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. Y

    Linear Algebra - Singular Value Decomposition Problem

    Homework Statement Find the SVD of Homework EquationsThe Attempt at a Solution I'm stuck My question is why in the solution it originally finds u_2=[1/5,-2/5]' but then says u_2=[1/sqrt(5),-2/sqrt(5)]'. I don't see what math was done in the solution to change the denominator from 5...
  2. KT KIM

    I Matrix Representation of Linear Transformation

    This is where I am stuck. I studied ordered basis and coordinates vector previous to this. of course I studied vector space, basis, linear... etc too, However I can't understand just this part. (maybe this whole part) Especially this one which says [[T(b1)]]c...[[T(bn)]]c be a columns of...
  3. F

    I General solution to linear homogeneous 2nd order ODEs

    Given a linear homogeneous 2nd order ODE of the form $$y''(x)+p(x)y'(x)+q(x)=0$$ the general solution is of the form $$y(x)=c_{1}y_{1}(x)+c_{2}y_{1}(x)$$ where ##c_{1},c_{2}## are arbitrary constants and ##y_{1}(x), y_{2}(x)## are linearly independent basis solutions. How does one prove that...
  4. micromass

    Insights How to Self-study Algebra: Linear Algebra - Comments

    micromass submitted a new PF Insights post How to Self-study Algebra: Linear Algebra Continue reading the Original PF Insights Post.
  5. RoboNerd

    Question about conservation of linear momentum

    Homework Statement [pardon my crude drawing] Say I have a ball that hits the a rod at rest pivoted around hinge A. The ball in the diagram is the solid black, and the rod is the thin line in the diagram and is rotating around the square-shaped hinge at the bottom. I am not an artist but...
  6. M

    Linear Tranformation: Find the kernel of T

    Homework Statement Let T: P4--->P3 be a linear transformation given by T(p)=p'. What is the kernel of T? Homework EquationsThe Attempt at a Solution T(a0+a1+a2x2+a3x3+a4x4)=a1+2a2x+3a3x2+4a4x3 Ker(T)= { T(p)=0} so, a1+2a2x+3a3x2+4a4x3=0 then a1=2a2x+3a3x2+4a4x3 Ker(T)= { (-2,1,0,0)...
  7. smodak

    Linear Algebra Found a real nice free linear algebra book

    https://www.math.ucdavis.edu/~anne/linear_algebra/
  8. smodak

    I Stupid Question about a Notation in Linear algebra

    I know the arrow -> means a map. For example, defines a linear map. But I cannot figure out what does a X mean? I know that above denote a inner product map. whatever is on the left of the : is defined by whatever is on the right. but what is the x symbol? What is the correct way to read...
  9. P

    Linear Algebra -- Projection matrix question

    Homework Statement Let A be an n×n matrix which has the property that A^2 =A. (i) Write down the most general polynomial in AHomework EquationsThe Attempt at a Solution My biggest problem is that I don't even understand what the question is asking Is it just sum (alphaA^n)=0 but A^n=A...
  10. G

    MHB How to define this linear transformation

    > Admit that $V$ is a linear space about $\mathbb{R}$ and that $U$ and $W$ are subspaces of $V$. Suppose that $S: U \rightarrow Y$ and $T: W \rightarrow Y$ are two linear transformations that satisfy the property: > $(\forall x \in U \cap W)$ $S(x)=T(x)$ > Define a linear transformation $F$...
  11. G

    I How this defines a linear transformation

    Admit that V is a linear space about \mathbb{R} and that U and W are subspaces of V. Suppose that S: U \rightarrow Y and T: W \rightarrow Y are two linear transformations that satisfy the property: (\forall x \in U \cap W) S(x)=T(x) Define a linear transformation F: U+W \rightarrow Y that...
  12. M

