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. T

    All possible planes, given two points

    Homework Statement Find the equation of all planes containing the points P(2, -1, 1) and Q(1, 0, 0) Homework EquationsThe Attempt at a Solution I use PQ to get a vector, (-1, -1, 1). I some how need to use another vector so I can use the cross product to find the planes. So i let another...
  2. N

    How to set up two linear actuators to share loading?

    I have a structure that needs to be pushed by a set of actuators. It is because 1 actuator's loading may not be sufficient and changing it to a more powerful model will just sacrifice my space which is not favorable. Can anyone tell me if it is a common way of doing it? If so, how do I manage...
  3. pellman

    I Aren't all linear operators one-to-one and onto?

    Let W be a vector space and let A be a linear operator W --> W. Isn't it the case that for any such A, the kernel of A is the zero vector and the range is all of W? And that it is one-to-one from linearity? I ask because an author I am reading goes through a lot of steps to show that a certain...
  4. Alanay

    How do I graph -66/-99 from -12/11 x 10 + 54/11?

    Okay, so I'm down to the last equation. -12/11 x 10 + 54/11 I get -66/-99. Is this right? If so how do I put it into the graph. -12/11 x 10 = -120/110 + 54/11 = -66/99 (I think I've went wrong somewhere)
  5. Ryan Reed

    How calculate iris radius and cavity radius in accelerator?

    In linear accelerators that use a disk loaded structure (traveling wave), how would you calculate the iris(disk hole) radius, cavity radius, and disk thickness according to the wavelength
  6. M

    MHB Decide h so that the linear system has infinite solutions

    Hi! I'm need some help with this question: Decide $h$ so that the linear system $Ax=b$ has infinite solutions. $$A=\pmatrix{ 5 & 6 & 7 \cr -7 & -4 & 1 \cr -4 & 4 & 16 \cr}$$ $$b=\pmatrix{ 6 \cr 30 \cr h \cr}$$ I solved a similar question before but with A being a 2x2 matrix (and B a 2x1) and...
  7. D

    Matrices and Systems of Linear Equations

    Homework Statement Homework EquationsThe Attempt at a Solution No clue really. I went ahead and tried to simplify this by turnining it into an echelon matrix. But I am sort of stuck now, since I can't divide by -k because I don't know whether or not it is equal to 0?
  8. G

    How to plot the linear system solutions with multiple solutions?

    Homework Statement Solve the linear system of equations: ax+by+z=1 x+aby+z=b x+by+az=1 for a,b\in\mathbb R and plot equations and solutions in cases where the system is consistent. Homework Equations -Cramer's rule -Kronecker-Capelli's theorem The Attempt at a Solution Using Cramer's rule, we...
  9. G

    Solution set: S = {(8 + 7z, 6 + 5z, z, 1) : z ∈ ℝ}

    Homework Statement Plot the solution set of linear equations x-y+2z-t=1 2x-3y-z+t=-1 x+7z=8 and check if the set is a vector space. 2. The attempt at a solution Augmented matrix of the system: \begin{bmatrix} 1 & -1 & 2 & -1 & 1 \\ 2 & -3 & -1 & 1 & -1 \\ 1 & 0 & 7 & 0 & 8 \\...
  10. G

    MHB Proving $(T^2-I)(T-3I) = 0$ for Linear Operator $T$

    Problem: Let $T$ be the linear operator on $\mathbb{R}^3$ defined by $$T(x_1, x_2, x_3)= (3x_1, x_1-x_2, 2x_1+x_2+x_3)$$ Is $T$ invertible? If so, find a rule for $T^{-1}$ like the one which defines $T$. Prove that $(T^2-I)(T-3I) = 0.$ Attempt: $(T|I)=\left[\begin{array}{ccc|ccc} 3 &...
  11. G

    MHB Linear Subspaces: Properties and Examples

    For the brief explanation: $\mathcal{P}$ contains $0$ by choice $p(x) = 0$ and polynomial plus a polynomial is a polynomial, and a scalar times a polynomial is a polynomial. So $\mathcal{P}$ is a non-empty subset of $\mathcal{C}^{\infty}$ that's closed under addition and scalar multiplication...
  12. G

    MHB Linear transformation and its matrix

    1. Show that the map $\mathcal{A}$ from $\mathbb{R}^3$ to $\mathbb{R}^3$ defined by $\mathcal{A}(x,y,z) = (x+y, x-y, z)$ is a linear transformation. Find its matrix in standard basis. 2. Find the dimensions of $\text{Im}(\mathcal{A})$ and $\text{Ker}(\mathcal{A})$, and find their basis for the...
  13. J

    I Is the Linear Ehrenfest Paradox Accurate for Circular Motion?

