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. NP04

    Linear Thermal Expansion with respect to time

    ΔL= αLoΔT ΔL = (23*10^-6)(0.2480 m)(28.30°C) ΔL = 1.61 * 10^-4 Period is 1s, so each second the pendulum moves 1.61 * 10^-4 m 1.61 * 10^-4 m/s *(60s/1min)*(60min/1hr)*(24hr/1day) = 13.95 m/day T = s (1:1 ratio) 13.95 seconds. But the answer is actually 69.3 s. Is the equation T = 2π√L/g...
  2. JD_PM

    Matrix representation of a linear mapping

    I know that to go from a vector with coordinates relative to a basis ##\alpha## to a vector with coordinates relative to a basis ##\beta## we can use the matrix representation of the identity transformation: ##\Big( Id \Big)_{\alpha}^{\beta}##. This can be represented by a diagram: Thus note...
  3. ozcliff

    Conversion of angular momentum to linear momentum

    The balls used in the game of lawn bowls are biased so that they travel in a curved path of decreasing radius. When a bowl in motion collides at a glancing angle with another bowl at rest, it -appears- to increase its velocity. Due to conservation of linear momentum the post-collision velocity...
  4. U

    Linear Force to Torque Conversion

  5. A

    Matrix which reverses Gram-Schmidt - Linear Algebra

    My idea was to write out the formulas for the orthogonal q vectors in terms of the input vectors using the basics of gram-schmidt. Then, I would rewrite those equations suhc that the a vectors were written in terms of the q vectors. And then, try to find some matrix which would capture the...
  6. W

    Getting this Array to be in 2D instead of 1D for Python Linear Regression

    import matplotlibimport matplotlib.pyplot as plt import numpy as np from sklearn import datasets, linear_model import pandas as pd # Load CSV and columns df = pd.read_csv("C:\Housing.csv") Y = df['price'] X = df['lotsize'] # Split the data into training/testing sets X_train = X[:-250] X_test =...
  7. barryj

    Linear and angular momentum problem: Ball hitting a rod

    Homework Statement:: Ball of mass mb and velocity vb hits rod of length L , Rod pivots about the center. What is the angular momentum aafter impact? Homework Equations:: I = 1/12 (mR^2) I = mR^2 See the attached figure. I understand the concept of linear and angular momentum separately but I...
  8. J

    I Fulvio Melia's new argument for a linear cosmology

    I would be interested in what people think of Fulvio Melia's new cosmological paper in which he argues that the comoving frame is locally inertial only if we have a linearly expanding Universe (or Minkowski spacetime)...
  9. Math Amateur

    MHB Norms for a Linear Transformation .... Browder, Lemma 8.4 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding Lemma 8.4 ... Lemma 8.4 reads as follows: In the...
  10. E

    Some quick questions concerning uniform acceleration and linear motion

    Hello, I hope you are all having a great day ! I've got a physics test in a couple of days and I have some questions:1. In a calculation, if the acceleration is in m/s², I presume the speed also has to be in m/s and not in km/h ? 2. So with this graph (v with t), I have to find the total...
  11. S

    Calculating Average Linear Velocity: 1.66m/s to 1.73m/s

    First i need the average linear velocity 100L/min = 0.1m3/min 0.00167m3/sec Cross Sec Area = Pie r2 =pie(0.0175m)2 9.61x10-4 m2 0.0016m3/s / 9.61x10x-4m2 = 1.66m/s But my notes say this should be 1.73 m/s Once i have the average linear velocity i can calculate Re. Help appreciated
  12. Math Amateur

    MHB Understanding Browder's Remarks on Linear Transformations

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 8: Differentiable Maps ... ... and am currently focused on Section 8.1 Linear Algebra ... ... I need some help in order to fully understand some remarks by Browder in Section 8.1, page...
  13. T

    Questions about using an Inertia Wheel to apply a compressive linear force

    Summary: How to define dimensions of an inertia wheel Firstly, I have been working in aviation industry, making me to involve in the Physics a lot. For the last few weeks, There have been a problem to solve since the manfacturer had not wanted to help us by not sharing information. We have...
  14. E

    Projectile motion with linear drag

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  15. D

    Calculating the average linear speed of all points in a rotating sphere

    Is there a way to calculate the average linear speed of all points in the volume of a sphere rotating on a single axis? Since points closer to the axis of rotation and the poles move slower than points further out, would the average speed be a simple function of r/2 and pi? It would seem that...
  16. binbagsss

    A QHE: rotational invariance, no terms linear in E or B

    'Let’s first see what all of this means in the context of d = 3 + 1 dimensions. If we have rotational invariance then we can’t write down any terms linear in E or B. The first terms that we can write down are instead ...' Why is this? I don't understnad . My thoughts would be pictruing the set...
  17. D

    Find n and m values of a linear Transformation if given a matrix A

  18. E

    Does an impulse contribute to both linear and angular momentum?