    Linear Algebra- Dependent or independent

    Homework Statement Let(v1, v2, v3) be three linearly independent vectors in a vector space V. Is the set {v1-v2, v2-v3, v3-v1} linearly dependent or independent? Homework Equations Linearly independent is when c1v1+c2v2+...+ckvk=0 and c1=c2=...ck=0 The Attempt at a Solution c1(v1-v2)+...
  13. Y

    Linear Algebra - Find Orthogonal Matrix Q that diagonals

    Homework Statement I'm told to find the matrix Q of the matrix A Homework EquationsThe Attempt at a Solution So my problem is that in the answer key they have S = (1/3)... and I have no idea where this 1/3 comes from. I get an equivalent answer for X_1, X_2, and X_3 S = [X_1, X_2, X_3] but...
  14. A

    Linear algebra or intro to programming courses have higher failure rate?

    which has higher failure rate: linear algebra or intro to programming. I understand failure rate might be skewed: math majors, who are gifted and interested in math, are required to take linear algebra but not intro to programming, while CS majors are required to take both. Which class do CS...
  15. T

    Linear combination and orthogonality

    Given the non-zero vectors u, v and w in ℝ3 Show that there is a non-zero linear combination of u and v that is orthogonal to w. u and v must be linearly independant. I am not really sure at all. But I have done this: This is a screenshot of what I have done. Basicly, I assumed in the end...
  16. B

    I Factor of how much linear size scale has changed

    Question: By what factor has the linear size-scale of the Universe changed between when the light was emitted from the galaxy and now? The question gives the red shift z and I have the equation linear size scale = 1 / 1 + z and I know how to calculate the linear size scale. But I'm a little...
  17. Dusty912

    Complex-Valued Solutions of Second-Order Linear Equations

    Homework Statement Suppose y(t) is a complex-valued solution of y'' +py' + qy=0 where p and q are real numbers. Show that if y(t)=yre(t) + iyim(t), where yre(t) and yim(t) are real valued, then both yre(t) and yim(t) are solutions of the second-order equation. Homework Equations We can use the...
  18. Zarmina Zaman Babar

    A Fourier Transform of Piecewise linear spline wavelet

    Fourier Transform of Piecewise linear spline wavelet is defined by 1-|t|, 0<t<1; 0, otherwise, is (sinc(w/2))^2. Can anyone please show me the steps. Thanks
  19. T

    Linear and Angular Momentum on a wooden gate

    Homework Statement A uniform, 4.5-kg, square, solid wooden gate 1.5 m on each side hangs vertically from a frictionless pivot at the center of its upper edge. A 1.1-kg raven flying horizontally at 5.0m/s flies into this door at its center and bounces back at 2.0m/s in the opposite direction...
  20. G

    Collision involving angular and linear motion

    Homework Statement a small 0.199 kg block slides down a frictionless surface through height h = 0.608 m and then sticks to a uniform vertical rod of mass M = 0.398 kg and length d = 2.23 m. The rod pivots about point O through angle θ before momentarily stopping. Find θ. point O is at the end...
  21. joe_cool2

    Linear Algebra - Gram-Schmidt Process

    Homework Statement Let R2 have the Euclidean inner product and use the Gram-Schmidt process to transform the basis {u1,u2} into an orthonormal basis. u1 = (1,-3) u2 = (2,2) Homework Equations Gram-Schmidt process: \\v_1 = u_1 \\v_2= u_2 - \frac{\left ( \left \langle u_2, v_1\right \rangle...
  22. Y

    Engineering Linear Algebra - Analysis of purely resistive DC circuit

    Homework Statement I'm trying to solve this circuit when I try to simulate and run a dc sweep i get that it can't be solved. When I try to find the answer using linear algebra I get this answer after I throw it into MATLAB with the warning “[w]arning: Matrix is close to singular or...
  23. Y

    Linear Algebra - Elimination Matrix when Permutation Needed

    Homework Statement I feel like I should know the answer to this, so I believe this to be an easy question. Say have matrix A, and I store the elimination matrices E_1,1 E_2,1 etc. and somewhere in the elimination process I have to use a permutation matrix to swap rows. My question is when I...
  24. Y

