Linear Definition and 1000 Threads

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. ZX.Liang

    I The brightness and linear polarization of pulsars

    I have a question to ask everyone. Is there a positive correlation between the linear polarization degree of pulsars and their brightness in statistics? Or, is it true that the brighter the pulsar, the greater the linear polarization component?
  2. S

    Looking for a low-latency force-based actuator - do they exist?

    Without going into too much detail, I have a project which I am considering my options for, and one design would have potential if I can find some sort of force-based linear actuator. I suspect that there might be some sort of magnetic-based system out there which operates on a changing...
  3. M

    I Understanding linear expressions

    I understand linear expressions map straight lines. How can I identify which expressions are linear; is x/(1-x^3) isn't linear but 1/(1-x^3) is linear? Thanks Martyn
  4. billtodd

    A Scalars, Vectors, Matrices,Tensors, Holors....

    Is there something beyond Holors? :cool:
  5. D

    Unit normed linear functional on a space of sequences

    Let ##X=\{(x_i)\in \ell^\infty | x_i=1 \text{ for } i\leq n, x_i=0 \text{ for } i>n, n\in \mathbb{N}\}##. For every sequence ##(x_i)## in ##X##, ##(x_i)## has the property that the first ##n## entries are 1s and the rest are 0s. So, every sequence in ##X## trivially converges to ##X## and hence...
  6. F

    I Does a rotating accelerometer moving linearly measure linear velocity?

    If an accelerometer is rotating as it moves forward through the air, what force is acting on the accelerometer? Say we remove gravity and just concentrate on the horizontal axis. I know there is centripetal and tangential acceleration that shows up as an offset in the sin wave, but what causes...
  7. DaTario

    I How (un)reasonable is graphic linear regression in a monolog graphic?

    Hi All, Consider two different data sets whose contexts are not related. One is ploted in a linear graphic and the other in a monolog graphic. Both data sets end up looking the same in each graphic. How (un)reasonable is to proceed to a graphic method of linear regression in a monolog graphic...
  8. M

    Proving stability of linear system equations

    For this problem, My solution is to find the characteristic equation of the system by putting the system into a matrix. This gives ##\lambda^2 + 2f \lambda + f^2 + 1 = 0## Then each eigenvalue is ##\lambda_1 = -f - i## and ##\lambda_2 = -f + i## I then want to find the Jacobian, however, I...
  9. D

    Find an example of a linear functional with some properties

    I considered ##X=\mathbb{R}^n## and quickly realized any linear functional like ##f=a_1x_1+\cdots a_nx_n## would attain a maximum on the boundary. I regret to say that my knowledge of topology is still very limited, and did a lot of experimenting with a pen and paper without fruitful results...
  10. H

    I A step in a proof of linear dependence of ODE solutions

    Please consider the following step in a proof: After transposing the matrix, its coefficients still are functions of ##x##. Why then the solution ##a, a_1, a_2## is constant?
  11. N

    B What unique contributions to math did linear algebra make?

    I've been struggling to understand what was the key insight or insights that linear algebra brought to math, or what problems it allowed the solving of that couldn't be solved before. To make a comparison with calculus, I understand that calculus' two key insights were finding a method to...
  12. M

    Fundamental matrix for complex linear DE system

    For this problem, I am trying to find the fundamental matrix, however, the eigenvalues are both imaginary and so are the eigenvectors. That is, ##\lambda_1 = 4i, \lambda_2 = -4i## ##v_1 = (1 + 2i, 2)^T## ##v_2 = (1 - 2i, 2)^T## So I think I just have an imaginary matrix? This is because the...
  13. M

    Fundamental matrix linear system equivalent to linear matrix system

    My working is , Consider case where the there are two linearly independent solutions ##x'(t) = c_1x' + c_2y' = A(c_1x + c_2y)## ##(x'~y')(c_1~c_2)^T = A(x~y)(c_1~c_2)^T## Then cancelling coefficient matrix I get, ##(x'~y')= A(x~y)## ##Φ'(t) = AΦ(t) ## by definition of 2 x 2 fundamental matrix...
  14. M

