What is Linearly: Definition and 226 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. B

    Proving that Columns are Linearly Dependent

    Homework Statement Let A be an m x n matrix with m<n. Prove that the columns of A are linearly dependent. Homework Equations Its obvious that for the columns to be linearly dependent they must form a determinate that is equal to 0, or if one of the column vectors can be represented by a...
  2. P

    Number of linearly independant motions

    Homework Statement http://postimg.org/image/uc0c581cb/ For (b), how many points of linearly independent motion are present in this system? Homework EquationsThe Attempt at a Solution The solution says there is rotation of J1, between K1 and D1, to the right of K2 and finally J3 But isn't...
  3. Dethrone

    MHB Linearly Independent: Why ${}\left\{X+Y, Y+Z, Z+W, W+X\right\}$ Isn't

    Let ${}\left\{X, Y, Z, W\right\}$ be an independent set in $\Bbb{R}^n$, is the following set independent? ${}\left\{X+Y, Y+Z, Z+W, W+X\right\}$ My textbook says it isn't, but I'm not sure why. Let $\lambda_1, \lambda_2, \lambda_3, \lambda_4$ be scalars, then...
  4. B

    MHB Linear Dependence in \mathbb{R}^4?

    Question: If \textbf{v}_1,...,\textbf{v}_4 are in \mathbb{R}^4 and \{\textbf{v}_1, \textbf{v}_2, \textbf{v}_3\} is linearly dependent, is \{\textbf{v}_1, \textbf{v}_2, \textbf{v}_3, \textbf{v}_4\} also linearly dependent? My Solution: http://s29.postimg.org/4wvwjlkqd/Linearly_Independent_Sets.png
  5. C

    MHB Linearly independence using Wronskian?

    Hi, so I am given this problem: Using the Wronskian, show that 1, x, x^2,..., x^(n-1) for n>1 are linearly independent. The wronskian is not zero for at least one value in the interval so it is linearly independent, I just do not know how to show it properly.Thank you! :D
  6. 22990atinesh

    Meaning of linearly dependent equations

    How the system of equations ##y=-2x+1## ##y=x+1## ##y=2x+1## are linearly dependent. In wiki its written for the above system of equations "one equation linearly dependent on the others"
  7. H

    EMF Generated When Magnet Moves Linearly Across Its Field

    Does an emf create across the body of a magnet when the magnet is linearly moving across its own field. If ABCD is a rectangular magnet of thickness t, AB is the north pole and CD is the south pole. If the magnet moves in the direction of t, that is perpendicular to ABCD plane, will there be an...
  8. grassstrip1

    Linearly Independent Sets and Spans in R4

    Hey everyone I just had a quick thought that was bothering me. For a set to be a basis it must be linearly independent and span the vector space. I've seen cases however of only two vectors forming a basis for R4 I don't see how two vectors could span 4 space or am I missing something. Thanks!
  9. F

    MHB Different ways to determine if functions are linearly independent

    Is calculating the determinant the Wronskian the only way to show that a set of functions is linearally independent? For example could you build a matrix where the numbers represent the coefficents to the polynomial functions and if it rref's to the identity matrix wouldn't this show it's...
  10. W

    Are These Vectors Linearly Independent?

    http://tinypic.com/r/euht06/8 Hi everyone! It was the question given to us in today's test & I was wondering if anyone can tell me what is the correct answer. The teacher will post the answers by the end of this week but I am feeling anxious to know the answer. Thanks in advance. P.S. Sorry...
  11. QuantumCurt

    Determine if the set of functions is linearly independent

    Homework Statement Determine if the given set of functions is linearly independent or linearly dependent.Homework Equations $$S=x~sin~x, ~ x~cos~x$$The Attempt at a Solution My first instinct was to use the Wronskian. $$W[y_1(x), y_2(x)]=\begin{vmatrix} x~sin~x & x~cos~x\\ x~cos~x+sin~x &...
  12. K

    For which value of k are the vectors linearly dependent?

