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. C

    Linear independence of Coordinate vectors as columns & rows

    Summary:: x Question: Book's Answer: My attempt: The coordinate vectors of the matrices w.r.t to the standard basis of ## M_2(\mathbb{R}) ## are: ## \lbrack A \rbrack = \begin{bmatrix}1\\2\\-3\\4\\0\\1 \end{bmatrix} , \lbrack B \rbrack = \begin{bmatrix}1\\3\\-4\\6\\5\\4 \end{bmatrix}...
  2. Santiago24

    Linear Algebra I need textbook recommendations to learn linear algebra by myself

    Hi PF community, recently i learned about Calculus in one variables and several, so now i'd like to study linear algebra by myself in a undergraduate level, in order to do that i need some textbooks recommendations. I'll be waiting for your recommendations :).
  3. T

    Calculate resultant force of a linear mechanism

    Hi all, I'm currently designing a linear to linear transform mechanism (wedge cam design). Would like to check how much force required to push/ slide the roller down. To ease calculation, i have include some assumption: F2 = 1kg @ maximum compressed angle = 20degree F1 required = ? and how...
  4. L

    A Non linear diff. eq. first order

    I tried to solve for y' the algebric second grade equation but I don't know how to solve the diff. equation either. Any help? Thanks. -- lightarrow
  5. Tony Hau

    Boundary conditions of linear materials

    The question is as follows: Solutions given only contain part a) to c), which is as follows: So I now try to attemp d), e) and f). d) The magnetic field of a uniformly magnetized sphere is: $$ \vec B =\frac{2}{3}\mu_{o} \vec M = \mu_{o}\vec H$$ $$\frac{2}{3}\vec M = \vec H$$ The perpendicular...
  6. cianfa72

    Circuit theory - Resistor function for dynamic non linear circuits

    Hi, I'm reading the following paper (L. Chua) about the state-of-art of dynamic non linear circuit analysis -- Chua_Dynamic_Circuits I've a doubt about Theorem 2 on section 3.2 On the Existence of the Resistor Function that establishes sufficient conditions for the existence of network...
  7. K

    A linear combination of states that diagonalize the Hamiltonian

    He told me I "need to show that the Hamiltonian matrix elements you get by using those states have nonzero elements only on the diagonal." I understand what and how a diagonal matrix works, but what I don't understand is what those states are. Are they states I put in my "quantum mechanical...
  8. S

    How Do Vector Spaces of Linear Maps Differ from Standard Vector Spaces?

    Solution 1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##. From...
  9. V

    Optimizing Resources with Linear Programming: A Step-by-Step Approach

    I am comfortable solving these types of problems, however I am having trouble setting up this problem and am unsure if I am doing this correctly so would someone be able to assist me? I first wrote an equation for the people: 10x +8y +11z. Then I made an equation for the flour: 4x + 12y + 8z...
  10. yucheng

    Linear Equations: how to deduce this inequality is true?

    The solution from my book: From $$\frac{3}{x+2}<\frac{13}{12}<\frac{3}{x} \tag1$$ It follows that ##13x<36<13(x+2)## x<3, i.e. x = 1 or 2. By checking, x=1 is not the solution and x = 2 satisfies the equation. However, how does the author deduce (1)?
  11. N

    B Area Increasing as a linear dimension increases -- Looking for intuition on this

    I am working on related rates problems involving figuring out how area of a square increases per second based on how much one side increases per second (or how the area of a circle increases based on increase of the radius, etc.). I was wondering about the practical significance of problems like...
  12. J

    Linear chain of oscillators and normal coordinates

    Hello, I hope the equation formatting comes out right but I'll correct it if not. So far, I have attempted to write ##\ddot{a}_k(t) = \sum_{n}(u^{k}_n)^*\ddot{q}_n(t) ##. Then I expand the right hand side with the original equation of motion, and I rewrite each coordinate according to its own...
  13. A

    Rotational Work versus Linear Work

    In the link below, there is a force tangent to a wheel that travels distance Ds. There is also a torque that moved thru an angle. Why do they consider them equal? There are two works, shouldn't they be added together...
  14. K

    Linear algebra inner products, self adjoint operator,unitary operation

    b) c and d): In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct? e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find two.Otherwise does b-d above look correct? Thanks in advance!
  15. F

    I Proving linear independence of two functions in a vector space

    Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...
  16. mertcan

    Linear programming and resolution theorem

    Hi everyone hope you are well, I would like to express what I have done for this question: Proving and employing caratheodory theorem we can say that any point in polyhedron can be expressed as a convex combination of at most n+1 points (where n is the space dimension) in same polyhedron that...
  17. M

    MHB Linear system of equations: Echelon form/Solutions

    Hey! 😊 I am looking at the following exercise but I think that I miss something. The statement is the following: We are given the following system of equations: \begin{align*}2a-2c+d-2e=&-2 \\ -2c-2d+2e=&\ \ \ \ \ 3 \\ d+2e=&-2\end{align*} 1) Is the system in echelon form? Justify. 2)...
  18. Butterfly41398

    Can you help me evaluate the integral in this linear differential equation?

