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. mollwollfumble

    B Linear dark features on the Sun

    I’ve been watching =AT2AcUch2CYj2r40rQ431p1Rt1SQ-YzVLo-ryJHYFIT8s9ZlMJ5pcWH9lN17FRuz2eLTlNPVb4fP2yk4PPLqjJK_6lpeWheAbATLZd7foljjUg2sOFxm-mZbhhcw3G9ApCzWIWNjgBGcGjoXMqdW6Jv-Rb8P2yA']https://www.youtube.com/watch?v=l3QQQu7QLoM What is the scientific name of the enigmatic long linear dark features...
  2. M

    I Know a simple, linear, complex, eigenvalue BVP?

    Hi PF! I'm trying to find a 1D, linear, complex, 2nd order, eigenvalue BVP: know any that admit analytic solutions? Can't think of any off the top of my head. Thanks!
  3. T

    Rotation around a non fixed axis + linear motion of a system

    I have had a thought experiment in my head for a while now and I am unable to find clear enough examples/info that deal with similar issues, to solve it on my own. This is why I hope that someone in this forum can at least point me towards a solution or provide hints as to where should I be...
  4. X

    Introduction to Linear Algebra: Solving Real-World Problems

    Summary:: Linear algebra 1.Let a a fixed vector of the Euclidean space E, a is a fixed real number. Is there a set of all vectors from E for which (x, a) = d the linear subspace E / 2. Let nxn be a matrix A that is not degenerate. Prove that the characteristic polynomials f (λ) of the matrix A...
  5. hairey94

    Linear Motion and Linear Momentum

    Linear Motion Equation to get the common velocity of the block and bullet just after collision: v2=u2+2gs, I set v=0 at max height of s = 0.004 m and g = -9.81 m s-2 I got u = 0.28 m/s Then I calculate the impulse of the block using formula J = mv - mu, where v=0 and u=0.28 m/s and I got J = 1.4...
  6. Y

    MHB Objective function of a linear program with multiple variables.

    Hello, Please I need help to find the objective function of a linear program (attachement : example). I tried to figure it out from the formula provided in (attachement : formula) but I couldn't understand it, it's written (MIN(lambda)wj) I think it's the key to resolve my question ! ( Full file...
  7. Y

    Objective function of a linear program with multiple variables

    Hello, Please I need help to find the objective function of a linear program (attachement : example). I tried to figure it out from the formula provided in (attachement : formula) but I couldn't understand it, it's written (MIN(lambda)wj) I think it's the key to my question ! ( Full file is...
  8. M

    Deriving the 1-D Linear Convection Equation

    With the assumptions of Inviscid flow, no pressure gradient and no body force terms in 1-D Navier Stokes becomes 1-D nonlinear convection equation; And if we assume velocity of wave propagation is constant value c, equation becomes 1-D linear convection equation; This is online derivation and...
  9. M

    Conservation of angular momentum and its counterpart for linear momentum

    Hi, I have just joined the forum. Thank you all for being a part of such places so that people like me can get answers to the questions on their minds! --------------------------- I have been trying to understand how a quadcopter yaws. Referring to the figure below which is bird's eye view of...
  10. G

    To find the relative velocities of linear and circular motion

    Could I please ask for help regarding the final part of the following question: It is the very last part, to find v in terms of u. So I have that the velocity of the midpoint of XY is: V_m = (u/2) i + (u/2) j I let the position vector of P be: r_p = cos(wt) i + sin(wt) j (w = angular...
  11. hackedagainanda

    Question about this linear equation with fractions

    I use 2x -4 as the LCD and turn 8/(x - 2) - (13/2) = 3 into 16 - 13x - 4 = 3, I then get 12 - 13x = 3 which leads me to 13x = -9 so x = -9/13 which is the wrong answer. Where did I make a mistake?
  12. B

    I Which method should I use for a linear fit of data with error on both z and y?