    Homework Statement For which value of k are the vectors linearly dependent? v1=[1,0,1] v2=[1,1,2] v3=[1,2,k] <--Originally written as column vectors Homework EquationsThe Attempt at a Solution None of the vectors look like multiples of each other to me...so time to row reduce. Since each...
  13. QuantumCurt

    Finding maximum number of functions that can be linearly independent

    Homework Statement Hey everyone. I'm in an introductory differential equations class, and I think this homework problem has got me stumped. The functions y1(x), y2(x), ... , yn(x) are linearly independent on an interval I. c1y1(x)+c2y2(x)+...+cnyn(x)=0 for all x in I, implies that...
  14. ajayguhan

    Linearly varying force and bending moment problem.

    I have attached both the question and my attempt at solution. My problem is reducing the equation further to get the value of a/L
  15. D

    Stability of linearly perturbed linear nonautonomous system

    I have a linear time-varying linearly perturbed ODE of the form: \dot{x} = [A(t)+B(t)]x where A(t) is a bounded lower-triangular matrix with negative functions on the main diagonal, i.e. 0>a^0\ge a_{ii}(t). The matrix B(t) is bounded, so that ||B(t)|| \le \beta. The question is...
  16. N

    Show that non-zeros rows in REF are linearly independent

    Homework Statement Given 2 1 1 0 0 0 1 1 0 0 0 3(i) Show that the rows of A are linearly independent. (ii) Show that the nonzero rows of any matrix in row echelon form are linearly independent.The Attempt at a Solution i) REF gives 1 0 0 | 0 0 1 0 | 0 0 0 1 | 0 0 0 0 | 0 x1 = x2 = x3 =x4 =...
  17. N

    Can something moving linearly without rotation have anguar momentum?

    I was doing a question in Taylor book (example 3.3) where a sticky putty is thrown at a stationary wheel. To solve it we use conservation of angular momentum. What I am confused about is that the wheel is initially at rest and has no angular momentum initially. But when the putty is thrown at it...
  18. J

    Two linearly independent vectors in a plane that don't span the plane

    Homework Statement Say we have the plane, x+2y+4z=8 (part of a larger problem) Homework Equations The Attempt at a Solution The vectors (8,0,0) and (0,0,2) both lie in the plane. They are linearly independent. But (0,4,0) lies in the plane and is not a linear combination of the first two...
  19. D

    Are fractional polynomials linearly independent?

    i.e., does the set of functions of the form, \{ x^{\frac{n}{m}}\}_{n=0}^{\infty} for some fixed m produce a linearly independent set? Either way, can you give a brief argument why or why not? Just curious :)
  20. A

    Hydrostatic Fluid- Linearly Accelerating Slope

    Homework Statement An open rectangular tank contains water up to about half of its depth. This tank accelerates at a=2.20 m/s^{2} up a slope α (alpha), which causes the free water surface to form an angle θ (theta) with the original horizontal plane. What is the α angle of the slope?Homework...
  21. rogeralms

    Linearly polarized electric field

    Homework Statement Consider a harmonic, electromagnetic plane wave traveling along a line from the origin to the point (3,5,6). It is linearly polarized and its electric field lies in a plane perpendicular to the direction of travel of the wave. The wavelength of the wave is 2.0mm and it has a...
  22. R

    Finding Equation for Linearly Changing Density

    Homework Statement I need to find the Kinetic energy of a bar rotating about its center of mass. I know the bar as length 3b and it's center of mass is located at 2b, the bar density changes linearly along it's length. Homework Equations T=1/2 W^2 I The Attempt at a Solution...
  23. W

    Linearly accelerating hydrostatic fluid

    Homework Statement A cart is acclerating to the right with a=3m/s^2. Fluid is in hydrostatic state. Find the force on the back wall. Cart goes .8m into the page. In the image dotted line is free surface when cart is stationary. Homework Equations \vec{\nabla P}=\rho (\vec{g} -...
  24. V

    Why are the columns of Q linearly independent?

    Why column 1 is M^2*v? How can we know? Please see attached. Many thanks.
  25. M

    Set of vectors, linearly dependent or independent?

