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. M

    Linear actuator use in all electric mini-digger

    Hi all, I've got this idea in my head of building a mini-loader like a http://www.multione.com/mini-articulated-loaders/ . I'm interested in replacing the hydraulics with linear actuators and wondered if anyone here could shed some thoughts/advice. Given the driveline would also be fully...
  2. Mr Davis 97

    Linear independence of polynomials of different degree

    Homework Statement Let S be a set of nonzero polynomials. Prove that if no two have the same degree, then S is linearly independent. Homework EquationsThe Attempt at a Solution We will proceed by contraposition. Assume that S is a linearly dependent set. Thus there exists a linear dependence...
  3. Mr Davis 97

    Linear Independence of a Set of Vectors

    Homework Statement Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. Homework EquationsThe Attempt at a Solution I think that that it would be easier to prove the logically equivalent statement: Prove that a set S of vectors...
  4. Dethrone

    MHB Optimizing Linear Regression Cost Function

    I'm trying to optimize the function below, but I'm not sure where I made a mistake. (This is an application in machine learning.)$$J(\theta)=\sum_{i=1}^n \left(\sum_{j=1}^{k}(\theta^Tx^{(i)}-y^{(i)})_j^2\right)$$ where $\theta$ is a $n$ by $k$ matrix and $x$ is a $n$ by 1 matrix...
  5. michaelgtozer

    I 1st year linear algebra question

    given P(-1,1,2), Q(-3,0,4), R(3,2,1), find an equation of the line through P that is parallel to the line through Q and R. All the words after the given three points really confuse me and I just need some help on where to start to tackle this problem. Thanks
  6. D

    Engineering What Voltage Should Be Applied to Achieve a Change in Capacitor Energy of 4uJ?

    Homework Statement This is the second problem from our test preparation cycle and i partially solved and understood it. The circuit in the picture is in the state of switch being open (STATE 1: SWITCH OPEN). Calculate ##E1## so that the change in the electrical energy of the capacitor is...
  7. D

    Engineering Understanding thevenin's circuit in linear circuits

    Homework Statement This is one of the problem i had had some help in solving but some things remain unclear to me and i will highlight them by posting the exact question in the brackets next to the line of solution so that you can see what bothers me. Hopefully you can help me better understand...
  8. M

    MHB Injective linear mapping - image

    Hey! :o Let $F$ be a field and $V,W$ finite-dimensional vector spaces over $F$. Let $f:V\rightarrow W$ a $F$-linear mapping. We have to show that $f$ is injective if and only if for each linearly independent subset $S$ of $V$ the Image $f(S)$ is linearly independent in $W$. I have done the...
  9. Eclair_de_XII

    How would you explain to a dummy how a diff. eq. is linear?

    Homework Statement This is the definition I got from my book: "A crucial classification of differential equations is whether they are linear or nonlinear. The ordinary differential equation ##F(t,y,y',...,y^{(n)})=0## is said to be linear if F is a linear function of the variables ##y...
  10. N

    A Is asymptotic safety linear with respect to the wavefunction

    AS is a local quantum field theory, so I'm assuming that it is? What do you guys think?
  11. U

    Linear momentum in oblique collisions and generally

    In an oblique collision my understanding is that linear momentum is conserved in all directions (x, y, normal, tangential). But in a constrained oblique collision, does this change? For example if we had a block lying between two frictionless surfaces with an angled face ( a slope on one face)...
  12. Adgorn

    Linear algebra problem: linear operators and direct sums

    Homework Statement Homework Equations N/A The Attempt at a Solution I proved the first part of the question (first quote) and got stuck in the second (second quote). I defined Im(E1) as U and Im(E2) as W and proved that v=u+w where v ∈ V, u ∈ U and w ∈ W. After that however I got stuck at...
  13. T

    A Is linear polarization not expected from an astrophysical maser

    I was reading this text where they state: "Many cosmic masers are observed to have both circular and linear polarization although the reasons for linear polarization are not well understood." Just curious, aren't there any molecules such as OH, CH4, or water masers that have transitions that...
  14. G

