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. greg_rack

    Conservation of linear momentum, undergrad particle dynamics

    Hi all, I'm opening this thread because of my uncertainty in how to correctly approach this exercise. My first thought was that, since the plate is subject to friction with the floor, it is going to stop, thus the final moment is 0. Hence, from the conservation of linear moment: $$m_Av_A+\sum...
  2. R

    B Proof involving two linear equations

    Given ## a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0 ##, if ##(x_1,y_1)## and ##(x_2,y_2)## are pairs of real numbers satisfying: ## ax_1 + by_1 = e, cx_1 + dy_1 =f ## ## ax_2 + by_2 = e, cx_2 + dy_2 = f ## then ## (x_1,y_1) = (x_2,y_2). ## Here is my attempt at a proof, I have gotten stuck...
  3. S

    Polymer linear bearing grooves

    All polymer linear bearings i have seen have a lot of groves in the running direction (at least the ones for round shafts). Is this just to minimize the surface area in contact with the shaft to minimize the friction? Or do they have another function like clearing dust and particles? And why...
  4. D

    Linear speed of sphere as it passes through lowest point

    Let ##m_s = 0.05, m_{s_1} = 0.02, m_r = 0.12, L = 0.8.## be the masses of the two spheres, mass of the rod, and length of the rod. Then the work done by gravity when the rod reaches the vertical position is ##(m_s(L/2) - m_{s_2}(L/2))g## and the kinetic energy equals ##\frac{1}2 (\frac{1}{12}...
  5. Purpleshinyrock

    Does element x present a linear resistance?

    ANSWER: element x does not present a linear resistance because it isn't constant as i and v increases. Is my answer correct?
  6. M

    Algebra Looking for my first textbook on Linear Algebra Need suggestions

    First of all, I attached pictures of the very last algebra textbook that I have finished studying. I'm going the self taught route. I really loved this book because it had lots of examples, practice exercises, quizzes and even tests! It also had answers in the back. It's currently my favorite...
  7. J

    MHB Finding a Basis for a Linear Subspace Orthogonal to a Given Point P in R^3

    I have a given point (vector) P in R^3 and a 2-dimensional linear subspace S (a plane) which consists of all elements of R^3 orthogonal to P. The point P itself is element of S. So I can write P' ( x - P ) = 0 to characterize all such points x in R^3 orthogonal to P. P' means the transpose...
  8. J

    Linear momentum of the Klein Gordon field

    The correct answer is: #P = \int \frac{dp^3}{(2\pi)^3}\frac{1}{2E_{\vec{p}} \big(a a^{\dagger} + a^{\dagger}a\big)# But I get terms which are proportional to ##aa## and ##a^{\dagger}a^{\dagger}## I hereunder display the procedure I followed: First: ##\phi = \int...
  9. Caleydos

    Magnetic field produced by linear current in COMSOL

    Hello, I have used an edge current of 10 A through a 0,45 cm (lenght) wire inside an air sphere. The thing is that, according with Ampere law, the magnetic field (B) produced at a 1 mm of distance from the wire shall be 0,002 T, and I am obtaining much higher values in this simulation (around...
  10. P

    Using Monopods for city travel utilizing linear induction motors

    This is another open ended question, exploring a space of design concepts, in similar spirit to this. I want to explore monopods with regard to travel in densely populated cities(even possibly intercity travel). The main idea is to use small personalized pods to travel in tubes(or tracks). The...
  11. S

    I Angle-Preserving Linear Transformations in 2D Space for Relativity

    I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the ##ct## axis and the worldline of an observer moving relative to me will be at some angle w.r.t. the ##y## axis. When we switch...
  12. fresh_42

    A Center of a linear algebraic group

    Let ##G\leq GL(n)## be a linear algebraic group of dimension ##m,## and ##C## its ##c##-dimensional center. What do we know about lower and upper bounds of ##c=c(m)\,\text{?}## Clearly ##c(0)=0, c(1)=1## and ##n^2\geq c(m)\geq 1## for ##m\neq 0.## By Schur's Lemma we also know ##c(n^2)=1##. Did...
  13. coolul007

