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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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/
  6. N

    I Are there any linear quantum gravity theories?

    Are there any linear quantum gravity theories out there with respect to the wave function?
  7. 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 . | . . | ...
  8. 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...
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. L

    MHB Find the General Expression for a Linear Transformation

    I don't quite get this question, how is it done ?
  17. 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...
  18. 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...
  19. 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...
  20. 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...
  21. L

    Linear 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}##<=>...
  22. 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...
  23. 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?
  24. 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})...
  25. 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?
  26. 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...
  27. 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...
  28. 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.
  29. 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...
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. 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...
  35. 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...
  36. A

    Comparing direct and iterative solution of linear equations

    I want to understand which of these is computationally expensive (in the sense of computational time) which is more accurate. Also I want to understand which of these two problems (computations time + accuracy) of iterative methods are addressed by multi-grid methods?
  37. M

    MHB Can the vectors be written as a linear combination?

    Hey! :o We have the vectors $\overrightarrow{a_1}=\begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix}, \overrightarrow{a_2}=\begin{pmatrix}-1 \\0 \\ 2\end{pmatrix}, \overrightarrow{a_3}=\begin{pmatrix}7 \\ 8 \\ 6\end{pmatrix}$. I have shown that these vectors are linearly dependent: $\begin{bmatrix}...
  38. jamalkoiyess

    Linear Algebra How Does Linear Algebra Help with Differential Equations?

    Hello PF, I have just finished my first semester in college and did Calc. 3. Now for the spring semester i have to take differential equations and i have been given the advice that linear algebra comes in handy when dealing with DEs. So can anyone recommend a good introduction for linear algebra...
  39. T

    Compare these two Linear Algebra courses

    Hi! First off, I am actually a math / econ major. I hope I'm still welcome here I am trying to figure out if it's worth it to take both of these courses or just one of them. I have not taken LA before. Course 1: Addition, subtraction and scalar multiplication of vectors, length of vector...
  40. C

    I Question On Linear Independence

    We were going over linear independents in class and my professor said that if y1 and y2 are linearly independent then the ratio of y2/y1 is not a constant, but he never explained why it is not a constant.
  41. T

    I Linear regression on data collection error

    Hi I've collected few sets of data and obtained significant different linear regression (R^2) in 2 particular sets of data . Does that indicates the 2 sets of data is not validated which might due to data collection error? For example, 20 sets of data contain linear regression of 0.900+...
  42. A

    MHB Proving Linear Independence

    In a problem I am working on, it is given that $V_1, V_2, ... , V_n$ are mutually perpendicular vectors in a space defined with a certain scalar product. I need to prove or disprove that $V_i$ are linearly independence regardless of any definition of scalar product. I think the solution should...
  43. PhotonSSBM

    Potenital of a cylindrical, linear dialectric in a E-Field

    Homework Statement A very long cylinder of linear dialectric material is placed in an otherwise uniform electric field ##E_0##. Find the resulting field within the cylinder. (The radius is ##a##, the susceptibility ##X_e##, and the axis is perpendicular to ##E_0##) Homework Equations Boundary...
  44. A

    Two Coaxial Linear Magnetic Materials

    Homework Statement Consider a two-layered cylindrical wire with inner-layer permeability μ1 and outer-layer permeability μ2. A line current I runs through the center in the z direction. Calculate the bound currents and the magnetic field produced by the bound currents. Homework Equations [1]...
  45. StarWarsNerd

    B Calculating Linear and Angular Quantities from Applied Force

    In the image above, a centroid with radius 1 is depicted. F1 is pointing directly at point A (which is the center of the circle), and F2 is pointing directly at point B. The radius for finding the torque would be the perpendicular between the center of the object and the force vector, so r1...
  46. W

    Linear Differential Equations and Linear Operator Problem

    Homework Statement I'm not sure how to approach this. The question involves linear operators and a non-homogenous differential equation. Here is the question: https://s15.postimg.org/cdmw80157/Capture.png Homework Equations They are given in the question The Attempt at a Solution I really...
  47. Valour549

    I 2nd Order Linear ODE w/ nonconstant coefficient

    xy'' + 2xy' - y = 0 Honestly no clue where to start, Wolfram Alpha gives a rather complex answer lol (http://www.wolframalpha.com/input/?i=xy%27%27%2B2xy%27-y%3D0)
  48. M

    I Is the set {e^x, x^2} linearly independent?

    Hello all. I have a question about linear dependency. Suppose we have a set ##S## of functions defined on ##\mathbb{R}##. ##S = \{e^x, x^2\}##. It seems very intuitive that this set is linear independent. But, we did something in class I'm unsure about. Proof: Let ##\alpha, \beta \in...
  49. nysnacc

    Are the Wronskians set up and compared in this linear independent check?

    Homework Statement Homework Equations The Attempt at a Solution I set up the Wronskian?? and compare them?
  50. O

    2-D Linear Momentum, Collision Problem

    Good evening! I'm having trouble completing an assignment related to inelastic collisions and momentum. Homework Statement Here is the statement: You are in great shape on this hot July morning. You are working as an intern for the Montreal Police (SPVM) Collision Investigation unit. You...
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