times{(i+4j)}=3k## Homework Equations ##u=(ai+bj+ck)## The Attempt at a Solution I am having trouble visualising what to do here. So my throught so far is that, if i do ##(ai+bj+ck)\times {(i+4j)}=(0i-0j+(4a-b)k) ## now if I...
  28. Bobs

    I Solve 2x+3y=0: Find x & y Points for Line

    Hi, I have 2x+3y=0 equation, I need to find x and y points to draw a line on plane. Can anyone help me?
  29. B

    What is the relation between angular and linear acceleration

    I am wondering, when solving rigid body exercises, how can I express the relationship between linear and angular acceleration for a general case? E.g. what would be the linear acceleration in function of the angular one of a 1m rod that is rotating through a fixed point 0.6 m away from its mass...
  30. K

    I Linear transformation of a given coordinate

    I have a question about weights of a basis set with respect to the other basis set of one specific vector space. It seems the weights do not covert linearly when basis sets convert linearly. I've got this question from the video on youtube "linear transformation" Let's consider a vector space...
  31. H

    I Linear and nonlinear physical theories

    Classical physics is a nonlinear theory, but how is it that? Why is it nonlinear? Also quantum mechanics is a linear theory so that the sum of the solutions of the schrödinger equation is itself a solution. But I'm not sure I grasp this completely. Why is quantum mechanics linear while...
  32. nightingale123

    What Values of ##\lambda## Allow Non-Trivial Solutions in This Linear System?

    Homework Statement 3.For which values of ##\lambda## does the following system of equations also have non trivial solutions Homework EquationsThe Attempt at a Solution What I tried doing first is to put all variables on the same side and got ## v+y-\lambda*x=0\\ x+z-\lambda*y=0\\...
  33. A

    Is This Matrix Both Onto and One-to-One?

    Homework Statement Say I have a matrix: [3 -2 1] [1 -4 1] [1 1 0] Is this matrix onto? One to one? Homework EquationsThe Attempt at a Solution I know it's not one to one. In ker(T) there are non trivial solutions to the system. But since I've confirmed there is something in the ker(T), does...
  34. Ian Baughman

    Courses Numerical Linear Algebra or Modern Algebra

    So I am working out a course schedule for my last two years of undergrad and have room for only one more math class but do not know which would be more beneficial. The two courses are Intro to Modern Algebra or Numerical Linear Algebra. I am working towards a bachelors degree in physics and plan...
  35. A

    Proving Completeness of Continuous Basis Vectors

    Homework Statement Consider the vector space that consists of all possible linear combinations of the following functions: $$1, sin (x), cos (x), (sin (x))^{2}, (cos x)^{2}, sin (2x), cos (2x)$$ What is the dimension of this space? Exhibit a possible set of basis vectors, and demonstrate that...
  36. Adgorn

    Linear functionals: Φ(u)=0 implies Φ(v)=0, then u=kv.

    Homework Statement Suppose u,v ∈ V and that Φ(u)=0 implies Φ(v)=0 for all Φ ∈ V* (the duel space). Show that v=ku for some scalar k. Homework Equations N/A The Attempt at a Solution I've managed to solve the problem when V is of finite dimension by assuming u,v are linearly independent...
  37. F

    Finding Linearly Independent Vectors in Subspaces

    Homework Statement The vectors ##a_1, a_2, a_3, b_1, b_2, b_3## are given below $$\ a_1 = (3~ 2~ 1 ~0) ~~a_2 = (1~ 1~ 0~ 0) ~~ a_3 = (0~ 0~ 1~ 0)~~ b_1 = (3~ 2~ 0~ 2)~~ b_2 = (2 ~2~ 0~ 1)~~ b_3 = (1~ 1~ 0~ 1) $$ The subspace of ## \mathbb R^4 ## spanned by ##a_1, a_2, a_3## is denoted by...
  38. 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...
  39. Mr Davis 97

    Eigenvalues of transpose linear transformation

    Homework Statement If ##A## is an ##n \times n## matrix, show that the eigenvalues of ##T(A) = A^{t}## are ##\lambda = \pm 1## Homework EquationsThe Attempt at a Solution First I assume that a matrix ##M## is an eigenvector of ##T##. So ##T(M) = \lambda M## for some ##\lambda \in \mathbb{R}##...
  40. Mr Davis 97

    T/F Question of linear independence

    Homework Statement T/F: Let ##T: V \rightarrow W##. If ##\{v_1,v_2,...,v_k \}## is a linearly independent set, then ##\{T(v_1), T(v_2),..., T(v_k) \}## is linearly independent. Homework EquationsThe Attempt at a Solution This seems to be true, because we know that ##a_1v_1 + a_2v_2 + \cdots +...
  41. B

    MHB Function - one linear, one rational - is the following True or False

    The function ƒ(x) is a linear function and g(x) is a rational function. These functions have the following values: ƒ(3) = 7 g(3) = 5.6 ƒ(4) = 5 g(4) = 6.7 There is a solution to the equation ƒ(x) = g(x) between x = 3 and x = 4 that must be closer to 3 than 4. TRUE or FALSE?
  42. M

    I Is the F-test Calculation Different for Longitudinal Data in Linear Regression?

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

    Linear Algebra - what is Re and Im for complex numbers?

    Homework Statement http://prntscr.com/eqhh2p http://prntscr.com/eqhhcg Homework EquationsThe Attempt at a Solution I don't even know what these are, it is not outlined in my textbook. I'm assuming I am is image? But how do you calculate image even? As far as I'm concerned I am has to do wtih...
  44. FallenApple

    A Dichotomize or not? Logistic vs Linear

    Say I want to see if a continuous variable, say strength, is associated with another continuous variable, say height. Clearly I can just use linear regression with height as the response variable and strength as the predictor variable. But I can also split height into a tall category and a...
  45. D

    Gauss' gun/magnetic linear accelerator

    Hi, I'm currently doing my extended essay for the IB in physics and I'm basing mine around the Gauss gun (similar post that explains it really well here https://www.physicsforums.com/threads/physics-behind-gaussian-accelerator-magnetic-linear-accelerator.320389/). What I was wondering was how I...
  46. 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...
  47. Whiteboard_Warrior

    I Parameterization of linear operators on the holomorphisms

    Let D represent the differentiation of a single-parameter holomorphism, with respect to its parameter x. It's clear that for any sequence of holomorphisms g on x, sigma[k=0,inf](g[k](x)*D^k) is a linear operator on the space of holomorphisms. Is this a complete parameterization of the linear...
  48. M

    MHB Integrating Factors for Solving Linear Differential Equations

    Can i have help with this linear differential equation ? First, i divided by (1-x^2) to be like dy/dx + p(x)y= q(x). But i could not obtain Q(x). Any help will be welcomed.
  49. L

    Conservation of Linear momentum

    Homework Statement A 500g putty ball moving horizontally at 6m/s collides with and sticks to a block lying on a friction-less horizontal surface. If 25% of the kinetic energy is lost, what is the mass of the block? Homework Equations initial (i) = final (f) m1v1+m2v2 = m1v1+m2v2...
  50. 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...
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