    B What Are Solutions for Universal Linear Recursions?

    https://www.testsite.cocoams.org/linrec.pdf
  14. steve1763

    A Quantum linear code/ Dual Code (CSS) proof

  15. guyvsdcsniper

    Solving a Linear Combination Problem

    I have attached my work to this thread. Could someone help me with this Linear Algebra problem. This is my first week so I do not know many advanced ways to solve these problems. I could not figure out how to get this matrix into rref, so I solved it the following way. Is the way I used...
  16. E

    How does a 3 stage linear mechanical actuator work?

    How does a 3 stage linear actuator mechanical work. I can only find a regular linear actuator mechanical but I'm unsure how will the last stage go up and down. Anyone got a poor 3d drawing for a better understanding.?
  17. Flash27

    I Building a Linear Induction Motor | Tips & Tricks

    I am currently trying to create a linear induction motor for fun and am having some trouble getting it to start oscillating or move at all. I am using this video as a reference... I am using 3D printed PLA as the structure for the copper to wind around, 26 GA Craftware USA copper wire, 5/8"...
  18. L

    I Solve second order linear differential equation

    Consider the second order linear ODE with parameters ##a, b##: $$ xy'' + (b-x)y' - ay = 0 $$ By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form: $$ \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned} $$...
  19. JD_PM

    Statements about linear maps | Linear Algebra

    First thing to notice is that ##L## and ##L \circ L## are precisely equal linear maps. What we know $$L \ \text{is injective} \iff \ker(L)=\{0\}$$ $$\ker L' = \{ x \in \Im(L) \ | \ L'(x)=0\}$$ $$\Im(L)=\{ x \in V \ | \ \exists \ v \in V \ \text{such that} \ L(v)=x\}$$ Besides, we notice...
  20. Monsterboy

    Linear size of everything in the Universe has been doubled overnight

    I found this question in a textbook, not sure if this question has been asked before. Not sure if the author just wanted to make the reader think or he had anything specific in mind that he wanted the readers to understand. Most of the people immediately conclude that the speed of light doesn't...
  21. R

    Design a linear phase low pass filter

    Design a linear phase low pass filter whose specification is:1) Maximum tolerance in the passband equal to 0.01% (linear) in the passband;2) Cutoff frequency at ω_c = 0.3π and transition band at 0.05π;3) Minimum reduction of the 0.95 rejection band.
  22. JD_PM

    Finding a complementary subspace ##U## | Linear Algebra

    We only worry about finite vector spaces here. I have been taught that a subspace ##W## of a vector space ##V## has a complementary subspace ##U## if ##V = U \oplus W##. Besides, I understand that, given a finite vectorspace ##(\Bbb R, V, +)##, any subspace ##U## of ##V## has a complementary...
  23. C

    I How do shockwaves in a 1D linear lattice work?

    I am struggling to understand shocks in a one dimensional lattice with a linear spring connecting the masses. Say I have a one dimensional lattice with a linear spring constant, k and lattice spacing a. If the particles in the lattice has mass, m then my speed of sound c is a*sqrt(k/m). That is...
  24. JD_PM

    Given subspaces ##U \& W##, show they are equal | Linear Algebra

    Show that ##U = span \{ (1, 2, 3), (-1, 2, 9)\}## and ##W = \{ (x, y, z) \in \Bbb R^3 | z-3y +3x = 0\}## are equal. I have the following strategy in mind: determine the dimension of subspaces ##U## and ##W## separately and then make use of the fact ##dim U = dim W \iff U=W##. For ##U## I would...
  25. A

    Linear Algebra - LU Factorization

    Hello all, I have a problem related to LU Factorization with my work following it. Would anyone be willing to provide feedback on if my work is a correct approach/answer and help if it needs more work? Thanks in advance. Problem: Work:
  26. JD_PM

