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. H

    I Explanation of all "its linearly independent derivatives"

    I'm studying Differential Equations from Tenenbaum's, and currently going through non-homogeneous second order linear differential equations with constant coefficients. Method of Undetermined Coefficients is the concerned topic here. I will put forth my doubt through an example. Let's say we are...
  2. C

    One set v is a linear combination of u. Prove u is linearly dependent

    Hi Everybody, I am having some difficulties on the prove this problem. I picked a nice example when I was trying to think about the proof. Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear...
  3. Mr_Allod

    Depletion width of linearly doped PN-junction

    Hello there, I have derived the expressions for electric field and potential to be the ones above, then for continuity at ##x = 0## I set the electric fields and potentials to be equal to yield the expressions: $$Sx_p^2 = Kx_n^2$$ $$V_{bi} = V_n - V_p = \frac {q}{3\epsilon} \left( Sx_p^3 +...
  4. S

    Determining value of r that makes the matrix linearly dependent

    for problem (a), all real numbers of value r will make the system linearly independent, as the system contains more vectors than entry simply by insepection. As for problem (b), no value of r can make the system linearly dependent by insepection. I tried reducing the matrix into reduced echelon...
  5. karush

    MHB Q01 are linearly independent vectors, so are....

    Let A be invertible. Show that, if $\textbf{$v_i,v_2,v_j$}$ are linearly independent vectors, so are \textbf{$Av_1,Av_2,Av_3$} https://drive.google.com/file/d/1OuHxfUdACbpK4E5aca2oBzdaxGR0IYKv/view?usp=sharing ok I think this is the the definition we need for this practice exam question...
  6. karush

    MHB 115.C51 find a linearly independent set T so that T=S

    $\tiny{115.C51}$ find a linearly independent set T so that $\langle T\rangle =\langle S\rangle$ $S=\left\{ \left[\begin{array}{r}2\\-1\\2\end{array}\right], \left[\begin{array}{r}3\\0\\1\end{array}\right], \left[\begin{array}{r}1\\1\\-1\end{array}\right]...
  7. karush

    MHB Linearly Dependent Vectors: Find h Value and Justify

    $\tiny{311.1.7.11}$ ok I am going to do several of these till I get it... Find the value(s) of h for which the vectors are linearly dependent. Justify $\left[\begin{array}{rrrrrr} 2\\-2\\4 \end{array}\right], \left[\begin{array}{rrrrrr} 4\\-6\\7 \end{array}\right], \left[\begin{array}{rrrrrr}...
  8. C

    Frequency of EM waves produced by linearly accelerating charges

    I was wondering about EM waves produced by linearly accelerating charges, as opposed to oscillating charges. With oscillating charges, the frequency of the wave depends on the frequency of the oscillation of the charge. But what determines the frequency of the wave produced by a linearly...
  9. M

    MHB Show that these elements are linearly independent

    Hey! 😊 Let $1\leq n\in \mathbb{N}$, let $\mathbb{K}$ be a field, $V$ a $\mathbb{K}$-vector space with $\dim_{\mathbb{K}}V=n$ and let $\phi:V\rightarrow V$ be linear. - Let $0\neq v\in V$ and $1\leq m\in \mathbb{N}$, such that $\phi^{m-1}(v)\neq 0=\phi^m(v)$. Show that $\phi^0(v)=v, \phi (v)...
  10. M

    MHB Statements with linearly independent vectors

    Hey! 😊 Let $\mathbb{K}$ a field and let $V$ a $\mathbb{K}$-vector space. Let $1\leq m, n\in \mathbb{N}$ and $n=\dim_{\mathbb{K}}V$. Let $v_1, \ldots , v_m\in V$ be linearly independent. Let $\lambda_1, \ldots , \lambda_m, \mu_1, \ldots , \mu_m\in \mathbb{K}$ such that...
  11. B

    MHB Creating Linearly Dependent Square Matrix from Cam Mechanism Equation

    I have an equation that comes from an especific topic of cam mechanisms and it goes like this: $$ 2M[tan(B)-B] - \beta Ntan(B) - 2\pi\sqrt{1 - N^2} = 0 \ \ \ \ \ \ \ \ \ (1) $$ For this it doesn't matter what each variable means. I'm trying to create a 3x3 matrix with a determinant equal to...
  12. brotherbobby

    Proving that the two given functions are linearly independent

    Summary:: I attach a picture of the given problem below, just before my attempt to solve it. We are required to show that ##\alpha_1 \varphi_1(t) + \alpha_2 \varphi_2(t) = 0## for some ##\alpha_1, \alpha_2 \in \mathbb{R}## is only possible when both ##\alpha_1, \alpha_2 = 0##. I don't know...
  13. brotherbobby

    I Proving functions are linearly dependent

    We can make the first three functions add up to zero in the following way : ##\sin^2 t+\cos^2 t-\frac{1}{3}\times 3 = \varphi_1(t) + \varphi_2(t) - \frac{1}{3} \varphi_3(t) = 0##. However, look at ##\varphi_4(t) = t## and ##\varphi_5(t) = e^t##. How does one combine the two to add up to zero? I...
  14. M

    MHB Are the vectors linearly independent?