    Here is a linear version of the Ehrenfest paradox with the goal of understanding the observations of someone in motion in the scenario, then solicit your views on whether the calculations are correct and whether one can extend it to circular motion.Consider a one dimensional train of proper...
  14. Y

    Linear Algebra - Hooke's Law Problem

    Homework Statement For the system of springs a) Assemble the stiffness matrix K and the force-displacement relations, K*u = f b) Find the L*D*L^T factorization of K. Use Matlab to solve c) Use the boundary conditions and applied forces to find the displacements Homework EquationsThe Attempt...
  15. Y

    Linear Algebra - Left Null Space

    Homework Statement I am given the follow graph and asked to find the left null space Homework EquationsThe Attempt at a Solution First I start by transpose A because I know that the left null space is the null space of the incidence matrix transposed. I then reduce it to reduce row echelon...
  16. D

    Understanding Linear Momentum of Waves with No Mass

    Hi people, I studying electromagnetic waves (intermediate) and I don't understand how the expression for linear momentum of a wave is obtained, if the wave doesn't carry any mass. In particular, I have to explain why the radiation pressure on a perfect absorber is half that on a perfect...
  17. T

    I Linear Transformation notation

    I'm confused about the notation T:R^n \implies R^m specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?
  18. G

    MHB How can the Wronskian be used to determine linear independence?

    I'm asked to check whether $\left\{1, e^{ax}, e^{bx}\right\}$ is linearly independent over $\mathbb{R}$ if $a \ne b$, and compute the dimension of the subspace spanned by it. Google said the easiest way to do this is something called the Wronskian. Is this how you do it? The matrix is: $...
  19. T

    B What are the values in a vector?

    I'm trying to understand the concept of vectors. Vectors have magnitude and a direction. When I read vector with some values \textbf{x} = \left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right) = \left(\begin{array}{c}1\\2\\3\end{array}\right) I'm not sure what these values are. Are the values...
  20. R

    Finding Coordinate Matrix for Linear Transformation T

    Homework Statement Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question. Consider the...
  21. A

    Difference between Lyapunov and linear stability criteria

    Dear all, Consider the connection of two electrical circuits. Both circuits, Z1 and Z2, are stable and only one of them is non-passive. I.e., the eigenvalues are located in the LHP but Re{Z2(jw)}<0 in a frequency range. For studying the closed-loop stability, you represent the linear system by...
  22. R

    Linear Algebra matrix linear transformation

    Homework Statement Consider the linear transformation T from V = P2 to W = P2 given by T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2 Let E = (e1, e2, e3) be the ordered basis in P2 given by e1(t) = 1, e2(t) = t, e3(t) = t^2 Find the coordinate matrix...
  23. G

    Linear algebra: Prove the statement

    Homework Statement Prove that \dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L) for every subspace \mathbb{F} and every linear transformation L of a vector space V of a finite dimension. Homework Equations -Fundamental subspaces -Vector spaces The Attempt at a Solution Theorem: [/B]If...
  24. G

    Sum of eigenvectors of linear transformation

    Homework Statement Find all values a\in\mathbb{R} such that vector space V=P_2(x) is the sum of eigenvectors of linear transformation L: V\rightarrow V defined as L(u)(x)=(4+x)u(0)+(x-2)u'(x)+(1+3x+ax^2)u''(x). P_2(x) is the space of polynomials of order 2. Homework Equations -Eigenvalues and...
  25. D