    I Quantum Superposition, Linear Combinations and Basis

    Hello, just a quick question. I am aware that a a state in a space can be written as a linear combination of the basis kets of that space ψ = ∑ai[ψi] where ai are coefficients and [ψi] are the basis vectors. I was just wondering is this a linear superposition of states or just a linear...
  15. squelch

    Determining if Systems are Linear

    Homework Statement For each of the following, determine if the system is linear. If not, clearly state why not. (a) ##y''(t)+15y'(t)+sin(y(t)))=u(t)## (b) ##y''(t)-y'(t)+3y(t)=u'(t)+u(t)## (c) ##y'(t)=u(t)## and ##z'(t)=u(t)-z(t)-y(t)## Homework Equations None The Attempt at a Solution...
  16. V

    I Linear algebra ( symmetric matrix)

    I am currently brushing on my linear algebra skills when i read this For any Matrix A 1)A*At is symmetric , where At is A transpose ( sorry I tried using the super script option given in the editor and i couldn't figure it out ) 2)(A + At)/2 is symmetric Now my question is , why should it be...
  17. B

    I Robust linear estimation to guess parameters

    Hi guys! I am working on a project and I need to perform a robust linear estimation as described here: https://postimg.org/image/t9b9fybtt/ I have to do this to the equation 43: https://postimg.org/image/x4eniiuzl/ I have assumed that the final function that I have to minimize, basing on both...
  18. W

    Show: Elementary row operations don't affect solution sets

    Homework Statement Show that elementary row operations don't affect solutions sets in linear systems Homework Equations - The Attempt at a Solution It's pretty easy to come up with a random linear system and perform ERO on them and showing that solutions are not affected, but is there all...
  19. M

    MHB Solving Linear Mapping Problems with Matrix $M$ in $\mathbb K^2$

    Hey! :o Let $\mathbb{K}$ be a field. Find a matrix $M\in \mathbb{K}^{2\times 2}$ such that for the linear mapping $f:\mathbb{K}^2\rightarrow \mathbb{K}^2, x\mapsto Mx$ it holds that $f\neq 0$ and $f^2:=f\circ f=0$. Let $V$ be a $\mathbb{K}$-vector space and $\psi:V\rightarrow V$ be a...
  20. M

    Making an exponential function linear

    Homework Statement a= eD/R*T*G make a linear equations and calculate the value for D and G R=8,3 and constant D,G=constant T= variable Homework Equations y=ax+b y=numberax*bThe Attempt at a Solution ax= E/(R*T) x= 1/T a= E/R y= (E/R)*x+G I don't know how to move on and if this is even correct/
  21. N

    I Are there any linear quantum gravity theories?

    Are there any linear quantum gravity theories out there with respect to the wave function?
  22. A

    Linear Regression with Measurement Errors

    Hello, I have a set of data, two columns, and each datum has its measurement error like illustration shows below: x | y --------------|----------------- x1+/-xe1 | y1+/-ye1 . | . . | ...
  23. B

    I How can robust linear estimation be carried out in a non-linear context?

    Hi and thanks to everyone for his/her attention. I have to minimize a function that depends on several parameters. The aim of minimizing that function is to actually guess these parameters, which are unknown. The thing is that the author of the pdf from which I have to make the calculations...
  24. W

    How does linear velocity of a spinning object affect its rebound

    Consider the case where a basketball rotating about its center of mass strikes a wall head-on. The velocity given it parallel to the wall due to the spin is caused by the friction between the wall and the ball as the ball is sliding against the wall during the collision. Friction is proportional...
  25. Hypercube

    Linear Algebra / Linear Maps (Transformations)

    This isn't really a homework question, I just need help understanding the example: ===================== ==================== So transformation takes complex n-tuple as input, and it seems output is also a complex n-tuple (which is what makes it "operator"). But permutations of n entries is...
  26. D

    Engineering Solving Linear Circuit Homework: Calculating ΔP for Ideal Generator E1

    Homework Statement In the circuit above ##E_1=E_2=E_3##, ##Ig_1=Ig_2=Ig_3## and ##R_1=R_2=R_3=R_4##. In the case when the switched is closed the power of the ideal generator ##E_2## equals ##P_{E_2}=-30W##. Then the switch opens. Calculate the ##ΔP_{E_1}##. Homework Equations 3. The Attempt...
  27. R

    A Linear enercy transform (LET) and momentum

    I have some understanding of Liner Energy Transfrom (LET), but when I compare it with other terminologies such as momentum, then it gets little contradicting. For example, I know that if talk about particles and ions, then a gold ion (AU) has much higher LET than an Iron particle (Fe). And...
  28. Euler2718

    I Can Nonlinear Equations be Linearized Using Free Parameters?