    Hey! :o We have that the vectrs $\vec{v},\vec{w}, \vec{u}$ are linearly independent. I want to check if the pairs $\vec{v}, \vec{v}+\vec{w}$ $\vec{v}+\vec{u}$, $\vec{w}+\vec{u}$ $\vec{v}+\vec{w}$, $\vec{v}-\vec{w}$ are linearly indeendent or not. Since $\vec{v}, \vec{w}, \vec{u}$...
  15. karush

    MHB 16.1 Show that e^{2x}, sin(2x) is linearly independent on + infinity -infinity

    16.1 Show that $e^{2x}$, sin(2x) are linearly independent on $(-\infty,+\infty)$ https://www.physicsforums.com/attachments/9064 that was the example but... \begin{align*} w(e^x,\cos x)&=\left|\begin{array}{rr}e^x&\cos{x} \\ e^x&-\cos{x} \\ \end{array}\right|\\ &=??\\ &=?? \end{align*}
  16. karush

    MHB 12.6 linearly dependent or linearly independent?

    Are the vectors $$v_1=x^2+1 ,\quad v_2=x+2 ,\quad v_3=x^2+2x$$ linearly dependent or linearly independent? if $$c_1(x^2+1)+c_2(x+2)+c_3(x^2+2x)=0$$ is the system $$\begin{array}{rrrrr} &c_1 & &c_3 = &0\\ & &c_2 &2c_3= &0\\ &c_1 &2c_2& = &0 \end{array}$$ I presume at this point observation can...
  17. physics girl phd

    Solving Electric Potential Difference in a Linearly Varying Field

    Homework Statement I'm trying to do a problem two ways, and things aren't consistent, finding the electric potential difference in a linearly varying field. The electric potential difference between two points is often summarized in texts as ΔV = Vf - Vi = - ∫ E⋅ds where the lower bound of...
  18. hilbert2

    A Linearly independent function sets

    It is well known that the set of exponential functions ##f:\mathbb{R}\rightarrow \mathbb{R}_+ : f(x)=e^{-kx}##, with ##k\in\mathbb{R}## is linearly independent. So is the set of sine functions ##f:\mathbb{R}\rightarrow [-1,1]: f(x) = \sin kx##, with ##k\in\mathbb{R}_+##. What about...
  19. QuasarBoy543298

    I Why are linearly ordered R and R/{0} not isomophic?

    i saw a proof that said “in R/{0} , the set [-1,0) has an upper bound ,but no least upper bound. no such set exists in linearly ordered R” ,but i could not understand it.
  20. F

    Distance traveled with linearly increasing kinetic energy

    Homework Statement An object of mass 50 kg gains 20,000,000 joules every second. Devise formulae to find the distance covered at any given point in time, and the time necessary to cover a certain distance. Homework EquationsThe Attempt at a Solution E = 20,000,000 * t V = (E * 2/50)1/2 That's...
  21. Russell Patterson

    I Do orbital dynamics scale linearly?

    I need to scale my solar system model and it's not working. I'm making an augmented reality app that shows a sample solar system rotating. It works great! (Thanks to this $7 program https://assetstore.unity.com/packages/tools/physics/gravity-and-orbits-solar-system-105300 . The Unity engine...
  22. R

    Faraday's law and linearly time dependent B field

    Homework Statement A positron is moving in a circular orbit of radius r = 2cm within a uniform magnetic field B0 = 50##\mu##T. The magnetic field varies over time according to the expression: B = 700t + Bo and, therefore, each orbit can be considered almost circular. (a) Calculate the...
  23. J

    Decomposition of linearly polarized field MRI

    Homework Statement Hi, I am having trouble understanding how the B1 field as described by (3.48) in the image attached in MRI which is described as a linearly polarized field is decomposed into it's final two circularly polarized field as described by (3.49) in the image attached. Homework...
  24. J

    Linearly Independent Eigen Vectors

    Solved (sorry i tried again and realized my E-values were wrong) 1. Homework Statement Homework Equations Find Eigen Values and then what? The Attempt at a Solution I got eigen values as 3 and -3. Now how to proceed? I got Eigen Vector as: 1, 1 for eigen value of 3 and eigen vector as 8, 2...
  25. M

