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. F

    I What conditions are needed to raise a linear operator to some power?

    Each operator has a domain, so for a power of an operator to exist, the domain of the operator must remain invariant under the operation. Is that correct? mentor note: edited for future clarity
  2. yecko

    Comp Sci Linear regression doubling time

    y=mx+c by z=log y axis m=9-2.5 / 35 = 13/70 z=13/70 * (t-1983) + 2.5 log y = 13/70 * (t-1983) + 2.5 # doubling time: t1=y1, t2=y2=2y1 log y1 = 13/70 * (t1-1983) + 2.5 ---{1} log (2*y1) = 13/70 * (t2-1983) + 2.5 ---{2} {2}-{1}: log2=13/70 * [(t2)-(t1)] [(t2)-(t1)] = log2/(13/70) # for log...
  3. S

    Linear Algebra uniqueness of solution

    My guess is that since there are no rows in a form of [0000b], the system is consistent (the system has a solution). As the first column is all 0s, x1 would be a free variable. Because the system with free variable have infinite solution, the solution is not unique. In this way, the matrix is...
  4. anemone

    MHB Linear combination of sine and cosine function

    Hi MHB! I recently came across a problem and I was thinking most likely I was missing something very obvious because I couldn't make sense of what was being asked, and I so wish to know what exactly that I failed to relate. Question: Find the minimum of $6\sin x+8\cos x+5$. Hence, find the...
  5. M

    Inversely proportional or a linear relationship?

    Would you call it an inverse relationship or a linear relationship, what is the correct terminology?
  6. F

    Linear Algebra What are good second course books in linear algebra for self-study?

    What are best second course(undergraduate) books in linear algebra for self-study?I have already read Introduction to Linear Algebra by Lang.
  7. PiEpsilon

    Freely hinged rods on a table - Linear velocities of CM after the impulse

    We know that impulse is $$\vec J = \vec F \Delta t = \Delta \vec p$$ Let ##l, m## be the length of single rod and its mass respectively. Analyzing torques and forces on each rod separately we have: Rod ##AC##: $$F\Delta t +N_x\Delta t = mV_{ac,x} \space\space\text{ eq. }(1)$$ $$F\Delta t\cdot...
  8. B

    Linear Programming Question

    I've tried formulating the LP model for the question above and would like to check if I'm doing anything incorrectly. Below is my LP model. Let X1 be the number of one-child family interviewed on weekdays morning Let X2 be the number of one-child family interviewed on weekdays evening Let X3...
  9. M

    Engineering Linear Filters for Removing Impulsive noise

    Hi, I was working on the following homework problem and just wanted to check whether my thoughts were along the right lines: "Explain briefly why a linear filter would not be appropriate for removing impulsive noise in a signal, and suggest an alternative filter." Attempt: When we use a linear...
  10. C

    Ohm's Law graphing inversed gradient value

    Hey all. This is about Ohm's Law (and specifically resistance). When you plot the change in current vs the change in voltage you should get a linear trend line (providing it is from an ohmic device). The gradient should be the resistance. My questions is why does the gradient value need to be...
  11. M

    I Help solving a System of Linear Equations

    I have the following equation to solve for the coefficients b: \sum\limits_{m,n=0}^N b_m b_n^* (x_{mn}+y_{mn}) e^{ ik (ux_{mn} + vy_{mn}) } = 0 which must be satisfied for all u and v in the interval [-1,1]. Here k is a constant, b is a vector of length N with unit norm ||b||^2=1, and x and y...
  12. F

    How to distinguish between linear and non-linear circuits?

    Many circuit analysis techniques only apply to linear circuits. I don’t quite understand how to distinguish between linear and non-linear circuits. I understand the mathematical concept of linearity. I understand why components like resistors, capacitors and inductors are linear. I don’t quite...
  13. F

    B Limitations of Multivariate Linear Regression

    Hello, With multivariate linear regression, there is a single dependent variable ##y## and multiple independent variables ##x_1##, ##x_2##, ##x_3##, etc. There is a linear, weighted relationship between ##y## and the various ##x## variables: $$ y = c_1 x_1 + c_2 x_2 + c_3 x_3 $$ The...
  14. karush

