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".

View More On Wikipedia.org
Recent contents Top users
  1. tomwilliam2

    When is a function linear and when is it affine?

    Homework Statement My son (aged 13) received his math test back, and got the following problem wrong: Which of the following functions is linear? A: f(x) = -3x + 5 B: g(x) = x^2 C: h(x) = 6x - (1/2) D: p(x) = 4x/5 The attempt at a solution He answered B, which is clearly wrong, but it wasn't...
  2. Aleoa

    I Why Is Reflection in a Hyperplane a Linear Function?

    Is it possible to understand intuitively (without using a formal proof ) why a reflection is a linear function ?
  3. L

    Setting up a matrix from a linear equation

    Homework Statement I need some help with a question on my assignment. It asks to set up a matrix from the linear equations, y=25x+70 and y=35x+40. Homework Equations How do I set this matrix up? The Attempt at a Solution I think that I have to rewrite it as 25x-y=-70 and 35x-y=-40. But then I...
  4. M

    Linear algebra, field morphisms and linear independence

    Homework Statement Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K. Homework Equations f1, ..., fn are field morphisms => Ker (fi) = 0 (injective) The Attempt at a Solution I...
  5. I

    MHB Significance test comparing slopes from linear fit

    I would like to compare four slopes I got from linear fits through five data points for each fit. The fitting results (for the slopes) are as follows: 1: 0.08885 ± 0.00991 2: 0.08744 ± 0.0118 3: 0.10288 ± 0.00669 4: 0.0926 ± 0.01285 My hypothesis is that they are all not significantly different...
  6. C

    I Linear Programming: how to identify conflicting equations

    I'm reading about the Simplex method to solve a Linear Program. If a program is rejected as being infeasible, is there a method to identify which equations are causing it to be infeasible and a technique for reducing the program in an optimal way? (I'm not sure what 'optimal' is exactly, but my...
  7. TaylorLL

    Rocket subject to linear resistive force -- two methods

    Homework Statement Consider a rocket subject to a linear resistive force, $$f = -bv$$, but no other external forces. Use Equation (3.29) in Problem 3.11 to show that if the rocket starts from rest and ejects mass at a constant rate $$k = -\dot{m}$$, then its speed is given by: $$v =...
  8. S

    Linear combination of random variables

    Homework Statement Let ##X_1 \sim N(3,2^2)## and ##X_2 \sim N(-8,5^2)## be independent. Let ##U=aX_1+bX_2##. What is the distribution of ##U## Homework EquationsThe Attempt at a Solution As they are independent, we can write the distribution of ##U## as the convolution of the 2. So I get...
  9. mkematt96

    Linear Abrasion Testing PVC/Polymer Blend Hoses

    Working on a research project to develop a test standard for abrasion testing pressure washer hoses. Ideally this could be used on any sort of hose. We have a set procedure that has worked on a couple hoses that looks like this: .Weight applied is constant .Cycle speed is constant .Constant...
  10. kvothe18

    I Linear fitting in physics experiments with errors

    Hello! I have a question that maybe has to do more with Mathematics, but if you do experimental physics you find it quite often. Let's assume that we want to measure two quantities x and y that we know that they relate to each other linearly. So we have a set of data points xi and yi...
  11. yecko

    Solving a Linear ODE using a power series

    Homework Statement Homework Equations Power series ODE The Attempt at a Solution [/B] Sorry for not typing all those things out from my phone.. How can I get C1? And how can I put the solution in the required format? (I don't know how to put it in summation sign... and i cannot even solve...
  12. lc99

    Simple Harmonic Motion with Linear Momentum

    Homework Statement Homework Equations T = 2pi * sqrt(m/k) mv =m2v2 (LM)[/B]The Attempt at a Solution [/B] So T2 depends on the mass and not velocity. So i can find T2 = 2pi * sqrt([m/2]/k) For A2 , i know that the amplitude before any collision is 1/2m1v1^2 = 1/2kA1^2 so solving that, i...
  13. Aleoa

    I Proving the Linear Transformation definition

    HI .I'm trying to prove that, for a linear transformation, it is worth that: f(a\bar{x}+b\bar{y})=af(\bar{x})+bf(\bar{y}) for every real numbers a and b. Until now, I have proved by myself that f(\bar{x}+\bar{y})=f(\bar{x})+f(\bar{y}). and , using this result i proved that: f(a\bar{v}) =...
  14. Clara Chung

    Linear fictitious force problem

    Homework Statement Homework EquationsThe Attempt at a Solution I understand the solution of the problem, but I want to know why don't my approach work. I didn't change the frame of reference. I use the regular method of solving a torque problem. (mgsinθ - mAcosθ)(L/2) = mL2/3 θ'' which is...
  15. DonDiablo

    Solving Differential Equations: Falling Objects & Linear Gravity

    Hy folks, Upfront I want to apologize for my writing and my dissability to use correct symbols to ease readability of the example. Ok now that that's done I just want to start upfront. If we set a usual example of an object falling from a tower with a height of x meters and assume that the...
  16. S