    Hello! I have some data of the form (x,y,z) which I know it is described by a function of the form: ##z=y(a+bx)##, where a and b are parameters to be fitted for. z and y have error associated to them while x doesn't (x is actually an integer going from 0 to 3 for each value of y). I tried to do...
  13. J

    MHB Linear Approximation: Check Your Answer

    Have I solved this linear approximation question correctly?
  14. MidgetDwarf

    I Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r

    ∈Was wondering if anyone here could help me with an explanation as to how Axler arrived at a particular step in a proof. These are the relevant definitions listed in the book: Definition of Matrix of a Linear Map, M(T): Suppose ##T∈L(V,W)## and ##v_1,...,v_n## is a basis of V and ##w_1...
  15. hackedagainanda

    Is there a better way to solve this linear equation with fractions?

    My method is converting 14 into ##\frac {42} {3}, = \frac 7 3x## Then $$ \frac {126} { 21}, = 6$$
  16. A

    Linear transformer driver explanation

    Ok, so to be honest I am not really sure to which subcategory this belongs, but I think it is high energy physics. So my question will be rather brief and vague , can you explain me the working principle of a linear transformer driver? Google wasn't of that much help with this one as it seems...
  17. R

    I This linear system has a solution? (Gauss' elimination)

    I don't know the terms so I'm sorry if the informations at summary above is unclear. But I add a detailed photo of my calculations below. I use Gauss' Elimination laws.
  18. C

    Calculating Force and Torque for Linear motion (image provided)

    This is how I understand the torque (T1) created at the axis of rotation due to the spring would be calculated. This is considering the spring is starting from a fully extended state and is not already compressed. This is considering that (A) has a distance of 2 in. and the force from the spring...
  19. Julio1

    MHB Solutions of the ODEs - 2 first order linear equations

    Find the general solution of the ODE: $\check{X_1}=X_1$ $\check{X_2}=aX_2$ where $a$ is a constant.
  20. Avatrin

    A Numerical Solution to Random Linear Non-Homogeneous ODE

    Hi I am trying to learn optimal estimation by reading Gelbs Applied Optimal Estimation, and I am having hard time with finding \Gamma defined as the following: $$ \Gamma_k w_k = \int_{t_k}^{t_{k+1}} e^{F(t_{k+1} - \sigma)} G(\sigma) w(\sigma) d\sigma$$ Here F is a known matrix. So is G, and w...
  21. wael_khayati

    How to Complete a Linear Inequality Assignment Without Full Instructions?

    hey so, this is an algebra assignment that we had to do and i really didn't understand the course material that well, but i managed to do the very first steps. anyways i was hoping you guys could help me finish the rest of this table. https://ufile.io/jowwrfj3 or you can see the file attached
  22. M

    I Weird covariance matrix for linear fit

    Hello! I really don't know much about statistics, so I am sorry if this questions is stupid or obvious. I have this data: ##x = [0,1,2,3]##, y = ##[25.885,26.139,27.404,30.230]##, ##y_{err}=[1.851,0.979,2.049,6.729]##. I need to fit to this data the following function: $$y = a (x+0.5)/4.186 +...
  23. A

    Stabilizer Leg Linear Actuator Force to Jack up a Truck's rear tyres

    Hi, I previously posted about the statically indeterminate truck problem. Thank you to everyone who helped me. However, I now realized that isn't the problem I need to solve. I need to know the force of the linear actuators to lift the rear tyres off the ground. Since the tyres will be...
  24. G

    MHB Linear Correlation: Is Evidence Sufficient?