    MHB 5.2a plot linear transformations

    ok we are supposed to go to here Find 3 different matrices that reflect the following transformations, report the matrix, the determinant, and the eigenvalues. 1. Rotation by $\dfrac{\pi}{4}$ 2. Shear along $x$ by a factor of $k$ 3. Reflection by the line $\theta$ there are some more but the...
  15. karush

    MHB 072 is Q(theta) a linear transformation from R^2 to itself.

    if $Q(\theta)$ is $\left[\begin{array}{rr} \cos{\theta}&- \sin{\theta}\\ \sin{\theta}&\cos{\theta} \end{array}\right]$ how is $Q(\theta)$ is a linear transformation from R^2 to itself. ok I really didn't know a proper answer to this question but presume we would need to look at the unit...
  16. PainterGuy

    Finding the general solution of this nonhomogeneous linear system

    Hi, I was trying to do the following problem. My attempt. Finding the reduced row echelon form for the system above. I do not see any way to proceed any further. The following is the solution presented in solution manual. How do I proceed to get the following answer?
  17. appletree23

    Help with linear algebra: vectorspace and subspace

    So the reason why I'm struggling with both of the problems is because I find vector spaces and subspaces hard to understand. I have read a lot, but I'm still confussed about these tasks. 1. So for problem 1, I can first tell you what I know about subspaces. I understand that a subspace is a...
  18. A

    Point Kinetics for Linear Insertion of Reactivity

    I want to use the point reactor kinetics equations to solve for the power as a function time when a reactivity of gamma*t is added to a reactor that is at equilibrium at time 0. I am also asked to consider the case where the transients are long-lived compared to the lifetime of prompt neutrons...
  19. G

    A small linear actuator with a very long stroke

    Hello, I am working on a mechanics project, on which I encounter a problem that I am sure others have faced before. I have tried to find answers here and there by typing keywords, but I can't find what I am looking for. Here is the context: Two horizontal plates are separated by a...
  20. J

    Linear Expansion - Finding temperature

    So if I am understanding the question correctly, I need to find the change in temperature that causes one rights outer radii to be the same as another rings inner radii. Now what I tried is two equations change length_1= a_1*original length_inner * change in temp change length_2= a_2*original...
  21. hquang001

    Linear momentum problem with n particles

    To find the mass in other pan, i need to find the force caused by beads on the pan ∴ KEinitial + PEinitial = KEfinal + PEfinal 0 + mgh = ½ mv^2 => v = 3.13 m/s ∴ The change in momentum : p2 - p1 = m ( v2-v1) = m( v - (-v)) = 2mv ∴ F = Δp / Δt = n. m. v How can i apply the rate of 100...
  22. karush

    MHB Matrices.......whose null space consists all linear combinations

    $ v=\left[\begin{array}{r} -3\\-4\\-5\\4\\-1 \end{array}\right] w=\left[\begin{array}{r} -2\\0 \\1 \\4 \\-1 \end{array}\right] x=\left[\begin{array}{r} 2\\3 \\4 \\-5 \\0 \end{array}\right] y=\left[\begin{array}{r} -2\\1 \\0 \\-2 \\7 \end{array}\right] z=\left[\begin{array}{r} -1\\0 \\2 \\-3...
  23. M

    MHB Fixed point,, Jacobi- & Newton Method, Linear Systems

    Hey! :giggle: Question 1 : Let $g(x)-=x-x^3$. The point $x=0$ is a fixed point for $g$. Show that if $x^{\star}$ is a fixed point of $g$, $g(x^{\star})=x^{\star}$, then $x^{\star}=0$. If $(x_k)$ the sequence $x_{k+1}=g(x_k)$, $k=0,1,2,\ldots$ show that if $0>x_0>-1$ then $(x_k)$ is...
  24. L

    Help with linear solenoid calculations (engine valve actuators)

    Hello, thanks in advance, I do many many things, jack of all trades but master of none. Would some one be willing to help me with some formulas and calcs for making my own linear solenoid? I can give the final specs I'm after then I could use advice on materials and other variables. I'd like...
  25. P

    A What is the Corollary of the Nucleus and Image Theorem?