    I am given the equations of Lorenz with respect to deterministic non-periodic flow: \frac{dX}{dt} = Pr(Y-X), X(0)=X_{0} \frac{dY}{dt} = -XZ + rX - Y, Y(0) = Y_{0} \frac{dZ}{dt} = XY-bZ, Z(0) = Z_{0} where Pr is the Prandtl number, r = Ra/Rac is the ratio of the Rayleigh number to its...
  29. D

    A Nonlinear regression which can be partially reduced to linear regression

    I encountered several times the following problem: Say I have a variable y dependent in a nonlinear way on m parameters ##\{x_i\}##, with ##i \in \{1,m\}##. However there is a linear relation between n>m functions ##f_j\in{x_i}##, i.e., ##y=\sum_j z_j f_j##. So I can get a solution of my problem...
  30. Mr Davis 97

    I Linear independence of functions

    Is there a difference between the linear independence of ##\{x,e^x\}## and ##\{ex,e^x\}##? It can be shown that both only have the trivial solution when represented as a linear combination equal to zero. However, the definition of linear independence is: "Two functions are linearly independent...
  31. L

    MHB Find the General Expression for a Linear Transformation

    I don't quite get this question, how is it done ?
  32. E

    How to maximum apparent power through a linear transformer?

    Homework Statement given a linear transformer rated for 50kVA, 11000V/120V How do I find the maximum apparent power? The prompt is to maximum apparent power while keeping the Load RMS voltage between 114V and 126V. Vsource has a peak voltage of 11000*sqrt(2). The circuit looks something like...
  33. I

    I What Blower Specifications Are Needed for an Effective Linear Air Track?

    I am a Physics teacher. I made a linear track of length 2m with 400 holes(two rows of 1mm dia). Somewhere I read household vacuum cleaner can be used as a blower. But it does not create sufficient pressure to lift the cart. What should be the specifications of the blower which can be used...
  34. A

    Fortran How to change data from linear scale to log scale

    I am trying to calculate and draw a relation between (IL) and the Growth rate (m). I have used Il as range from 10^15 to 10^18 in the calculation and when I draw the relation (x-axis ) is Il and (y-axis ) is m. I converted the IL to be in log scale. I noticed unacceptable output. So, I am...
  35. W

    2nd Order Linear ODE-Derivation of system-issue

    Homework Statement How exactly they combined equation1 and equation2 and got that system? I don't get that part. Homework Equations A*(dy/dt)= -k*y eq1 A*(dz/dt)=ky-kz eq2 The Attempt at a Solution I tried substituting the 1st ky in the 2nd equation and then differentiating but I don't...
  36. L

    How Does Gaussian Elimination Solve This System of Equations?

    Homework Statement system of equations is as follows x+y=1 2x+y-z=1 3x+y-2z=1 ##\begin{cases} x+y=1 |*(-1)\\2x+y-z=1\\3x+y-2z=1 \end{cases}## Homework Equations Gaussian elimination method technique The Attempt at a Solution ##\begin{cases} x+y=1 \\ x-z=0 |*(-2)\\ 2x-2z=0 \end{cases}##<=>...
  37. F

    MHB Find the diameter of a circle given linear velocity?

    [SOLVED] Find the diameter of a circle given linear velocity? Hello all! I need help with a certain type of problem. I do not know how I can find the diameter of a circular object given it's linear velocity. Here is an example problem, and I would love any explanation you could give me! Thanks...
  38. J

    Is rotational Work the same as linear Work?