    The accompanying table lists the numbers of Internet users per 100 people and numbers of scientific award winners per 10 million people for different countries. Construct a​ scatterplot, find the value of the linear correlation coefficient​ r, and find the​ P-value of r. Determine whether there...
  25. M

    MHB Kernel of Linear Map: Show $\ker \phi$ Equation

    Hey! :o Let $1\leq n,m\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $(b_1, \ldots , b_n)$ a basis of $V$. Let $W:=\mathbb{R}^m$ and let $\phi:V\rightarrow W$ be a linear map. Show that $$\ker \phi =\left \{\sum_{i=1}^n\lambda_ib_i\mid \begin{pmatrix}\lambda_1\\ \vdots \\ \lambda_n\end{pmatrix}\in...
  26. H

    Is there a more accurate way to estimate sales quantities?

    attached resolution attempt
  27. A

    MHB Introduction to linear algebra

    prove that $2+8{\sqrt{-5}}$ is unit and irreducible or not in $\mathbb Z+\mathbb Z{\sqrt{-5}}$.
  28. A

    MHB Introduction to linear algebra

    prove that u(z+zw)={+1,-1,+w,-w,+w^2,-w^2}
  29. LCSphysicist

    I Linear Subspace with Neutral Element in Brazilian Portuguese and Spanish

    W = {f(t) | f(0) = 2f(1)} The answer say yes, but i don't know how to prove the neutral element.
  30. JD_PM

    I Given two linear transformations L and K, show ##K = \lambda L## holds

    Let ##V## be a real vectorspace of finite dimension ##n##. Let ##L, K:V \rightarrow \Re## be linear transformations so that ##ker(L) \subset ker(K)##. Then there's a parameter ##\lambda \in \Re## so that ##K=\lambda L## a) Show that ##K=\lambda L## holds when ##K=0##. b) Suppose that ##K...
  31. S

    Need a compact linear drive for a hexapod table

    Hello everyone, I'm trying to make a hexapod table, in the same style as this: but I am trying to overcome the main flaw with this design - the minimum height being so high. My goal is to produce something which someone can stand on, in their home. It needs to become as flat as possible, and...
  32. B

    I Should I Remove the Outlier Point in My Linear Fit?

    Hello I have these points coming from different experiments: ##x = [-0.3, -0.2, -0.09, 0.01, 0.2]## ##y = [-8.15, -5.20, -3.32, 0., 5.65]## ##y_{err} = [0.1, 0.27, 0.35, 0.09, 0.44]## and I need to fit a straight line to them (based on theoretical arguments). I attached below the obtained...
  33. marciokoko

    Solve angle issues for opening a door with a linear actuator

    ok, setting the force issues aside (for now), Id like to use what I know about triangles to figure how to position my linear actuator in order to open and close this door correctly: The vertical side is 84cm, the door is 92cm. The measured stroke of the AL is 25cm. I would need to lower the...
  34. F

    Deriving the Adjoint / Tangent Linear Model for Nonlinear PDE

    I am trying to derive the adjoint / tangent linear model matrix for this partial differential equation, but cannot follow the book's steps as I do not know the math. This technique will be used to solve another homework question. Rather than posting the homework question, I would like to...
  35. K

    I Linear fit on the difference of data points

    Hello! I have some data points obtained from a measurement and one of them is defined as the reference point. I need to compute the difference between that reference point and all the others (including itself) and plot the difference as a function of another variable (which doesn't have an error...
  36. A

    MHB What Method Solves the ODE y''(x) + y'(x) + F(x) = 0?

    y''(x)+y'(x)+F(x)=0 Pleas me a idea
  37. D

    Conservation of linear and angular momentum

    Hi ; I have a few question regarding the conservation of linear and angular momentum. Would appreciate any help. 1 - When no external forces act are both linear and angular momentum conserved in all 3 directions separately or just the total linear/angular momentum conserved ? 2 - if I approach...
  38. T

    Describing the second law of motion using linear momentum

    Hi! This is a very simple question regarding terms of expressions. One law of motion is: F=ma Another, using L as the linear momentum, is: F = dL/dt If the first equation can be characterized (ignoring reference frames) as a "coordinate-based equation" (since is concerned with the second...
  39. Mondayman

    Linear Algebra What are some recommended second texts for self-studying linear algebra?