    I tried hard to understand what this author proposed, but I feel like I failed miserably. My attempt of solution is here: Item (a) is verified in the case where ##n = 2##, since ##F## being a linear transformation, by the Corollary of the Nucleus and Image Theorem, ##F## takes a basis of...
  26. Mr Davis 97

    Dimension of set of all linear maps that map three elements to zero

  27. M

    (CFD) Problem with C code for 1D linear convection equation

    This is my c program below. Again everything works as expected up until the highlighted lines or the second for loop. When i run my program to output the arrays contents i get bizarre numbers that are not correct. # include <math.h> # include <stdlib.h> # include <stdio.h> # include <time.h>int...
  28. M

    MHB Rank of composition of linear maps

    Hey! :giggle: Question 1: Let $C$ be a $\mathbb{R}$-vector space, $1\leq n\in \mathbb{N}$ and let $\phi_1, \ldots , \phi_n:V\rightarrow V$ be linear maps. I have shown by induction that $\phi_1\circ \ldots \circ \phi_n$ is then also a linear map. I want to show now by induction that if $V$ is...
  29. patric44

    Non linear BVP using a shooting algoritm with MATLAB?

    hi guys i was trying to use this shooting algorithm from Xue and Chen Scientific Computing with MATLAB book : to solve this non linear temperature distribution problem : i checked my Matlab function multiple times but i am keep getting a nonsense graph for the temperature , can someone cheak...
  30. MahdiI84

    Fortran Density of state of a linear lattice

    PROGRAM DensityOfState IMPLICIT NONE INTEGER :: omega , t0 , a0 ,kx omega=1 ; t0 =1 ; a0=1 REAL :: kx , eta , w , ek eta=0.0015 REAL,PARAMETER :: pi=3.14 COMPLEX , PARAMETER ::i=(0,1) COMPLEX :: Energy COMPLEX :: Density COMPLEX :: GreenFunc...
  31. karush

    MHB 307w.WLCA.C10 computations are linear?

    the matrix at the bottom is from eMH but thot this was the way to do it... maybe not!
  32. A

    Linear algebra projections commutativity

    Textbook answer: "If P1P2 = P2P1 then S is contained in T or T is contained in S." My query: If P1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}and P2 =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} as far as I...
  33. karush

    MHB Construct 3 Augmented Matrices for Linear Systems

    $\tiny{311.1.1.26}$ Construct 3 augmented matrices for linear systems whose solution set is $x_1=3, \quad x_2=-2, \quad x_3=-1$ ok the only thing I could think of is just rearrange the rows of an RREF matrix. albeit losing the triangle format hopefully no typos $\left[\begin{array}{rrr|rr} 1&...
  34. E

    Linear expansion of steel versus copper with increasing temperature

    Here I'm going to show all that I've understood - 1. 2. 3. What I've attempted - L = Lo (1+ α * ẟT) ẟT = 150°C - 15°C = 135°C (Steel) L = ẟL (1 + 11 * 10^-6 * 135) (Copper) L = ẟL (1 + 17 * 10^-6 * 135) This doesn't get me anywhere, obviously. Am I supposed to understand from the...
  35. Eclair_de_XII

    Am I using quotient spaces correctly in this linear algebra proof?

    %%% Assume that ##X/Y## is defined. Since ##\dim Y = \dim X##, it follows that ##\dim {X/Y}=0## and that ##X/Y=\{0\}##. Suppose that ##Y## is a proper subspace of ##X##. Then there is an ##x\in X## such that ##x\notin Y##. Let us consider the equivalence class: ##\{x\}_Y=\{x_0\in...
  36. I

    I Solving a complex linear system with parameters

    I have to study the solutions of the following system of three equations and three unknowns upon variation of parameters k and h. ix1+kx2-x3 = 1+i (k+i)x1+(1-i)x2-(ik-1)x3 = h kx1+(4+2i)x2-(k-3-3i)x3 = 1-i Obviously i is the imaginary unit. And as stated k and h are the parameters . I can't...
  37. Kaguro

    Circular polarizer followed by a linear analyser

    I think that the intensity of output of circularly polarised light falling on an analyser should not depend upon the axis of analyser. Because it is circularly polarized. It keeps rotating. So, the output should be linearly polarised. The amplitude of E field in the direction of analyser is a...
  38. LCSphysicist