    Matrices/Systems of Linear Equations

    Homework Statement Find the general solution: http://puu.sh/ngck4/95470827b1.png Homework Equations Method: Gaussian Elimination by row operations. The Attempt at a Solution http://puu.sh/ngcml/7722bef842.jpg I am getting the wrong answer( w = -27/5). The solutions provided to me says the...
  26. J

    Linear Algebra: Determine Span of {(1, 0, 3), (2, 0, -1), (4, 0, 5), (2, 0, 6)}

    Homework Statement Determine whether the set spans ℜ3. If the set does not span ℜ3 give a geometric description of the subspace it does span. s = {(1, 0, 3), (2, 0, -1), (4, 0, 5), (2, 0, 6)} Homework EquationsThe Attempt at a Solution I am having trouble with the second part of this problem...
  27. Duncan R

    Linear Algebra A search for a classic, out of print Linear Algebra textbook

    I'm looking for an excellent introductory linear algebra textbook for my second year pure mathematics course. My lecturer highly recommended Introduction to Linear Algebra by Marcus and Minc. She said she has searched for it for many years without success, as it is out of print. I love classic...
  28. Q

    Can I Successfully Take Calc 3 and Linear Algebra at the Same Time?

    Hi all! I have an important decision to make for the summer of 2016 and I need some advice from some who have taken these courses. I need one biology lab elective to graduate, but it is a field lab and it runs from from 5/13 - 6/19. Because it is a field lab, I will not be able to take other...
  29. D

    Solving Systems of Linear Equations (Echelon Matrices)

    Homework Statement find the general solution of the given system of equations: http://puu.sh/ncKaS/57a333f5b9.png Homework Equations Row Echelon Operations The Attempt at a Solution http://puu.sh/ncKcm/3e2b2bd5ab.jpg The correct answer given is x = 1, y = 1, z = 2, w = −3 I have done...
  30. B

    Precision Control of Linear Actuators Using Force Limitation and Feedback Loop

    This might be a simple and pretty basic question, but i have not succeeded on finding any relevant info online, so hopefully someone can help me out. Is it possible to pull and actuator and it resists being pulled with a preset amount of force? What I'm thinking is e.x you have set a preset a...
  31. P

    Linear algebra : Doing a proof with a square matrix

    Homework Statement Show that all square matrix (A whatever) can be written as the sum of a symmetric matrix and a anti symmetric matrix. Homework Equations I think this relation might be relevant : $$ A=\frac{1}{2}*(A+A^{T})+\frac{1}{2}*(A-A^{T}) $$ The Attempt at a Solution I know that we...
  32. Eric V

    Rotational Vs Linear Acceleration

    Hi guys, I'm having a debate with a mechanical engineer friend of mine, and I was wondering if you could help me solve it. I'm not much of a physicist, but honestly I think he might have this one wrong, I just can't remember my old physics classes well enough to calculate and be sure. The...
  33. Joa Boaz

    Rolling without slipping & linear acceleration vector

    Homework Statement Rolling without slipping A) Derive the linear acceleration vector equations for points A, B, C, and O in terms of R, ω, α and θ at this instant. B) R = 0.5 m, ω=-54 r/s and α = 0. Determine the MPH of the vehicle and the vector accelerations of points A, B, C, and O. C) R...
  34. Math Amateur

    MHB Yet Another Basic Question on Linear Transformations and Their Matrices

    I am revising the basics of linear transformations and trying to get a thorough understanding of linear transformations and their matrices ... ... At present I am working through examples and exercises in Seymour Lipshutz' book: Linear Algebra, Fourth Edition (Schaum Series) ... ... At...
  35. Math Amateur

    MHB (Very) Basic Questions on Linear Transformations and Their Matrices

    Firstly, my apologies to Deveno in the event that he has already answered these questions in a previous post ... Now ... Suppose we have a linear transformation T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2 , say ... Suppose also that \mathbb{R}^3 has basis B and \mathbb{R}^2 has basis B'...
  36. G

    Linear algebra: Find the matrix of linear transformation

    Homework Statement Check if L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x) is a linear transformation on \mathbb{R_2}[x]. If L(p)(x) is a linear transformation, find it's matrix in standard basis and check if L(p)(x) is invertible. If L(p)(x) is invertible, find the function rule of it's...
  37. C

    High voltage voltage controlled linear variable resistor

    I have some high voltage (300v+) analog circuits that I want to control digitally which requires the use of a voltage controlled linear resistor that can withstand high voltages. I don't expect the current levels to be that high. I originally settled on LDR optocouplers but it turns out they...
  38. G

    Does LIGO Ruling Out Linear Gravity Theories?