    Hello folks, I am currently finishing up a class on linear algebra, covering vector spaces, bases and dimension, geometry of n-dimensional space, linear transformations and systems of linear equations. I am only getting accustomed to proof writing for the first time in this course. However, I...
  40. K

    I Plot Linear Fit w/ Error Bounds: y=ax+b

    Hello! I made a linear fit, ##y=ax+b##, to some data points and I get the best parameters with their (1 sigma) errors: ##a\pm\delta a## and ##b\pm\delta b##. I want to plot this fit on top of my data points in such a way as to reflect the error on the parameters. The "main" fit is simply...
  41. Math Amateur

    MHB Linear Transformations & Matrices: Armstrong, Tapp Chs. 9 & 1 - Explained

    At the start of Chapter 9, M. A. Armstrong in his book, "Groups and Symmetry" (see text below) writes the following: " ... ... Each matrix A in this group determines an invertible linear transformation f_A: \mathbb{R} \to \mathbb{R} defined by f_A(x) = x A^t ... ... "I know that one may define...
  42. B

    Question about linear transformations

    Summary:: linear transformations Hello everyone, firstly sorry about my English, I'm from Brazil. Secondly I want to ask you some help in solving a question about linear transformations. Here is the question:Consider the linear transformation described by the matrix \mathsf{A} \in \Re...
  43. M

    Help with a Linear Algebra problem please

    For the following statement: V = R ≥ 1; x ⊕ y = max (x,y), with z = 1 My attempt is as follows: Should R3 be z ⊕ (x ⊕ y)? I am confused at to the notation of this rule. Moreover, I am struggling to find examples and answers of such problems in linear algebra online. Should I always view such...
  44. Avatrin

    3D geometry exercises with linear algebra

    Hi I have noticed that while I have the grasp of the theoretical underpinnings of linear algebra, I need work on applying it to geometric problems (think computer vision and rigid body motion). So, I am looking for a book that allows me to practice 3D geometry problems. Is there any obvious...
  45. cookiemnstr510510

    Linear operators, quantum mechanics

    Hello, I am struggling with what each piece of these equations are. I generally know the two rules that need to hold for an operator to be linear, but I am struggling with what each piece of each equation is/means. Lets look at one of the three operators in question. A(f(x))=(∂f/∂x)+3f(x) I...
  46. Fochina

    Proof: Relationship between a linear map and the associated matrix

    Hi! I don't understand how to demonstrate the following exercise. Let ##F: R^{n} \rightarrow R^{n}## be a linear map which is invertible. Show that if ##A## is the matrix associated with ##F##, then ##A^{-1}## is the matrix associated with the inverse of ##F##.
  47. Lauren1234

    I Linear Isomorphisms: Understand & Apply

    how would I go about answering the above question I need some pointers on how to start?
  48. Lauren1234

    I Proving Linear Transformation of V with sin(x),cos(x) & ex

    Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A). Define T:V→V,f↦df/dx. How do I prove that T is a linear transformation? (I can do this with numbers but the trig is throwing me).
  49. M

    MHB Give a linear map that satisfies given properties

    Hey! :o Let $v_1:\begin{pmatrix}1 \\ 1\\ 1\end{pmatrix}, \ \ v_2:\begin{pmatrix}1 \\ 0\\ 1\end{pmatrix}\in \mathbb{R}^3$. Let $w=\begin{pmatrix}1 \\ 0 \\2\end{pmatrix}\in \mathbb{R}^3$. If possible, give a linear map $\phi:\mathbb{R}^3\rightarrow \mathbb{R}^2$ such that $\phi...
  50. J

    Engineering Finding voltages in a linear circuit

    From the circuit I have: ##-v_b + v_a + V = 0## ##v_b - V = v_a## ##i_1 = (v_b - V)/R_1## ##I + i_2 = i_1## ##(v_b - V)/R_1 = I + v_b/R_2## From this last equation I get ##v_b = 10.8## and hence ##v_a = 5.8##. However, apparently that is wrong. (And hence my answers to #2 were all wrong as...
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