    Linear work is F*D and rotation is 1/2Iω2, but if a problem as me rotational energy (Rotation worl = KE?) of a wheel and I have the linear work, can I just set Wrotation=Wlinear?
  39. T

    I Preconditioning for linear equations

    I understand that for many iterative methods, convergence rates can be shown to depend on the condition number of the coefficient matrix A in the linear equation $$Ax=y.$$ Therefore, if a preconditioner satisfies $$P \approx A,$$ then by solving the transformed linear equation $$(AP^{-1})...
  40. V

    Linear Trig Equations: Solving sin(x + pi/4) = √2 cos x

    Homework Statement Solve sin (x + pi/4) = √2 cos x Homework EquationsThe Attempt at a Solution sinx*cos(pi/4) + cosx*sin(pi/4) = √2 cos x √2/2 sinx + √2/2 cosx = √2 cos x not sure if I am on the right track? or where would I go from here? would I bring √2 cos x to the left side?
  41. RJLiberator

    ODE: System of Linear Equations usuing Diff. Operator

    Homework Statement This is an ordinary differential equation using the differential operator. Given the system: d^2x/dt - x + d^2y/dt^2 + y = 0 and dx/dt + 2x + dy/dt + 2y = 0 find x and y equation Answer: x = 5ce^(-2t) y = -3ce^(-2t) Homework EquationsThe Attempt at a Solution We change...
  42. e2m2a

    Conservation of linear momentum when wheels are included

    I am not a student. This is not a homework question. I am a 64 year old man who wants to understand a principle of physics. Suppose there is a large block of matter, designated as m2, which lies on a flat surface that initially is at rest. At some point in time a small block of matter...
  43. Boudy

    A Obtaining Step Response for Linear Systems in Sinusoidal Frequency Domain

    The transfer function of a linear system is known in the sinusoidal frequency domain. It is given in its final form as a complex function of the angular frequency ω (not jω ). How to obtain the step response? Thanks in advance.
  44. MrsM

    Linear Algebra: characteristic polynomials and trace

    The question is : Is it true that two matrices with the same characteristic polynomials have the same trace? I know that similar matrices have the same trace because they share the same eigenvalues, and I know that if two matrices have the same eigenvalues, they have the same trace. But I am...
  45. ramzerimar

    Project: Speed and force control of a linear actuator

    My background is in mechanical engineering, but I'm working on a project for which I need to design a control system for a linear actuator. I want to design the actuator using a roller screw mechanism, which in turn will be conected to a 3 horsepower electric motor through gear reduction. I need...
  46. A

    Moment of inertia of a wheel: linear to angular motion

    Homework Statement You are asked to measure the moment of inertia of a large wheel for rotation about an axis perpendicular to the wheel at its center. You measure the diameter of the wheel to be 0.600 m. Then you mount the wheel on frictionless bearings on a horizontal frictionless axle at...
  47. C

    Linear Algebra Good reading on Applied Linear Algebra?

    I've been studying graduate level Linear Algebra from Steven Roman's Advanced Linear Algebra (Springer, GTM). It is a terrific book, but many of the concepts are extremely abstract so that I find it difficult to retain what I've learned. Can anyone point me to some books/reading on the...
  48. K

    Math proof: Linear Independence

    Homework Statement How can I show that if a vector (in a vector space V) cannot be written as a linear combination of a linearly independent set of vectors (also in space V) then that vector is linearly independent to the set? Homework Equations To really prove this rigorously it would make...
  49. E

    Courses What Math Course is Best Paired with Linear Algebra?

    I'm currently an applied math major. I'm creating a schedule for my next semester and I have the choice to take either complex variables or vector analysis with linear algebra and a college geometry course(elective of choice), but I don't know which pairing will be less stressful. I am currently...
  50. Euler2718

    Linear Dependence and Non-Zero Coefficients

    Homework Statement True or False: If u, v, and w are linearly dependent, then au+bv+cw=0 implies at least one of the coefficients a, b, c is not zero Homework Equations Definition of Linear Dependence: Vectors are linearly dependent if they are not linearly independent; that is there is an...
Back
Top