    As the discovery matches templates based on GR, and the regime is of very strong gravitational fields and very high speeds (relativistic speeds), and there is a 90% match between model and measured data, this does rule out linear or quasi linear alternative theories of gravity?
  39. Math Amateur

    MHB Matrices of Linear Transformations .... Example 2.6.4 - McInerney

    I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 2: Linear Algebra Essentials ... and in particular I am studying Section 2.6 Constructing Linear Transformations ... I need help with a basic...
  40. J

    Wave speed on a string of non-uniform linear mass density

    Homework Statement [/B] Consider a long chain of mass m and length L suspended from a tall ceiling. Like any string if one end is disturbed waves will travel along the string. However, the tension in the string is due to its own weight and is not uniform. As such the speed of the wave will be...
  41. G

    Linear algebra: Prove that the set is a subspace

    Homework Statement Let U is the set of all commuting matrices with matrix A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...
  42. W

    Testing for Linear Relation:r^2 vs H_0: slope =0

    Hi All, I am trying to understand better the tests used to determine the existence of a linear relation between two variables X,Y. AFAIK, one way of testing the strength of any linear relationship is by computing ##r^2##, where ##r## is the correlation coefficient; this measures the extend to...
  43. Math Amateur

    MHB Vector Spaces .... Linear Dependence and Indepence .... Basic Proof Required

    In Andrew McInerney's book: First Steps in Differential Geometry, Theorem 2.4.3 reads as follows:https://www.physicsforums.com/attachments/5252McInerney leaves the proofs for the Theorem to the reader ... I am having trouble formulating a proof for Part (3) of the theorem ... Can someone help...
  44. Sirsh

    Conservation of angular momentum & linear momentum

    Two things I'd like to discuss: 1. The conservation of angular momentum. If you have two discs rotating on the same fixed rigid axis, will these nullify each other? I.e. Create no net angular momentum? 2. How / is it possible to convert angular momentum to linear momentum in the sense to be...
  45. Travis McWilliams

    Linear force > Angular torque > Mechanical advantage

    Guys, I'm trying to figure some stuff out, but I'm stumped. I need to figure the what my output would be. I'm applying linear force via hydraulic cylinder. The cylinder will turn a "gear" and in turn turn another. The "gear" is not necessarily a gear, it and the hydraulic cylinder will mate...
  46. S

    Definition of Image of a linear transformation

    Homework Statement The image of a linear transformation = columnspace of the matrix associated to the linear transformation. More specifically though, given the transformation from Rn to Rm: from subspace X to subspace Y, the image of a linear transformation is equal to the set of vectors in X...
  47. R

    Linear system control with unmatched uncertainties

    Hi everyone, I have a problem about linear system control with unmatched uncertainties, the system is \dot{e}=Ae+Bu+w where \dot means differentail sign, u is the control input, w is the disturbance, A is already Hurwitz, is there any way to design u such that the disturbance w can be offset...
  48. ahmed habala

    Hi all -- I need a good reference about linear algebra

    hi all i need a good Reference about mathematics my level in mathematics as zero
  49. B

    Linear momentum and Angular momentum

    Homework Statement Two objects with mass of m are connected with a rod with length 2l and with no mass. The center of the rod is pinned so that it can spin. Object with mass M comes with speed v and sticks to m. There is no friction. 1) What is the angular speed w after collision? 2) FInd the...
  50. C

    Interpretation: Solution to a set of Linear Equations

    Hi, While solving a system of linear equations, there are three possible cases - unique / infinite / no solutions - to the system. One geometric interpretation is when one looks at a set of planes intersecting at one / many / no points respectively, for each of the above cases. While going...
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