    As an analogue, if 5J of work is done on an object then the linear KE might increase by 2J and the angular by 3J, so the work is divided between the linear and rotational forms. Now suppose there is a sphere sliding on a frictionless surface. If an impulse of magnitude 1Ns is applied to the...
  19. S

    I Find matrix of linear transformation and show it's diagonalizable

    The strategy here would probably be to find the matrix of ##F##. How would one go about doing that? Since ##V## is finite dimensional, it must have a basis...
  20. V

    Linear approximation and percentage error

    I found the linearization, L(x) = -0.0001x+0.2 and I found L(1/99) = 0.0199989899. Then I tried to put that value into my percentage error formula along with 1/99 and got: the absolute value of (1/99)-L(1/99) and then we divide that by our actual value which is 1/99, then I multiply everything...
  21. dazaireta

    Linear Polarization Interference: Can It Happen?

    Two waves are linearly polarized. The electric field of one wave is aligned with the x-axis and the other is aligned with the y-axis. In the absence of matter that might change the polarization, can these waves interfere with each other?
  22. T

    Coupled linear stochastic differential equations

    In order to solve for ##x##, I need to re-write the equation for ##dx## so it is independent of ##y## and ##dy##. However, I am having some issues with this. Can someone give me a push in the right direction?
  23. C

    I Systems of linear equations

  24. Nonametheone

    Can a virtual particle from a maget accelerate another particle on a linear path?

    Summary: Can a particle accelerator accelerate gaseous fuel? Does space-time differ for detonating ionized fuel moving near the speed of light within a magnetic field such as a particle accelerator?
  25. M

    Calculate linear charge density of rod & mag. of E-field at point

    Here is my work done for this problem, along with a diagram of the situation. I'm not worried so much about the arithmetic because our tests are only 50 min long so the problems they give us do not require heavy integration or calculus, but you need to know what goes where in the formula. That...
  26. S

    I Understanding linear transformation

    How can the function ##F(\mathbf{u})(t)=\mathbf{u}^{(n)}(t)+a_1\mathbf{u}^{(n-1)}(t)+...+a_n\mathbf{u}(t)##, where ##\mathbf{u}\in U=C^n(\mathbf{R})## (i.e. the space of all ##n## times continuously differentiable functions on ##\mathbf{R}##) be a linear transformation (from ##U##) to...
  27. X

    Analytical proof that L and C are linear devices?

    I mean I know they are linear since they obey the ohms law. But I don't quite understand the reasoning that since, say, V=Ldi/dt and taking a derivative is a linear operation therefore it is a linear device?? I can verify that sin'(x) = cos(x) or sin(x+90) so the signal is time shifted but...
  28. J

    I If T^2 = T, where T is a linear operator on V, T=I or T=0?

    I can't think of a counterexample.
  29. S

    Linear system of equations

    1) x = 3 - 4p + q x = 3 - 4y + z x + 4y - z = 3 2) x + 4y - z = 3 (i) let x = a and y = b, so z = a + 4b - 3 General solution: x = a y = b z = a+ 4b - 3 (ii) let x = r and z = t, so y = (3 - r + t) / 4 General solution: x = r y = (3 - r + t) / 4 z = t3) I don't understand this part. Is the...
  30. C

    Schools Where should I take my calculus and linear algebra online?

    Has anyone taken these two courses online in a self-paced course for credit? If so, where and how was it in terms of quality? How about price? Opinions/thoughts are much appreciated. I'm working and the closest community college is a commute away, so that's out. I'm finding $1100-3000~ for...
  31. Manasan3010

    Linear momentum or Angular Rotation

    I think the answer is ##\frac{mV}{M}## but I am not sure. Won't the cylinder tries to rotate due to the collision at one end? Is this anything related to Angular Momentum? The Answers given were,
  32. cianfa72

    I Gaussian elimination for homogeneous linear systems

    Hi, I ask for a clarification about the following: consider for instance a 10 x 12 homogeneous linear system and perform Gauss elimination for the first 8 unknowns. Suppose you end up with 5 equations in the remaining 12-8 = 4 unknowns (because in the process of the first 8 unknowns elimination...
  33. Eclair_de_XII

    If you were a linear algebra teacher, would you dock points for this?

    Let's say you were proctoring some test that required proofs of Jordan canonical forms and rational canonical forms. Would you dock points from a lazy student abbreviating the former as "J-canonical forms" and the latter as "##\mathbb{Q}##-canonical forms" in their proofs?
  34. K

    I Find Practical Resonance Frequencies in Linear Differential Equations

    Hi all, I would like to know what is the equation upon which I can use to determine the practical resonance frequencies in a system of second order, linear differential equations. First some definitions: What I mean by practical resonance frequencies, is the frequencies that a second order...
  35. K

    Linear Equations: Solving for 3 unknowns given 3 equations

    a+b=0 so.. a= -b -2a-2c = -1 = -2(-b)-2c = -1 = 2b-2c=-1 -a-3b+c= 1 = -(-b)-3b+c= 1 = b-3b+c=1 = -2b+c=1 i think this is right but i don't know where to go from here
  36. P

    Why is the Lorentz force linear with velocity?