    I tried it but I don't know how to evaluate the integral on the last equation. Help.
  19. Andrei0408

    Uniformly accelerated linear motion downwards

    Could you check if what I did is right and help me at e) and f)? Thank you!
  20. Ranku

    Circular and linear freefall

    Does circular freefall of an object involve cancellation of gravitational and inertial forces, as it does in linear freefall?
  21. L

    A Is the Frechet Derivative of a Bounded Linear Operator Always the Same Operator?

    I understand the Frechet derivative of a bounded linear operator is a bounded linear operator if the Frechet derivative exists, but is the result always the same exact linear operator you started with? Or, is it just "a" bounded linear operator that may or may not be known in the most general case?
  22. archaic

    Linear combination of random variables

    a) Total weight ##W=W_1+W_2+...+W_{25}##.$$E[W]=E[W_1]+E[W_2]+...+E[W_{25}]=25\times76=1900\,kg$$$$\sigma_W=\sqrt{V(W_1)+V(W_2)+...+V(W_{25})}=\sqrt{25\times(16)^2}=80\,kg$$ b) Since ##W## is a linear combination of normal distributions, the reproductive property tells us that ##W## is also...
  23. S

    Seeking a type of linear actuator, which may or may not exist

    As the brief summary explains, I'm looking for a linear actuator which moves quickly and can move a moderately low load (say, 10kg) but which also features an adjusting hard-stop which can withstand significantly heavier. So effectively, the actuator can move in and out, preferably quickly, but...
  24. I

    Comp Sci Searching Array for an Element using Linear Search

    I write a Linear Search code,then ı decided ask to user "r" and "int arr".I mean,User decide their r and arr numbers.I tried scanf("%d",r); command but doesn't work This code my first code before the decide Ask user : #include <stdio.h> int search(int arr[], int n, int r) { int i; for...
  25. R

    Size of bremsstrahlung radiation fields generated by linear accelerators

    Hi everyone, sorry we are talking about pulsed power accelerators again. In this context, it is about the HERMES III again. According to https://apps.dtic.mil/dtic/tr/fulltext/u2/a351472.pdf HERMES III, a 16 terawatt pulsed power accelerator at Sandia National Labs, which fired electron beams...
  26. LCSphysicist

    Linear algebra invertible transformation of coordinates

    ##A^{x'} = T(A^{x})##, where T is a linear transformation, in such way maybe i could express the transformation as a changing of basis from x to x' matrix: ##A^{x} = T_{mn}(A^{x'})##, in such conditions, i could say det ##T_{mn} \neq 0##. But how to deal with, for example, ##(x,y) -> (e^x,e^y)## ?
  27. LCSphysicist

    Killing’s equation in a linear space

    Be f a orthogonal transformation and g being in the canonical form. $$[f^{-1}]^{t}[g][f^{-1}] = [g']$$ So this equation of isometries implies that the diagonal of g is +- 1, but, apparently, if it is minus one it can't be a group, and i don't know why. To make the things clear, if det = +1, in...
  28. T

    Linear Actuator -- Why is this not working?

    I'm using the actuator that says its good for 270lb https://www.princessauto.com/en/detail/12v-dc-12-in-stroke-270-lb-linear-actuator/A-p8507840e I'm using the 4rth pully system from here Pulley diagram I'm using it to lift 40 lbs So from the diagram I figured that to lift 40 lbs I need an...
  29. S

    Find the linear charge density lambda and the radius of a charged semicircle

    Electric field for the semi-circle $$E = - \frac {πKλ} {2R} $$ In this case E is equals to 10 N/C Electric field for the straighten wire $$E = 2Kλ * ( 1 - \frac {2y} {\sqrt{4y^2 + L^2}})$$ In this case E is equals to 8 N/C What I'm searching is R, λ, and the length of the wire, so I think...
  30. E

    MHB Resource for learning linear algebra

    I want to take some courses that involve heavy math, so I have been learning maths on the khan academy site: precalculus, calculus, statistics etc. But one fundamental area of maths the khan academy site doesn't have is a course on linear algebra. I really need to learn and use linear algebra in...
  31. O

    How Do You Calculate Race Car Acceleration and Power Output?