    Homework Statement Check if the following set of vectors are linearly dependent or independent: A) V1= \stackrel{1}{1} V2= \stackrel{1}{3} B) V1= \stackrel{\stackrel{1}{2}}{3} V2= \stackrel{\stackrel{2}{1}}{3} C) V1= \stackrel{1}{3} V2= \stackrel{2}{1} V3= \stackrel{-1}{2} Homework...
  26. Y

    4-acceleration of a vector in a linearly moving train

    Consider the thought experiment in the diagram below. According to the authors, the vector will rotate even in the rest frame of the moving train relative to the body of the train. Is it possible to consider the spin axis of the gyroscope represented by the vector as a simple rod and...
  27. F

    MHB Set of eigenvectors is linearly independent

    I know eigenvectors corresponding to different eigenvalues are linearly independent but what about a set ${e_{1},...,e_{n}}$ of eigenvectors corresponding to different eigenvalues?
  28. P

    Proving linearly independent set

    1. Prove that if A is symmetric and B is skew-symmetric, then {A,B} is a linearly independent set. I am going to need some help to solve this. Not sure how to begin. Homework Equations The Attempt at a Solution
  29. K

    Linearly independence question

    Say I have a matrix ##A## that has linearly independent columns. Then clearly ##A^T## has lin. indep. rows. So what can we say about ##A^TA##? Specifically, is there anything we can say about the rows/columns of ##A^TA##? I'm thinking there has to be some sort of relation but I don't know what...
  30. S

    MHB Showing that two elements of a linearly independent Set Spans the same set

    Hi, i would like to have a hint for the following problem: Let $$v_1, v_2 \&\ v_3 $$ in a vector space V over a field F such that$$ v_1+v_2+v_3=0$$, Show that $\{v_1,v_2\}$ spans the same subspace as $\{v_2,v_3\}$ Thanks in advance
  31. G

    Are x and ix linearly dependant or independant? (i=√-1)

    The question is: are x and ix linearly dependent or independent? My first guess is that they should be linearly dependent since i is a constant. But when you apply the definition of linear independence i.e. when you solve ax+ibx=0 (where x≠0), you get a=-ib which shows that the only...
  32. D

    MHB Show that in every set p2 with more than three vectors is linearly dependent.

    i know S = { 1 , x, x^2} is linearly dependent set for p2. where (a_0, a_1, a_2) = (0,0,0) I wanted to use the Wronskian on { 1 , x, x^2, x^3} , but as I understand, it only proves linear independence and not the converse.
  33. U

    Prove mutually non-zero orthogonal vectors are linearly independent

    Homework Statement Let a1, a2, ... an be vectors in Rn and assume that they are mutually perpendicular and none of them equals 0. Prove that they are linearly independent. Homework Equations The Attempt at a Solution Consider βiai + βjaj ≠ 0 for all i, j => βiai + βjaj +...
  34. H

    Linearly independence of vector function

    Given two vectors x(t) = (e^t te^t)^T y(t) = (1 t)^T a) Show that x and y are linearly dependent at each point in the interval [0, 1] b) Show that x and y are linearly independent on [0, 1] I compute det([x y]) = 0, so they are linearly dependent how about part b. Isn't a)...
  35. 1

    For what values of a are these vectors linearly dependent?

    Homework Statement For what value(s) of a are the following vectors v1=[1,2,-1], v2=[0,1,3], and v3=[a,4,5] linearly dependent? Homework Equations Since linear dependence means that anyone of the vectors can be expressed as a linear combination of the other vectors: sv1v1+s2v2=v3 where s1and...
  36. S

    Homework: Matrix is invertible when rows are linearly independent

    Hi there, I have a homework where I have to do this: Prove that square matrix is invertible if the columns of the matrix are linearly independent. There is also a hint: You can help with the following statement: Linear transformation L: U->V is bijection when a vector space basis N for U...
  37. S

    Partial linearly distributed load on cantilever beam

    I have a partial linearly distributed load on a cantilever and would to determine the bending moment distribution along the beam. I usually do this by taking cuts along different sections of the beam and finding the expressions for shear force and bending moment. However when I do this...
  38. C

    Finding values for a linearly independent subset

    Homework Statement There is a vector space with real entries of all 2x2 matrices. You have to find what values of \alpha\inℝ make the set Z = \{ \begin{pmatrix} 1 & 2\\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 3 & 7\\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 2 & 6\\ \alpha & 0...
  39. M

    Linear algebra linearly independent

    Homework Statement Determine whether the following vectors are linearly independent in P3 "not sure what P3 stands for maybe polynomial of third degree?" 1,x2,x2-2 let p1(x)=1 p2(X)=x2 p3=x2-2 c1p1(x)+c2p2(x)+c3p3(x)=z where z=0x2+0x+0 I then create a matrix using the above...
  40. K

    Can kinetic energy be added linearly?

