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. N

    How Is a Salesperson's Wage Calculated Based on Sales?

    A pharmaceutical salesperson receives a monthly salary of $5000 plus a commission of 7% of sales. Write a linear equation for the salesperson’s monthly wage W in terms of monthly sales S. I will let W(S) = monthly wage W in terms of monthly sales S. $5000 plus a commission of 7% of sales =...
  2. M

    I Error on non-linearity in a linear fit

    Hello! If I have some data points, with error bars on both x and y, and I would like to fit them with a function f(x). How can I write the chi-squared in this case? For errors only on y, I would have ##\chi^2 = \sum_i(\frac{f(x)-y}{\sigma_y})^2##, but I am not sure how to include ##\sigma_x##...
  3. W

    I Bias in Linear Regression (x-intercept) vs Statistics

    Hi, In simple regression for machine learning , a model : Y=mx +b , Is said AFAIK, to have bias equal to b. Is there a relation between the use of bias here and the use of bias in terms of estimators for population parameters, i.e., the bias of an estimator P^ for a population parameter P is...
  4. A

    I Canonical Form for quadratic equations *with* linear terms

    Hello: I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables: $$ax^2+by^2+cxy+dx+ey+f=0$$ Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part): $$ w^TDw+[d \ \ e]w+f=0$$ $$w^TDw+Lw+f=0$$ where $$...
  5. Z

    Linearizing an equation for Mixed-integer linear programming (MILP)

    Summary:: Linearizing an equation for MILP Hi All I have Linear programming problem where I need to linearise the following function: Y = A * log2(1 + (Bx/Cx^2)). where A, B and C are constants. Can you please help or advise?
  6. cianfa72

    Two-port linear network general representation AV + BI = 0

    Hi, as follow-up to this thread I've a question about general representation of a two-port network. Basically it is ad hoc built four-terminal linear network (using controlled sources + nullator-norator pair): for it I found a general representation ##AV + BI = 0## as in the picture above. If...
  7. dextercioby

    I Linear & Rotational Motion: Speed of Tip in Lab Frame

    I have a pencil of Iron of length ##L## rotating about its center in a plane at constant angular velocity ##\omega##. The tip of the pencil in Newtonian mechanics has linear velocity ##\frac{\omega L}{2}##. It can exceed ##c##, of course. Now let us complicate this. Assume the center of the...
  8. S

    MHB Calculating probability from linear regression parameters

    I'm a bit stuck on how to calculate the probability in part b from the linear regression parameters. I tried plugging the parameter values into the linear regression model: Y =β0+β1X+ε, ε∼N(0,σ) So P(Y=y| X=40) = 2.85 + 0.07 * 40 + 1^2 P(Y=y|X=40) = 5.65 But I don't think this is the...
  9. P

    Looking for Linear Motor Exercises

    Homework Statement:: Hi, I'm looking for an exercise involving a linear motor, but on my physics book there are none Relevant Equations:: Lenz Law, Maxwell's equation Hi, everyone I'm looking for an exercise involving a linear motor, but on my physics book there are none. Do you happen to...
  10. Uchida

    Conservation of Linear Momentum of Rigid Body

    I solved it by two methods: ----------------------------------------------------- First, by conservation of linear momentum, using the vector velocities of each particle: In the imminence of the impact, the velocity of all the three particles are the same, \vec v_0 = - \sqrt{2gh} \hat j...
  11. snypehype46

    Computing OPEs of linear dilaton CFT

    I'm trying to do the following question from David Tong's problem sheets on string theory: > A theory of a free scalar field has OPE $$\partial X(z)\partial X(w) = \frac{\alpha'}{2}\frac{1}{(z-w)^2}+...$$. Consider the putative candidate for the stress energy tensor $$T(z) = \frac{1}{\alpha '}...
  12. P

    I Why the linear combination of eigenfunctions is not a solution of the TISE

    The linear combination of the eigenfunctions gives solution to the Schrodinger equation. For a system with time independent Hamiltonian the Schrodinger Equation reduces to the Time independent Schrodinger equation(TISE), so this linear combination should be a solution of the TISE. It is not...
  13. M

    Nonlinear spring made from many different linear springs in series?

    Can I create a nonlinear spring, for a nonlinear oscillator, by putting many linear springs in series and parallel?
  14. A

    How to determine if a transformation is linear

    Hello! I need to check if this transformation (not sure if it is the right word in English) from ## R^3 to R^3 ## is linear f(x1,x2,x3) = f(sin(x1),x2+x3,0). Now we are given that the transformation is linear if this you can prove this statement. $$f(\lambda * u + \mu * v) = \lambda * f(u) +...
  15. F

    Linear Algebra What are good books for a third course in Linear Algebra?

    What are the suitable books in linear algebra for third course for self-study after reading Linear Algebra done right by Axler and Algebra by Artin?
  16. 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
  17. 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...
  18. 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...
  19. 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...
  20. M

    Inversely proportional or a linear relationship?

    Would you call it an inverse relationship or a linear relationship, what is the correct terminology?
  21. 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.
  22. 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...
  23. B

    Is My Linear Programming Model Formulated Correctly?

    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...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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...
  31. 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?
  32. 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...
  33. 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...
  34. 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...
  35. 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...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. Mr Davis 97

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

  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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!
  47. 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...
  48. 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&...
  49. 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...
  50. 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...
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