    A new development race car of mass 400 kg accelerates from 20 km/h to 200 km/h over a distance of 120 metres. Frictional and wind resistance forces can be assumed to be 1000N during the acceleration. Determine the following: (i)The average acceleration from 20 km/h to 200 km/h (ii)The time...
  32. rbh

    Linear motion, motorcyclist braking time

    I tried using equations listed above to get the answer, but I get stuck with a and Vi. What am I doing wrong? The answer is 6.7 seconds.
  33. K

    I Trying to get a better understanding of the quotient V/U in linear algebra

    Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...
  34. K

    Linear algebra, find a basis for the quotient space

    Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...
  35. K

    I Linear regression with error term

    I'm not a statistician, but this has been bothering me for a bit. Suppose we have the simple model Y= aX + b + U where Y,X and U are taken to be random variables representing the explanatory variable, the independent variable and the error term respectively. In the case of a stochastic...
  36. H

    Solve this integral involving a quadratic and linear air resistance equation

    Hi, I'm trying to solve this integral and then isolate V, but I can't get the right answer. I don't know where is my errors. I probably muffed the integral. ##-bv -cv² = m\frac {dv}{dt}## ## \int_0^t dt = - m \int_{Vo}^v \frac {dv}{bv+cv^2} ## I get this after the integration ##t =...
  37. L

    MHB Connecting linear algebra concepts to groups

    The options are rank(B)+null(B)=n tr(ABA^{−1})=tr(B) det(AB)=det(A)det(B) I'm thinking that since it's invertible, I would focus on the determinant =/= 0. I believe the first option is out, because null (B) would be 0 which won't be helpful. The second option makes the point that AA^{−1} is I...
  38. J

    A What is your opinion of Linear Canonical Transformations?

    Hello everyone, I looked for the best physics forum to ask this question because the subject interests me a lot. The authors of this preprint (https://arxiv.org/abs/1804.10053) seem to claim that their approach (using linear canonical transformations) is a new alternative way for the...
  39. MexChemE

    My first proof ever - Linear algebra

    First, a little context. It's been a while since I last posted here. I am a chemical engineer who is currently preparing for grad school, and I've been reviewing linear algebra and multivariable calculus for the last couple of months. I have always been successful at math (at least in the...
  40. K

    Show that a (1,2)-tensor is a linear function

    I know that a tensor can be seen as a linear function. I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.
  41. U

    Other Elementary Linear Algebra book

    Hello I am looking for an introductory linear algebra book. I attend university next year so I want to prepare and I want to become an engineer. I have a good background in the prerequisites, except I don't know anything about matrices or determinants. I am looking for the more application side...
  42. Feodalherren

    Simple Linear DiffEq, not understanding the book

    The way I want to solve it is the way that I always want to solve separable linear diffEqs: after some trivial algebra and an easy integral I end up with t = (-1/2) ln (20-2x) +C Easy enough, solve for x(t) yields x(t) = 10 - (1/2)e^(-2t) + C Solve for C when x(0) = 3 yields C = -13/2 But...
  43. ToucanFodder

    Is the Calculation of Linear Density k for a Uniformly Charged Bar Correct?

    I attached a drawing of the problem for a better understanding and my attempted solutions. The first point is fairly simple but there's something that I can't figure out. dq=λdx=kxdx Q=∫ k x dx from 0 to L -> Q=k[x^2/2]0-L -> Q=(L^2/2)k -> k=2Q/L^2 This is what I came up with. I integrated...
  44. Ozen

    Minute Of Angle Linear Adjustment Help

    I am working on developing a micro reflex sight and finally got the optical science of it sorted out, but now I reached a new impasse (hopefully the last really difficult one!). The device has to have an adjustment mechanism so the user can align the reticle with the bullet impact point, which...
  45. Demandish

    Introductory Linear Algebra Texts

    I am currently enrolled in Multivariate Calculus and am looking to get build up a solid base of mathematics for undergraduate physics curriculum. I am looking for a Linear algebra book that will aid me in my quest. I currently own Axler's Linear Algebra Done Right, but I fear it is too...
  46. maxim

    Non magnetic linear displacement system for moving a Hall Sensor

    Dear mechanical expert I have to realize a linear displacement system for a Hall sensor that has to slide along the central axis of a narrow cylinder and with which high intensity magnetic field measurements (1-14 Tesla) have to be made. The field is produced by a commercial vertical magnet...
  47. H

    Levers, torque and the (non?) linear relationship

    I'm unsure what equation to use to achieve this. My solution was to draw a scale model and measure the distance offset from the bar and how relative that was to the same length bar bent to 90 degrees. My answer was 70.8% assuming a linear relationship between length of lever and force (is that...
  48. S

    Matrix concept Questions (invertibility, det, linear dependence, span)

    I have a trouble showing proofs for matrix problems. I would like to know how A is invertible -> det(A) not 0 -> A is linearly independent -> Column of A spans the matrix holds for square matrix A. It would be great if you can show how one leads to another with examples! :) Thanks for helping...
  49. K

    B System of linear equations: When are we guaranteed a unique solution?

    I was reading about systems of linear equations. Even if we have the same number of unknowns and equations, we may still have infinitely many or no solutions. But if in addition to that the determinant of the matrix of coefficients does not vanish, then does it necessarily imply that we have a...
  50. gullpacha

    B Why is a one-variable linear equation called linear?

    i get preparation for my university entering exam and i studying linear equation and they define it linear equation are those which graph line now my question is that why one variable linear equation called linear for example y=3 that actually give us just line why we draw line for this...
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