    Proving statements about matrices | Linear Algebra

    Hi guys! :) I was solving some linear algebra true/false (i.e. prove the statement or provide a counterexample) questions and got stuck in the following a) There is no ##A \in \Bbb R^{3 \times 3}## such that ##A^2 = -\Bbb I_3## (typo corrected) I think this one is true, as there is no squared...
  27. J

    Cam forces acting on a linear motion bearing with nonlinear forces

    I have a machine I am designing that for all intensive descriptions, is a simple press designed to compress loose product into a puck like shape. The press force comes from a roller bearing mounted to a piston shaft, the rod sliding through a rigid linear bearing and the piston on the end of...
  28. H

    MHB Application of Linear differential equation in solving problems

    A rumour spreads through a university with a population 1000 students at a rate proportional to the product of those who have heard the rumour and those who have not.If 5 student leaders initiated the rumours and 10 students are aware of the rumour after one day:- i)How many students will be...
  29. patric44

    I Dimension of a Linear Transformation Matrix

    hi guys I was trying to find the matrix of the following linear transformation with respect to the standard basis, which is defined as ##\phi\;M_{2}(R) \;to\;M_{2}(R)\;; \phi(A)=\mu_{2*2}*A_{2*2}## , where ##\mu = (1 -1;-2 2)## and i found the matrix that corresponds to this linear...
  30. N

    Write a Linear Model For this Data

    The average weight of a male child’s brain is 970 grams at age 1 and 1270 grams at age 3. (Source: American Neurological Association) (a) Assuming that the relationship between brain weight y and age t is linear, write a linear model for the data. (b) What is the slope and what does it tell...
  31. N

    Write a Linear Equation for the Value of this Printer Over Time

    Write a linear equation. A school district purchases a high-volume printer, copier, and scanner for $24,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value V of the equipment during the 10 years it...
  32. N

    Write A Linear Equation for Monthly Salary + Commission

    Write a linear equation for the application. A pharmaceutical salesperson receives a monthly salary of $5000 plus a commission of 7% of sales. Write a linear equation for the salesperson’s monthly wage W in terms of monthly sales S. Solution: I am looking for W(S). S = monthly sales Let...
  33. N

    What is the Linear Equation for Dollar Value of a Product Over Time?

    You are given the dollar value of a product in 2016 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t = 16 represent 2016.)1...
  34. N

    Linear Model for Male Child Brain Weight from Age 1 to 3: y = 150t + 300

    The average weight of a male child’s brain is 970 grams at age 1 and 1270 grams at age 3. (Source: American Neurological Association) Assuming that the relationship between brain weight y and age t is linear, write a linear model for the data. I am going to start by saying y = mx + b. Let x...
  35. N

    Form a Linear Equation for the Value of this equipment during the 10 years it will be in use

    A school district purchases a high-volume printer, copier, and scanner for $24,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value V of the equipment during the 10 years it will be in use. Let t =...
  36. N

    How Is a Salesperson's Wage Calculated Based on Sales?

    A pharmaceutical salesperson receives a monthly salary of $5000 plus a commission of 7% of sales. Write a linear equation for the salesperson’s monthly wage W in terms of monthly sales S. I will let W(S) = monthly wage W in terms of monthly sales S. $5000 plus a commission of 7% of sales =...
  37. M

    I Error on non-linearity in a linear fit

    Hello! If I have some data points, with error bars on both x and y, and I would like to fit them with a function f(x). How can I write the chi-squared in this case? For errors only on y, I would have ##\chi^2 = \sum_i(\frac{f(x)-y}{\sigma_y})^2##, but I am not sure how to include ##\sigma_x##...
  38. W

    I Bias in Linear Regression (x-intercept) vs Statistics

    Hi, In simple regression for machine learning , a model : Y=mx +b , Is said AFAIK, to have bias equal to b. Is there a relation between the use of bias here and the use of bias in terms of estimators for population parameters, i.e., the bias of an estimator P^ for a population parameter P is...
  39. A

