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. M

    I "Rationale" for Homogeneous vs. Nonhomogeneous Differential Equations?

    Hi; I am missing something. I can follow the technicality of a homogenous linear equation has all coefficients of zero and the "contra" for non homogenous equations. I just can't figure out the relevance of the consequences of outcome. If I am not being clear maybe I can be guided as to how...
  2. S

    I A wonderful flow chart for taxonomy of matrices

  3. F

    I Coefficient sign flip in linear regression with correlated predictors

    Hello Forum, I have read about an interesting example of multiple linear regression (https://online.stat.psu.edu/stat501/lesson/12/12.3). There are two highly correlated predictors, ##X_1## as territory population and ##X_2## as per capita income with Sales as the ##Y## variable. My...
  4. F

    Intro to Linear Algebra - Nullspace of Rank 1 Matrix

    The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...
  5. Daniel Guh

    AP Physics C Mechanics: Linear Momentum for Colliding Billiard Balls

    I'm guessing this question can be solved using the law of conservation of momentum Vi = 5 m/s (5 m/s) M = (4.33 m/s) cos30 M + V sinθ M I don't know what to do after this... I'm also not sure if I use the sin and cos correctly.
  6. Ahmedbadr132

    Power Efficiency of Linear Piezo Electric Motors

    Hi I want to know the power efficiency of linear piezo electric motors in percentile.
  7. F

    I Linear Model with independent categorical variable

    Hello, I have been pondering on the following: we have data for blood pressure BP (response variable) and data about age and gender (categorical variable with two levels). We can build two linear regression models: $$BP=b_0+b_1 age+b_2 gender$$ $$BP=b_0+b_1 age$$ The first model does not take...
  8. C

    Help Needed Proving Implication for Linear Functional on Banach Space

    Dear everybody, I am having some trouble proving the implication (or the forward direction.) Here is my work: Suppose that we have an arbitrary linear functional ##l## on a Banach Space ##B## is continuous. Since ##l## is continuous linear functional on B, in other words, we want show that...
  9. C

    Use linear regression to find Planck’s constant

    I am trying to find Planck's constant using Excel given the data: Frequency [Hz] Photon Energy [J] 7.5E+14 4.90E-19 6.7E+14 4.50E-19 6E+14 4.00E-19 5.5E+14 3.60E-19 5E+14 3.30E-19 4.6E+14 3.00E-19 4.3E+14 2.80E-19 4E+14 2.65E-19 3.75E+14 2.50E-19 I am using Linear...
  10. B

    A Linear framedragging effect

    Imagine two equal masses, m, moving through flat spacetime with opposite 3-momenta, as seen from an inertial frame in the COM. In the massless case of two parallel, non-colinearly infinitely long moving bundles of light, Bonnor Beams, the beams are curved if the momenta are opposite, and stay...
  11. Euge

    POTW A Linear Operator with Trace Condition

    Let ##V## be a finite dimensional vector space over a field ##F##. If ##L## is a linear operator on ##V## such that the trace of ##L\circ T## is zero for all linear operators ##T## on ##V##, show that ##L = 0##.
  12. F

    I Linear regression, feature scaling, and regression coefficients

    Hello, In studying linear regression more deeply, I learned that scaling play an important role in multiple ways: a) the range of the independent variables ##X## affects the values of the regression coefficients. For example, a predictor variable ##X## with a large range typically get assigned...
  13. F

    I Expected coefficient change from simple to multiple linear regression

    Hello forum, I have created some linear regression models based on a simple dataset with 4 variables (columns). The first models simply involve one predictor variable: $$Y=\beta_1 X_1+\beta_0$$ and $$Y=\beta_2 X_2+ \beta_0$$ The 3rd model is multiple linear regression model involving the 3...
  14. C

    Showing that this equation is a solution to the linear wave equation

    For this problem, Where equation 16.27 is the wave equation. The solution is I don't understand how they got the second partial derivative of ##y## with respect to ##x## circled in red. I thought it would be ##1## since ##v## and ##t## are constants Many thanks!
  15. H

    Mathematica How to check a particular solution of System of Linear ODEs?

    If I have been given a system of inhomogeneous linear ODEs, $$ \vec{x'} = \begin{bmatrix} 4 & -1 \\ 5 & -2 \\ \end{bmatrix} \vec{x} + \begin{bmatrix} 18e^{2t} \\ 30e^{2t}\\ \end{bmatrix} $$ I have found its particular solution to be: $$ 1/4 \begin{bmatrix} -31e^{2t} - 25e^{6t} \\ 85e^{2t} -...
  16. F

    I Linear regression and random variables

    Hello, I have a question about linear regression models and correlation. My understanding is that our finite set of data ##(x,y)## represents a random sample from a much larger population. Each pair is an observation in the sample. We find, using OLS, the best fit line and its coefficients and...
  17. C

    Calculating Linear Charge Density of a Cylinder

    For part a: I know that linear charge density is the amount of charge per unit length, and we are given the volume charge density. Since we are given the volume, we can obtain the length by multiplying the volume by the cross sectional area, so C/m^3 * m^2 = C/m. The cross sectional area of a...
  18. Euge