    According to special relativity, velocity cannot be added to another velocity linearly. But I was thinking, what about kinetic energy? K.E. = mv2/2. As velocity increases, the object's mass also increases. The way I see it, the increase in mass "compensates" for the less-than-expected increase...
  41. P

    {u1, ,uk} is linearly dependent iff {[u1]_B, [uk]B} is.

    Hi, Homework Statement Given a vector space V and its basis B = {v1, v2, ..., vn}, I was asked to prove that: a group of vectors {u1,...,uk} in V is linearly dependent if and only if {[u1]B,...[uk]B} is linearly dependent. Homework Equations The Attempt at a Solution I proved...
  42. D

    Are AC and DC parameters related linearly?

    I've a small wind turbine that is rated at about 24V. The turbine produces AC voltage but it's got an internal recifier converting this to DC. I need to create an embedded system that measures the output of this turbine displaying AC voltage, current and power output. I did some research...
  43. K

    Linearly graded pn junctions

    Hi, my question is: Given a linearly graded on-junction with the following doping conditions a=10^21 I have to calculate Emax, Vbi, W and the junction capacitance with Va=0.2V and Vr=-5V This is for Silicon @ T=300k ---- From what I understand, I have to use an iterative method...
  44. E

    Linear Algebra - Linearly Independent Functions

    Forgive me ahead of time, I don't really know how to use LaTeX, (it's on my to do list). Homework Statement Given the vector space C([0,pi]) of continuous, real valued functions on the given interval, as well as the inner product <f,g>=integral(f(t)*g(t))dt from 0 to pi: a) Prove the set...
  45. T

    Show that a set of functions is linearly independent

    Hello everybody I have to show that this set of vectors a = (e-t, e-it, et, eit ) is linearly independent. My attempt : f(x) = k1 * e-t + k2 * e-it + k3 * et + k4 * eit f '(x) = k1 * -e-t + k2 * -ie-it + k3 * et + k4 * ieit f ''(x) = k1 * e-t + k2 * -e-it + k3 * et + k4 * -eit f(0) =...
  46. M

    Linearly polarized light on a quarter wave plate

    Homework Statement A linearly polarized beam propagates in the z-direction with its E-field oscillating in the y-direction. It is incident on a quarter wave plate (QWP) located in the x-y plane at the origin. a. How should the fast and slow axes of the QWP be oriented if the beam emerging from...
  47. S

    Function to generate linearly independent vectors

    Hi, I want to whether there is a function (/matrix) such that it can generate a m-dimensional vector such that this generated vector will always be linearly independent of the set of vectors the function has already generated. My problem can be written in pseudocode format as follow. I...
  48. S

    Is a set with a 0 vector linearly independent?

    I don't know how to write out matrices nicely on this forum, but suppose you have some matrices:[1 0 3] [2 0 4] [0 0 5] This would, by definition, be linearly dependent, spanning a plane in r3..is this correct? Since c1=0, c2=anything, c3=0 where c1v1+c2v2+c3v3=0 With this: [1 0 3 5] [3 0...
  49. N

    Linearly Independent Equations => 1 solution?

    Homework Statement Prove that if you have 3 linearly independent equations in 3 variables, then there exists only 1 solution to the system. Homework Equations linear independence implies none of the equations can be expressed as a linear combination of the other equations. The...
  50. C

    Is the set T = {w1,w2,w3} linearly dependent?

    Homework Statement Suppose that S = {v1, v2, v3} is linearly independent and w1 = v2 w2 = v1 + v3 and w3 = v1 + v2 + v3 Determine whether the set T = {w1,w2,w3} is linearly independent or linearly dependent. Homework Equations Let c1, c2, c3=scalars c1w1+c2w2+c3w3=0...
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