    I Canonical Form for quadratic equations *with* linear terms

    Hello: I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables: $$ax^2+by^2+cxy+dx+ey+f=0$$ Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part): $$ w^TDw+[d \ \ e]w+f=0$$ $$w^TDw+Lw+f=0$$ where $$...
  40. Z

    Linearizing an equation for Mixed-integer linear programming (MILP)

    Summary:: Linearizing an equation for MILP Hi All I have Linear programming problem where I need to linearise the following function: Y = A * log2(1 + (Bx/Cx^2)). where A, B and C are constants. Can you please help or advise?
  41. cianfa72

    Two-port linear network general representation AV + BI = 0

    Hi, as follow-up to this thread I've a question about general representation of a two-port network. Basically it is ad hoc built four-terminal linear network (using controlled sources + nullator-norator pair): for it I found a general representation ##AV + BI = 0## as in the picture above. If...
  42. dextercioby

    I Linear & Rotational Motion: Speed of Tip in Lab Frame

    I have a pencil of Iron of length ##L## rotating about its center in a plane at constant angular velocity ##\omega##. The tip of the pencil in Newtonian mechanics has linear velocity ##\frac{\omega L}{2}##. It can exceed ##c##, of course. Now let us complicate this. Assume the center of the...
  43. S

    MHB Calculating probability from linear regression parameters

    I'm a bit stuck on how to calculate the probability in part b from the linear regression parameters. I tried plugging the parameter values into the linear regression model: Y =β0+β1X+ε, ε∼N(0,σ) So P(Y=y| X=40) = 2.85 + 0.07 * 40 + 1^2 P(Y=y|X=40) = 5.65 But I don't think this is the...
  44. P

    Looking for Linear Motor Exercises

    Homework Statement:: Hi, I'm looking for an exercise involving a linear motor, but on my physics book there are none Relevant Equations:: Lenz Law, Maxwell's equation Hi, everyone I'm looking for an exercise involving a linear motor, but on my physics book there are none. Do you happen to...
  45. Uchida

    Conservation of Linear Momentum of Rigid Body

    I solved it by two methods: ----------------------------------------------------- First, by conservation of linear momentum, using the vector velocities of each particle: In the imminence of the impact, the velocity of all the three particles are the same, \vec v_0 = - \sqrt{2gh} \hat j...
  46. snypehype46

    Computing OPEs of linear dilaton CFT

    I'm trying to do the following question from David Tong's problem sheets on string theory: > A theory of a free scalar field has OPE $$\partial X(z)\partial X(w) = \frac{\alpha'}{2}\frac{1}{(z-w)^2}+...$$. Consider the putative candidate for the stress energy tensor $$T(z) = \frac{1}{\alpha '}...
  47. P

    I Why the linear combination of eigenfunctions is not a solution of the TISE

    The linear combination of the eigenfunctions gives solution to the Schrodinger equation. For a system with time independent Hamiltonian the Schrodinger Equation reduces to the Time independent Schrodinger equation(TISE), so this linear combination should be a solution of the TISE. It is not...
  48. M

    Nonlinear spring made from many different linear springs in series?

    Can I create a nonlinear spring, for a nonlinear oscillator, by putting many linear springs in series and parallel?
  49. A

    How to determine if a transformation is linear

    Hello! I need to check if this transformation (not sure if it is the right word in English) from ## R^3 to R^3 ## is linear f(x1,x2,x3) = f(sin(x1),x2+x3,0). Now we are given that the transformation is linear if this you can prove this statement. $$f(\lambda * u + \mu * v) = \lambda * f(u) +...
  50. F

    Linear Algebra What are good books for a third course in Linear Algebra?

    What are the suitable books in linear algebra for third course for self-study after reading Linear Algebra done right by Axler and Algebra by Artin?
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