    POTW Linear Systems of ODE

    Solve the linear system of ODE ##x' = 2x + 3y##, ##y' = -3x + y## with initial conditions ##x(0) = 1, y(0) = 2##.
  19. _maxim_

    Mini linear actuator for syringe

    Dear, I would like to create a controlled system to inject 1-2 ml of liquid (a chemical reagent) into a solution contained in a glass flask. The only request made is to use a reliable linear piston of minimum size, equal to or slightly larger than the injected volume (1 cm3) so that it can...
  20. F

    I Linear Regression and OLS

    Hello, Simple linear regression aims at finding the slope and intercept of the best-fit line to for a pair of ##X## and ##Y## variables. In general, the optimal intercept and slope are found using OLS. However, I learned that "O" means ordinary and there are other types of least square...
  21. S

    I A = norm-preserving linear map (+other conditions) => A = lin isometry

    I'm studying "Semi-Riemannian Geometry: The Mathematical Langauge of General Relativity" by Stephen Newman. Theorem 4.4.4 in that book: The proof of part 2 is given like this: Seems a bit incomplete. I'd like to know if my approach is correct: $$\langle A(v+tw),A(v+tw)\rangle=\langle...
  22. J

    B Rotation is absolute, linear motion is relative?

    Can you explain with example what mean rotation is absolute and linear motion is relative?
  23. abdulbadii

    The linear spring having even forcing pull/push

    How do we have linear spring direction (mostly a spherical spring) to have pull/push force evenly across some points within a range? Or is it possible to create spring material with anomaly property capable of performing so?
  24. S

    I Intuition for why linear algebra is needed in quantum physics

    I'm watching a nice video that tries to explain how linear algebra enters the picture in quantum physics. A quick summary: Classical physics requires that physical quantities are single-valued and vary smoothly as they evolve in time. So a natural way to model classical physical quantities is...
  25. G

    B Projected linear separation between companion stars

    A quasar with a bolometric flux of approximately 10−12 erg s−1 cm−2 is observed at a redshift of 1.5, i.e. its comoving radial distance is about 4.4 Gpc. Assume that the quasar in the previous question is observed to have a companion galaxy which is 5 arcseconds apart. What is the projected...
  26. bphillips921

    Trying to determine linear actuator force needed

    I'm designing a pivot lift system to lift my movie projector up when not in use. I've designed the parts and begun 3D printing, but am concerned that the 3D print won't be strong enough, so I may bite the bullet and have the parts CNC machined. If I do that I want to be certain that I have the...
  27. D

    I The Price of Beer - Linear Algebra Problem

    I came across the following problem somewhere on the web. The original site is long gone. The problem has me stumped. May be sopmeone can provide some insight. (The problem seems too simple to post in the "Linear/Abstract Algebra" forum.) The Cost of Beer It was nearing Easter, and a group...
  28. PhysicsRock

    Show a function is linear

    I don't really know how I am supposed to approach that. In general, I know how to show that a function is linear, which is to show that ##f(\alpha \cdot x) = \alpha \cdot f(x)## and ##f(x_1 + x_2) = f(x_1) + f(x_2)##. However, for this specific function, I have no idea, since there is nothing...
  29. A

    Electromagnetic linear momentum for a system of two moving charges

    When you write out the equations of motion for a system of two isolated charges, you can add both of the equations and get the increase in the particles linear momentum on one side. On the other side, you get the sum of all the forces between the particles. I understand that this sum of forces...
  30. H

    Finding linear acceleration of a spool and cable

    My angular acceleration is wrong but all I had done was torque which was 110 NM / I = 930 kg-m^2 and calculated 0.118 rad/s^2. But because this is wrong I am stuck and I have no idea how to find angular velocity to plug into the equation to find linear acceleration.
  31. B

    : Calculation of Linear Bearing (Carriage) Load

    Hi Guys, please please assist me! I need to pick for a project at my work linear carriages which should work at this configuration : As you can see, there is a piston which moves the plate on 4 bearings from the side (yes, I know, not the best configuration, but that's what we have given the...
  32. P

    Linear algebra problem with a probable typo

    Well, my guess is that there is something wrong with the factors chosen, because ##\left\Vert \left(0,1,0\right)\right\Vert =1## and \begin{align} \left\Vert F\left(0,1,0\right)\right\Vert &=\left\Vert...
  33. A

    Help with random variable linear estimation

    Hi all, I have a problem on linear estimation that I would like help on. This is related to Wiener filtering. Problem: I attempted part (a), but not too sure on the answer. As for unconstrained case in part (b), I don't know how to find the autocorrelation function, I applied the definition...
  34. F

    I Linear Models vs Nonlinear Models

    Hello, Models can be linear and nonlinear and I just learned that a "linear model" is not just a model where the dependent variable ##Y## is connected to independent variables ##X## raised to the 1st power... The model is called "linear" because the coefficients/parameters are not raised to...
  35. V

