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. G

    MHB Linear Programming word problems

    a small business makes laptops and notepads. In any given month labour costs must not exceed £1350 and material costs must be a maximum of £1150. Relevant information: Laptops cost £50 in materials to make and labour costs £50, they make £110 profit on each laptop. notepads cost £25 in...
  2. CynicusRex

    Studying Guidelines to studying linear algebra and statistics.

    I'll try to be concise. I've been out of math for years and never truly learned to understand it. Until now. I want to put the growth mindset theory to the test and see if I can handle physics (or any STEM field) on a university difficulty. To verify if I'm up to it and even have the slightest...
  3. H

    B Mixed states V superposition V linear combinations?

    Can someone explain the difference using concrete examples. I will attempt to explain my current understanding by example; A H atom has different energy levels which can be exactly described by algebraic functions with quantum numbers n, l etc. An electron can be excited from say the ground...
  4. S

    Linear Motion and Free Fall: Solving Projectile Motion Problems

    I. Free Fall Motion A mass m = 544 g is thrown straight up with an initial speed of 3.50 m/s from a height of h = 2.50 m Neglecting a drag force, determine: 1. The acceleration of the object while it moves up. 2. The acceleration of the object at the highest point. 3. The maximum height...
  5. Kmol6

    Linear momentum - Bullet fired vertically

    Homework Statement A bullet is fired vertically into a 1.40 kg block of wood at rest directly above it. If the bullet has a mass of 29.0 g and a speed of 510 m/s, how high will the block rise after the bullet becomes embedded in it? Homework Equations 1. m1v1 +m2v2 = mfvf 2. x=xo +vot...
  6. F

    So what am I supposed to learn in linear algebra

    I am self teaching the subject but I am unsure of what is the whole point and picture
  7. E

    Linear Differential equation problem

    Homework Statement Solution of the differential equation ##(\cos x )dy = y (\sin x - y) dx , 0 < x < \dfrac{\pi}{2} ## is Homework EquationsThe Attempt at a Solution Only separation of variables, homogenous and linear DEs are in the syllabus, therefore it must be one of those. It obviously...
  8. mpoli

    Linear Programming Case Study - Case Problem

    Homework Statement Linear Programming Case Study - Case Problem ( Page # 109 Decision making methods) “The Possibility” Restaurant? In the case problem, Angela and Zooey wanted to develop a linear programming model to help determine the number of beef and fish meals they should prepare each...
  9. G

    Matrix of linear transformation

    Homework Statement Let A:\mathbb R_2[x]\rightarrow \mathbb R_2[x] is a linear transformation defined as (A(p))(x)=p'(x+1) where \mathbb R_2[x] is the space of polynomials of the second order. Find all a,b,c\in\mathbb R such that the matrix \begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0...
  10. Geofleur

    A Isomorphism between a linear space and its dual

    I have been trying to prove the following theorem, for a finite dimensional vector space ## X ## and its dual ## X^* ##: Let ## f:X\rightarrow X^* ## be given by ## f(x) = (x|\cdot) ##, where ## (x|\cdot) ## is linear in the first argument and conjugate linear in the second (so I am using the...
  11. W

    I Proof that the general solution of a linear equation is....

    any particular solution plus the general solution to the homogeneous equation. I'm having difficuilty understanding this proof from my lecture notes Theorem : Let T : V → W be a linear transformation. Let w ∈ W and suppose T(u0) = w T(v) = 0. where v ∈ V (the kernel ) to prove: T(u) = w...
  12. John Pang

    Does Energy Conservation Apply to Linear Momentum Collisions?

    Just a conceptual question : During a collision of two objects say A and B with the same mass, is the object with a higher velocity before collision never has its velocity increased after the collision, while the object with a lower initial velocity never has its velocity further decreased...
  13. C

    Linear algebra, can A be one-to-one given a case

    Homework Statement Given an nxn matrix, if a b exists so Ax=b has no solutions, can A be one-to-one? Homework Equations I understand that as a linear transformation, you need things such as (to be one-to-one as a linear trans) 1. n pivots 2. Only the trivial solution exists to Ax=0 Ax=b...
  14. Z

    MHB Linear Algebra: Analyzing A Linear Transformation

    Hey, I need help with part D2. My explanation is not right so I honestly do not know what I am suppose to write. My assignment is attached to this thread.
  15. Asad Albostami

    Mathematica Mathematica: ill-conditioned linear system

    Hi all, I have problem with regard to ill-conditioned linear system of solving sets of simultaneous equations using Mathematica program. I have tried my best to find a way to solve this but none was successful. I got results from m =1 and n =1 until m = 7 and n = 7, i,e. the systems are...
  16. 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...
  17. 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...
  18. 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...
  19. 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)
  20. 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
  21. 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...
  22. 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?
  23. 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...
  24. 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 \\...
  25. 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 &...
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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...
  31. 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...
  32. 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)?
  33. 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: $...
  34. 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...
  35. 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...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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...
  50. 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'...
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