    I Linear mapping of a binary vector based on its decimal value

    Given an ##N## dimensional binary vector ##\mathbf{v}## whose conversion to decimal is equal to ##j##, is there a way to linearly map the vector ##\mathbf{v}## to an ##{2^N}## dimensional binary vector ##\mathbf{e}## whose ##(j+1)##-th element is equal to ##1## (assuming the index starts...
  17. Pattarasak

    When I analyze elliptical polarization using a linear polarizer....

    hello everyone,I want to construct the elliptical polarization in Jones's Matrix form . So, I use the linear polarizer to analyze this polarization. where the Intensity ,which obtain by using the linear polarizer,come from? by integrating overall angle or the other way ?
  18. V

    Linear measurement of a curved part

    Below is a part proposed to be manufactured. The inspection of this part poses a problem, because vernier callipers only measure distances between parallel surfaces. I know there is some photographic method of measuring distances, however I would like to know is there no way of physically...
  19. rishi kesh

    Lorentz force and linear motors

    I have a picture below please have a look at it. The guy is using two conducting rods placed parallel to each other and placed below it are magnets. When battey is connected to rods and a conductor is placed over them it starts to move.That works according to flemmings left hand rule. I was...
  20. M

    MHB Why use point-slope form for linear equations?

    Why is y = mx + b called the slope-intercept form? Why is m the chosen variable for slope? Why is b the chosen variable representing the y-intercept?
  21. R

    MHB Solving Linear Equations d= Lh/R1+R2 for L

    How do i solve d= Lh/R1+R2. How do i also solve d= LM/R2+R1.
  22. avner yakov

    A Error estimation in linear regression

    I have a data set of 11 predictors and one response for 1000 observation and i want to do linear regression. I also have measurements errors of the predictors (also 11X1000 matrix) and i need to count for them in the total error estimation. how can i do that?
  23. mertcan

    Bland rule proof linear programming

    <Moderator's note: Continued from a technical forum and thus no template. Re-opening has been approved by moderator.> Hi, my question is related to simplex algorithm anticycling rule called Bland's rule. While I was working with the proof in the link...
  24. M

    MHB Angular and Linear Speed....Part 2

    Assume that the Earth is a sphere with a radius of 3,960 miles and a rotation of 1 revolution per 24 hours. A. Find the angular speed. Express your answer in units of radians/sec, and round to two significant digits. B. Find the linear speed of a point on the equator. Express the answer in...
  25. M

    MHB Angular and Linear Speed....Part 1

    You are given the rate of rotation of a wheel as well as its radius. For A-C, determine the following: A. The angular speed, in units of radians/sec. B. The linear speed, in units of cm/sec, of a point on the circumference of the wheel. C. The linear speed, in cm/sec, of a point halfway...
  26. Mr Davis 97

    I Solutions to equations involving linear transformations

    I have learned that for matrix theory, for ##A \vec{x} = \vec{b}##, if there exists a particular solution ##p##, then every solution looks like ##p+k##, where ##k \in \ker A##. Can someone help me find material on this online, but only in the context of general linear transformations? For...
  27. L

    MHB Question about proof of the linear independence of a dual basis

    This is from Kreyszig's Introductory Functional Analysis Theorem 2.9-1. Let $X$ be an n-dimensional vector space and $E=\{e_1, \cdots, e_n \}$ a basis for $X$. Then $F = \{f_1, \cdots, f_n\}$ given by (6) is a basis for the algebraic dual $X^*$ of $X$, and $\text{dim}X^* = \text{dim}X=n$...
  28. S

    Is the relationship between power, torque and RPM a linear relationship?

    Hi I have a problem dealing with rpm power and torque. I have a motor powering a gearbox which has a 20:1 ratio. The power and rpm are not an issue so i have not looked at those figures but the motor produces 240Nm of torque. Since it is a 20:1 gearbox, i have assumed the torque is 20 times...
  29. Math Amateur

    MHB Norm of a Linear Transformation: Proving Homogeneity From Definition - Peter

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows...
  30. Math Amateur

    MHB Help with Proof of Junghenn Proposition 9.2.3 - A Course in Real Analysis

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows: In the above...
  31. Math Amateur

    I Norm of a Linear Transformation .... Another question ....

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on ##\mathbb{R}^n##" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows: In the...
  32. C

    MHB Example of systems of the differential linear equations

    Dear Everybody, I have a question about an example: "Solve the system: $x'(t)=3x(t)-4y(t)+1$ $y'(t)=4x(t)-7y(t)+10t$ We write the system using the operator notation: $(D-3)[x]+4y=1$ $-4x+(D+7)[y]=10t$ We can eliminate x from this system by adding 4 times the first equation to $(D-3)$...
  33. T

    I ANOVA and Linear Regression Resource

    Hello, Can someone please let me know of a resource (book or other) that explains how to use ANOVA in linear regression? I didn't even know what ANOVA was until some days ago so I'm looking for something that explains it thoroughly with deductions. The resources I've read focused solely on...
  34. Math Amateur

    MHB Norm of a Linear Transformation .... Junnheng Proposition 9.2.3 .... ....