    Satellite mechanics: linear and rotational momentum

    [This is a continuation of OP's thread here: https://www.physicsforums.com/threads/satellite-mechanics-linear-and-rotational-momentum.1046963/ ] satellite mechanics: linear and rotational momentum I'm trying to better understand classical mechanics, and came up with a question: Say we have a...
  36. V

    I Satellite mechanics: linear and rotational momentum

    satellite mechanics: linear and rotational momentum I'm trying to better understand classical mechanics, and came up with a question: Say we have a squared satellite weighting 100kg, 1 meter on each side. it has a thruster on it's side, shown in picture thruster quickly ejects 100g of propellant...
  37. A

    Linear moving switch, decade counter, chaser

    I need a row of switches, about 30 to be exact and the idea I need is to have a "moving OFF position" much the same way a "LED chaser" works where you have a moving ON position. Only unlike in a LED chaser where one diode comes ON at any given time and the positions changes continually to...
  38. H

    I Second order non-homogeneous linear ordinary differential equation

    I shall not begin with expressing my annoyance at the perfect equality between the number of people studying ODE and the numbers of ways of solving the Second Order Non-homogeneous Linear Ordinary Differential Equation (I'm a little doubtful about the correct syntactical position of 'linear')...
  39. alpha

    Solving Linear Programming problems in python PulP

    Okay so far I have come up with the following: The objective function is: 30LADA + 40LAHO + 50SADA + 40SAHO + 110DANY + 125DACH + 105HONY + 115HOCH (to read this please see the table above as I have just used the first two letters of each except NY which I have represented by NY). In these I...
  40. alpha

    Linear Programming Problem

    So far I have figured out the following: maximize 0.02x1 + 0.01x2 +0.06x3 + 0.07x4 subject to: xi <= 200,000 xi >= 24,000 x3 <= x2 +x4 x1 >= x3 But when I try to solve this in solver, I get an error which means my contraints must be wrong. Any suggestions on what constaints to put?
  41. Wrichik Basu

    GitHub: How do I merge a PR but maintain linear history + signed commit

    Suppose, for a certain a repo, I create a branch protection rule for the main branch with the following: Require pull request before merging. This means that each feature or bugfix must be on separate branches, which will be later merged with main. Maintain linear history of the main branch...
  42. M

    Solve the simultaneous linear congruences

    Consider the following set of simultaneous linear congruences: ## 3x\equiv 2\pmod {5}, 3x\equiv 4\pmod {7}, 3x\equiv 6\pmod {11} ##. Observe that \begin{align*} &3x\equiv 2\pmod {5}\implies 6x\equiv 4\pmod {5}\implies x\equiv 4\pmod {5}\\ &3x\equiv 4\pmod {7}\implies 15x\equiv 20\pmod...
  43. M

    I Systematic uncertainties in linear fit

    Hello! I have some data points ##(x,y)##, with some uncertainties on y that are statistical ##dy_1## and some systematic ##dy_2##. I want to fit this data using a linear function. How exactly should I deal with the different types of uncertainties? Can I just add them in quadrature and perform...
  44. J

    Linear operator in 2x2 complex vector space

    Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2? ____________________________________________________________ An ordered basis for C2x2 is: I don't...
  45. J

    Linear Transformation from R3 to R3

    "There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case? Thank you.
  46. murshid_islam

    What would be a good book for learning Linear Algebra by myself?

    Summary: What would be a good book for learning Linear Algebra by myself in my situation (which is explained in my post below)? I did an undergraduate Linear Algebra course about 18 years ago. The textbook we used was Howard Anton’s “Elementary Linear Algebra”. The problem is that I never...
  47. guyvsdcsniper

    Linear Algebra in Dirac Notation

    I am trying to convert the attached picture into dirac notation. I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ> The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS. *Was going to type in LaTex but I...
  48. E

    B Attempt to solve this system of three linear equations

    The point (1, 5) is on the curve: y=ax^2+bx+c. This point gives the linear equation: 5 = a + b + c. A second point on the curve, (2, 10) gives the linear equation 10=4a+2b+c. A student called Erika thinks that the point (2, 19) is also on the curve. 5 = a + b + c. 10=4a+2b+c 19=4a+2b+c the...
  49. Graham87

    Intro to quantum mechanics - Spin and linear algebra

    So this expression is apparently in Sz basis? How can you see that? How would it look in Sy basis for example? The solution is following. They are putting Sz as a basis, bur how do you know that Sz is the basis here? Thanks
  50. M

    Obtain the eight incongruent solutions of the linear congruence

    Consider the linear congruence ## 3x+4y\equiv 5\pmod {8} ##. Then ## 3x\equiv 5-4y\pmod {8} ##. Note that ## gcd(3, 8)=1 ## and ## 1\mid (5-4y) ##. Since ## 3^{-1}\equiv 3\pmod {8} ##, it follows that ## x\equiv 15-12y\pmod {8}\equiv 7+4y\pmod {8} ##. Thus ## {(x, y)=(7+4y, y)\pmod {8}\mid 0\leq...