    I am working on derivation of Lorentz force. (I know that Lorentz force is in some sense definition of fields, but still there is nontrivial dependence on velocity). I want to derive that the force is linear in components of velocity, so for example $$F_x=q(E+Av_x + Bv_y + Cv_z ),$$where ##A...
  37. B

    Question about the derivation of linear magnification

    I tried assumed ##\theta \approx sin \theta \approx tan \theta##. By Snell's law(after approximation), $$n_1 \tan( i_1)= n_2 \tan( i_2)$$ If ##\tan (i_1)=\frac {h_o}{ u}## and ##\tan (i_2)=\frac {h_i}{ v}##,then $$m=\frac {h_i}{h_o}=\mod{\frac {v n_1}{un_2}}$$ Which is the expected...
  38. G

    MHB Solve linear algebra equation 2x – 1 = 9 – 3x

    Hi, Could someone help me with this question: 2x – 1 = 9 – 3x Thank you in advance!
  39. bagasme

    B Solving a Linear Programming Problem which Requires Integer Solutions

    Hello, In grade 11 of high school, I encountered this linear programming problem on my textbook: The "alternative solution" described in the textbook as follows: Let: - ##x## : amount of plant A - ##y## : amount of plant S - ##L## : garden area - ##L_x## : area of garden for one plant A -...
  40. S

    Converting linear state space into a transfer function

    My questions are now... Do the steps of converting this space to transfer function include any laplace ? or just we do get [SI-A]-1 and then transfer function is = C* [SI-A]-1 * B As [1 0] * [s-1/det -0.5/det ; 0.5/det s-0.5/det] * [0; 1] = -0.5/s^2+s+0.5 I mean do we need any laplace after that...
  41. cianfa72

    Linear 2-port network representation

    Hi, in the context of linear two-ports networks my (italian) textbook says that if its internal structure consists of passive one-port components with no independent current/voltage generators then the following (implicit) linear equations based representation always exist: ##\left[A\right] V...
  42. N

    MHB Linear profit, graphs and equations.

    I have no clue how to interpret this problem and where to start to get my values to plot my graphs and get my x, y and etc values as there is too much going on. Can someone shed some light of how to do this problem? I am not sure if is this how I solve this: How do I find that the HD...
  43. dRic2

    MATLAB Ltiview and negative step response of a linear system

    Hi, my professor gave me I code where he used to evaluate the answer of a linear system to a step increase in the input variables like this: MySystem = ss(A, B, C, D); ltiveiw('step', MySystem, 'r-', 300); My problem is that with this code I get the response only for a positive step. I'd...
  44. M

    MHB Linear Codes : Error detection

    Hey! :o The code words of a linear code $C$ have the length $n=5$. Writing the code words into a matrix to get the linear independent ones, we get the following: \begin{equation*}\begin{pmatrix}0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 &...
  45. karush

    MHB 1.3.11 Determine if b is a linear combination

    Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$ $$a_1\left[ \begin{array}{r} 1\\-2\\0 \end{array}\right], a_2\left[ \begin{array}{r} 0\\1\\2 \end{array}\right], a_3\left[ \begin{array}{r} 5\\-6\\8 \end{array}\right], b=\left[ \begin{array}{r}...
  46. karush

    MHB -m30 - 2nd order linear homogeneous ODE solve using Wronskian

    2000 Convert the differential equation $$\displaystyle y^{\prime\prime} + 5y^\prime + 6y =0$$ ok I presume this means to find a general solution so $$\lambda^2+5\lambda+6=(\lambda+3)(\lambda+2)=0$$ then the roots are $$-3,-2$$ thus solutions $$e^{-3x},e^{-2x}$$ ok I think the Wronskain...
  47. GrafZeppelim

    I Linear transformation T: R3 -> R2

    Homework Statement Find the linear transformation [/B] T: R3 --> R2 such that: 𝑇(1,0,−1) = (2,3) 𝑇(2,1,3) = (−1,0) Find: 𝑇(8,3,7) Does any help please?
  48. A

    I Linear Transformations: Why w1 is a Linear Combination of v

    Given w = T (v), where T is a linear transformation and w and v are vectors, why is it that we can write any coefficient of w, such as w1 as a linear combination of the coefficients of v? i.e. w1 = av1 + bv2 + cv3 Supposably this is a consequence of the definition of linear transformations, but...
  49. A

    Engineering Need help with a linear circuit problem

    Attached is my work for the voltages, and the problem itself.
  50. karush

    MHB 17.1 Determine if T is a linear transformation

    nmh{2000} 17.1 Let $T: \Bbb{R}^2 \to \Bbb{R}^2$ be defined by $$T \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix} 2x+y\\x-4y \end{bmatrix}$$ Determine if $T$ is a linear transformation. So if...
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