    Linear Algebra - Gram Schmidt & Normalization - Error in Sol

    Homework Statement So I think I found an error in the solution were it attempts to find q_2^ I'm asked find the orthornomal basis for the column space of matrix A. Homework EquationsThe Attempt at a Solution [/B] My question is in what it puts for q_2^ A_2 = [4/3 4/3 -2/3]^T ||A_2|| =...
  25. D

    Linear momentum problem (ballistic pendulum)

    Homework Statement A ballistic pendulum is a device that may be used to measure the muzzle speed of a bullet. It is composed of a wooden block suspended from a horizontal support by cords attached at each end. A bullet is shot into the block, and as a result of the perfectly inelastic impact...
  26. GetLastError

    Need help with toy design (linear + radial bearing)

    Hi, I'm building a toy and I'd like it to spin around a vertical shaft but also move freely up and down while it is rotating. Also, I'd like that the rotating body will get spinned using a gear which will be positioned in a fixed location on the shaft (at the bottom) while the rotating body...
  27. lep11

    When is this linear transformation an isomorphism?

    Homework Statement Let L: ℝ2→ℝ2 such that L(x1, x2)T=(1, 2 ; 3, α)(x1, x2)T=Ax Determine at what values of α is L an isomorphism. Obviously L is given in matrix form. The Attempt at a Solution First of all a quick check, dim (ℝ2)=dim(ℝ2)=2 Ok. An isomorphism means linear transformation which...
  28. RoboNerd

    Linear Momentum of system of two blocks

    1. Homework Statement The problem is shown in the photo above. I would like to discuss part B. Homework Equations conservation of linear momentum The Attempt at a Solutionthe solutions say that we should have the following approach: Mb*vb = [Mb + Ms] * vFinal. Plug in the numbers and get vF...
  29. Y

    Linear Algebra - Projection of Vector

    Homework Statement I feel like this is a easy question but it seems the answer key doesn't seem to be right. So say I have 2 vectors and I'm trying to find the projection of vector u perpendicular to the vector v Homework EquationsThe Attempt at a Solution So I don't remember doing...
  30. Schwarzschild90

    Evaluating U(l) with Applied Linear Algebra: A Gambler's Demise

    Homework Statement I ask for help in solving the exercises in this project on applied linear algebra. The problem outlined in the project is one in which we are tasked with modeling the demise of a gambler. I need help solving exercise 1 (in red) on page 6. I have pasted the exercise text...
  31. Y

    Linear Algebra - Solving AC RLC circuit

    Homework Statement So yeah I'm doing a project were I get to create a problem. I would like to learn how to solve a AC RLC circuit using linear algebra. I'm trying to find all of the currents on the edges of the graphs and find all of the voltages at the nodes connecting the edges. I don't...
  32. B

    How to graph a linear relationship between T and v of sound?

    Homework Statement I did an experiment involving the speed of sound at different temperatures. I placed two microphones at a fixed distance apart and I measured the time taken for a sound wave to travel between the two mics. I repeated this for different temperatures. I want to make a graph for...
  33. Z

    Linear equations algebra based problem

    Homework Statement I got following Question from a book: I am doing it using a different method but my answer is wrong. Can somebody please guide me, what is the problem with it?? The question is:In 1999, Diana read 10 English books and 7 French books. In 2000, she read twice as many French...
  34. S

    Difference between momentum and linear inertia

    Hi everyone, I have always been confused by momentum and linear inertia. What exactly is the difference between them? Aren't them both resistance to change in motion? Its hard to explain what I am confused with so here's an example: if there are 2 moving cars traveling at the same speed and car...
  35. kamhogo

    Light bulb & Diode: linear or non-linear elements.

    Homework Statement These questions are related to an experiment where we had to record the current for varying values of potential difference (10 readings), then repeat the process with the applied potential difference was reversed a) Why is the resistance of a light bulb not constant? b) Does...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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.
  50. 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...
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