    Linear transformation: Find the necessary quantity of T

    > Let ##C## be the disk with radius 1 with center at the origin in ##R^2##. > Consider the following linear transformation: ##T: (x,y) \to (\frac{5x+3y}{4},\frac{3x+5y}{4})## > > What is the lowest number such that ##T^{n}(C)## contains at lest ##2019## points ##(a,b)##, with a and b integers.So...
  39. M

    Engineering Filter Linear Phase Property for non-integer time delays

    Hi, I just have a quick question regarding the linear phase property of filters. It might be easier to provide some context before getting to the question, but feel free to skip to the bottom. Consider a system input as a discrete sequence obtained by sampling at t = 0, T, . . . , kT from an...
  40. karush

    MHB Is b a linear combination of a1, a2, and a3?

    $\tiny{311.1.3.12}$ Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$ $ a_1\left[\begin{array}{r} 1\\0\\1 \end{array}\right], a_2\left[\begin{array}{r} -2\\3\\-2 \end{array}\right], a_3\left[\begin{array}{r} -6\\7\\5 \end{array}\right], b=\left[\begin{array}{r} -7\\13\\4...
  41. S

    I Why linear in linear regression?

    In linear regression, one estimates parameters that are supposed to be linear with respect to the dependent variable, for instance ##y=\theta_0 e^x+\epsilon \ ,## or ##y=\theta_0+\theta_1 x_1+\theta_2x_2+...+\theta_n x_n+\epsilon \ . ## Is it not true that neither ##y(\theta_0)## nor...
  42. C

    Linear independence of Coordinate vectors as columns & rows

    Summary:: x Question: Book's Answer: My attempt: The coordinate vectors of the matrices w.r.t to the standard basis of ## M_2(\mathbb{R}) ## are: ## \lbrack A \rbrack = \begin{bmatrix}1\\2\\-3\\4\\0\\1 \end{bmatrix} , \lbrack B \rbrack = \begin{bmatrix}1\\3\\-4\\6\\5\\4 \end{bmatrix}...
  43. Santiago24

    Linear Algebra I need textbook recommendations to learn linear algebra by myself

    Hi PF community, recently i learned about Calculus in one variables and several, so now i'd like to study linear algebra by myself in a undergraduate level, in order to do that i need some textbooks recommendations. I'll be waiting for your recommendations :).
  44. T

    Calculate resultant force of a linear mechanism

    Hi all, I'm currently designing a linear to linear transform mechanism (wedge cam design). Would like to check how much force required to push/ slide the roller down. To ease calculation, i have include some assumption: F2 = 1kg @ maximum compressed angle = 20degree F1 required = ? and how...
  45. L

    A Non linear diff. eq. first order

    I tried to solve for y' the algebric second grade equation but I don't know how to solve the diff. equation either. Any help? Thanks. -- lightarrow
  46. Tony Hau

    Boundary conditions of linear materials

    The question is as follows: Solutions given only contain part a) to c), which is as follows: So I now try to attemp d), e) and f). d) The magnetic field of a uniformly magnetized sphere is: $$ \vec B =\frac{2}{3}\mu_{o} \vec M = \mu_{o}\vec H$$ $$\frac{2}{3}\vec M = \vec H$$ The perpendicular...
  47. cianfa72

    Circuit theory - Resistor function for dynamic non linear circuits

    Hi, I'm reading the following paper (L. Chua) about the state-of-art of dynamic non linear circuit analysis -- Chua_Dynamic_Circuits I've a doubt about Theorem 2 on section 3.2 On the Existence of the Resistor Function that establishes sufficient conditions for the existence of network...
  48. K

    A linear combination of states that diagonalize the Hamiltonian

    He told me I "need to show that the Hamiltonian matrix elements you get by using those states have nonzero elements only on the diagonal." I understand what and how a diagonal matrix works, but what I don't understand is what those states are. Are they states I put in my "quantum mechanical...
  49. S

    Vector space of linear maps

    Solution 1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##. From...
  50. V

    Optimizing Resources with Linear Programming: A Step-by-Step Approach

    I am comfortable solving these types of problems, however I am having trouble setting up this problem and am unsure if I am doing this correctly so would someone be able to assist me? I first wrote an equation for the people: 10x +8y +11z. Then I made an equation for the flour: 4x + 12y + 8z...
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