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows...
  35. Math Amateur

    I Norm of a Linear Transformation .... Junghenn Propn 9.2.3 ....

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on ##\mathbb{R}^n##" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows: In the...
  36. F

    Linear algebra matrix to compute series

    Post moved by moderator, so missing the homework template. series ##{a_n}## is define by ##a_1=1 ## , ##a_2=5 ## , ##a_3=1 ##, ##a_{n+3}=a_{n+2}+4a_{n+1}-4a_n ## ( ##n \geq 1 ##). $$\begin{pmatrix}a_{n+3} \\ a_{n+2} \\ a_{n+1} \\ \end{pmatrix}=B\begin{pmatrix}a_{n+2} \\ a_{n+1} \\ a_{n} \\...
  37. amerikantech

    Linear equation, point slope conversion

    Homework Statement convert: Y − 200 = −4 (X − 15) to X = −0.25 ⋅ Y + 65. with a given Δy/Δx = -36/4 Homework Equations point slope, slope intercept The Attempt at a Solution I understand point slope, slope intercept, and standard form, I understand how to convert one to the other, but I...
  38. R

    Linear motion question -- Drops falling from a dripping faucet

    Hello, I am a new member looking for the answer to a question I recently had on an exam. I will not know if I got it right for up to 6 weeks so am curious. Question is: A faucet drips water at 5 drops per second. calculate the distance in metres between the first and second drop after the first...
  39. L

    Linear Algebra - Find an orthogonal matrix P

    A problem that I have to solve for my Linear Algebra course is the following We are supposed to use Mathematica. What I have done is that I first checked that A is symmetric, i.e. that ##A = A^T##. Which is obvious. Next I computed the eigenvalues for A. The characteristic polynomial is...
  40. A

    I How to find a solution to this linear ODE?

    I want to find solution to following ODE $$ \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) $$ I have solved it with integrating factor method with ## I=\exp^{\int \frac{1}{D} \alpha^2 dt} ## as integrating factor and ##\frac{K}{S_s} = \frac{1}{D} ## I have...
  41. Math Amateur

    MHB Total Derivatives and Linear Mappings .... D&K Example 2.2.5 ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of Example 2.2.5 ... ... Duistermaat and Kolk's Example 2.2.5 read as follows: In the above text by D&K we...
  42. Math Amateur

    I Total Derivatives and Linear Mappings .... D&K Example 2.2.5

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of Example 2.2.5 ... ... Duistermaat and Kolk's Example 2.2.5 read as follows: In the above text by D&K we...
  43. A

    A How to simplify the solution of the following linear homogeneous ODE?

    During solution of a PDE I came across following ODE ## \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) ## I have to solve this ODE which I have done using integrating factor using following steps taking integrating factor I=\exp^{\int \frac{1}{D} \alpha^2...
  44. KFSKSS

    B Need some help with understanding linear approximations

    Hello. My problem is that I began with Linear Approximation and I'm terribly stuck. I have problems understanding its very concept and with calculations. (It may sound stupid but I'm autodidact and I'm studying mathematics in english (not my mother tongue) and sometimes it gets hard). It would...
  45. yecko

    Solving Linear ODEs: How to Obtain the Highlighted 1?

    Homework Statement How to obtain the "1" highlighted? Homework Equations multiply by μ then by dt (integration )to both sides The Attempt at a Solution [/B] lets just consider part "2y/t": ∫2y/t from pi/2 to t =ln(t^2/(pi^2/4)) from pi/2 to t =ln1-ln(t^2/(pi^2/4)) = -ln(t^2/(pi^2/4)) how...
  46. CopyOfA

    A Linear regression with discrete independent variable

    Hey, I have a problem where I have a discrete independent variable (integers spanning 1 through 27) and a continuous dependent variable (50 data points for each independent variable). I am wondering about the best method of regression here. Should I just fit to the mean or median? Is there a way...
  47. Anupama

    I Linear attenuation coefficient

    Is linear attenuation coefficient a constant for a given material or will it depend on the thickness of the material?
  48. S

    Covariance of partitioned linear combination

    Homework Statement Given random vector ##X'=[X_1,X_2,X_3,X_4]## with mean vector ##\mu '_X=[4,3,2,1]## and covariance matrix $$\Sigma_X=\begin{bmatrix} 3&0&2&2\\ 0&1&1&0\\ 2&1&9&-2\\ 2&0&-2&4 \end{bmatrix}.$$ Partition ##X## as $$X=\begin{bmatrix} X_1\\X_2\\\hline X_3\\X_4\end{bmatrix}...
  49. Math Amateur

    MHB Linear Mappings are Lipschitz Continuous .... D&K Example 1.8.14 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.8.14 ... ... The start of Duistermaat and Kolk's Example 1.8.14 reads as...
  50. M

    Inconsistent Linear Systems: Solving for k

    Homework Statement For what value(s) of k is the linear system with augmented matrix inconsistent? Homework Equations None. The Attempt at a Solution I know that inconsistent means no solution. So do I set k=2k and solve for k?
Back
Top