    Finding general solution to linear system

    For this problem, My working is, ##0v_1 + 0v_2 = 0##, however, does someone please know why the example says we cannot choose ##v_1 = (0, 0)## since from ##0v_1 + 0v_2 = 0## ##v_1, v_2 \in \mathbb{R}## i.e there is no restriction on what the vector components could be)? Thanks!
  15. S

    I Strange index notation for linear transformation matrix

    I'm reading Liang's book on General Relativity and Differential Geometry, and came across this part: I just want to have a crystal clear understanding of why this notation is chosen. Basis transformation would be an automorphism from ##V## to ##V##, and there's a result saying that the set of...
  16. chwala

    Work out the amount that Arjun paid in rent in 2019

    I am interested in a more direct linear equations approach...to solve, which i know is possible... A. My initial thinking was along the lines, let ##x## be the amount that Arjun paid in 2018 and let ##y## be the amount that Gretal paid in 2018 where A was the total amount paid in 2018 ...
  17. W

    How Does Impulse Relate to Momentum Change in Physics Problems?

    (bottom graph relates only to c) (a) i. The students can calculate the area under the graph to find the impulse exerted on the block. This is because the area under a force vs. time graph is the change in momentum or the impulse. ii. Knowing that the graph is linear and begins at around 3 N...
  18. S

    B Is Averaging Individual Slopes Equivalent to Linear Regression?

    So the linear regression formula is https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/statistics/regression-and-correlation/simple-linear-regression.html found here. Question - is the slope given by the regression formula mathematically equivalent to individually finding...
  19. D

    B Finite linear combination of continuous functions is continuous?

    ##G## and ##H## are real valued Lipschitz continuous functions. There exists a ##K_1,K_2\geq 0## such that for all ##s,t##, $$(s-t)^2\leq K_1^2 (G(s)-G(t))^2$$ and $$(s-t)^2\leq K_2^2 (H(s)-H(t))^2.$$ Is ##aG(t)+bH(t)## where ##a,b## are real constants also Lipschitz continuous? I tried showing...
  20. chwala

    I Solve the system of linear equations

    This is fine but i have my own approach as follows; ##\begin{pmatrix} -7 & -6 & -12 & -33 \\ 5 & 5 & 7& 24 \\ 1 & 0 & 4 & 5 \end{pmatrix} ## → Row 1 times 5 and row 2 times 7... ##\begin{pmatrix} -35 & -30 & -60 & -165 \\ 35 & 35 & 49& 168 \\ 1 & 0 & 4 & 5 \end{pmatrix}## →...
  21. F

    I Desirable estimator properties...

    Hello, An estimator is a random variable, i.e. a function that assigns a number to a random sample collected from a population with unknown parameters. More practically, an estimator is really a formula to calculate an estimated coefficient ##b## using the data from our single random...
  22. 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...
  23. S

    I A wonderful flow chart for taxonomy of matrices

    https://upload.wikimedia.org/wikipedia/commons/d/d1/Taxonomy_of_Complex_Matrices.svg
  24. 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...
  25. 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...
  26. 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.
  27. Ahmedbadr132

    Power Efficiency of Linear Piezo Electric Motors

    Hi I want to know the power efficiency of linear piezo electric motors in percentile.
  28. 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...
  29. 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...
  30. M

    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...
  31. B

    A How Does Linear Framedragging Affect Parallel Moving Masses in Flat Spacetime?

    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...
  32. 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##.
  33. 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...
  34. 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...
  35. M

    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!
  36. 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} -...
  37. 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...
  38. 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...
  39. Euge

    POTW Solve Linear System of ODEs: x',y' w/ Initial Conditions x(0),y(0)

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

    I Can Alternative Least Squares Methods Be Used in Linear Regression?

    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...
  42. 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...
  43. J

    B Rotation is absolute, linear motion is relative?

    Can you explain with example what mean rotation is absolute and linear motion is relative?
  44. 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?
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. PhysicsRock

    How to Show Linearity of a Function?

    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...
  50. 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...
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