    Find largest number of linearly dependent vectors among these 6 vectors

    Homework Statement Given the six vectors below: 1. Find the largest number of linearly independent vectors among these. Be sure to carefully describe how you would go about doing so before you start the computation. 2 .Let the 6 vectors form the columns of a matrix A. Find the dimension of...
  26. B

    Linearly Independent Sets in Abelian Groups

    Homework Statement [/B] ##X## is linearly independent if and only if every nonzero element of the subgroup ##\langle X \rangle## may be written uniquely in the form ##n_1 x_1 + ... n_k x_k## (##n_i \in \Bbb{Z} \setminus \{0\}##, and ##x_1,...,x_k \in X## are distinct). Homework Equations [/B]...
  27. J

    Beam Bending of a Linearly Varying Beam with 2 point loads

    I'm trying to design something and rather than solve it using FEA guess and check, I'm trying to mathematically solve it. The part I'm getting stuck at is having 2 point loads on a linearly varying cantilever beam. https://imgur.com/LtHCvCB The closest example I could find to this that was...
  28. G

    I Divergent Sums of Linearly Independent Elements

    Suppose we had an infinite series - z = ∑i = 1 to ∞ ( α1(i)x1 + α2(i)x2 + . . . + αm(i)xm ) - rewritten as the cumulative sequence - z(n) = α1(n)x1 + α2(n)x2 + . . . + αm(n)xm - where the xj are linearly independent and normalized (and serve as a finite basis across the sequence). If all...
  29. J

    I Argument that a maximum density Universe expands linearly

    The Friedmann equation for a spatially flat Universe is given by $$\Big(\frac{\dot R}{R}\Big)^2=\frac{8 \pi G}{3}\rho$$ where ##R(t)## is the proper radius of some spherical volume with us at its center. Let us assume that there is a mass ##M## inside this spherical volume of radius ##R##. The...
  30. cathal84

    I Proving a set is linearly independant

    I have two questions for you. Typically when trying to find out if a set of vectors is linearly independent i put the vectors into a matrix and do RREF and based on that i can tell if the set of vectors is linearly independent. If there is no zero rows in the RREF i can say that the vectors are...
  31. M

    Linearly stable / non-linearly unstable map example

    Homework Statement [/B] Give examples of fixed points of vector fields and maps that are stable in the linear approximation but are nonlinearly unstable Homework EquationsThe Attempt at a Solution [/B] I was able to find an example in a vector field that, when the Jacobian is found and the...
  32. Mr Davis 97

    Showing that exponential functions are linearly independent

    Homework Statement If ##r_1, r_2, r_3## are distinct real numbers, show that ##e^{r_1t}, e^{r_2t}, e^{r_3t}## are linearly independent. Homework EquationsThe Attempt at a Solution By book starts off by assuming that the functions are linearly dependent, towards contradiction. So ##c_1e^{r_1t}...
  33. D

    Determining the Max. Set of Linearly Independent Vectors

    (sorry for the horrible butchered thread title... should say "determination", not "determining") 1. Homework Statement In "Principles of Quantum Mechanics", by R. Shankar, a vector space is defined as having dimension n if it can accommodate a maximum of n linearly independent vectors (here is...
  34. H

    Loss of GPE of a linearly growing raindrop

    Consider a spherical raindrop that falls at a constant velocity and whose radius ##r## is proportional to the distance ##h## fallen, i.e., ##r=kh##. Find the loss of gravitational potential energy (GPE) after it has fallen a distance ##h##. The given answer is ##mg\frac{h}{4}## but my answer is...
  35. S

    MHB How can I find out if this matrix A's columns are linearly independent?

    How can I find out if this matrix A's columns are linearly independent? $\begin{bmatrix}1&0\\0&0\end{bmatrix}$ I see here that $x_1 = 0$ and similarly $x_2 = 0$ does this mean that this matrix A's columns are therefore linearly dependent? Also this is a projection onto the $x_1$ axis so is it...
  36. baby_1

    Linearly Chirped Fiber Bragg Grating group delay

    Homework Statement Here is my problem : I want to know where does group delay equation come form? Homework Equations As I checked reference 21 it only indicates The Attempt at a Solution I'm following how can I start and solve this problem
  37. B

    Wronskian and two solutions being independent

    In the uploaded file, question 11 says that in b) the solutions y1 and y2 are linearly independent but the Wronskian equals 0. I think it said that they are independent because it's not a fixed constant times the other solution (-1 for -1<=t<=0 and 1 for 0<=t<=1) but it clearly says in the...
  38. C

    Is a subset of linearly independent vectors indep?

    Homework Statement True or false. If v1... v4 are linearly independent vectors in 4D space, then {v1,v2,v3} is also linearly independent. There's a hint: Think about (c for different constant) cv1+cv2+cv3+0*v4=0 I know that linear independence means there's only the trivial solution... which...
  39. H

    Vertical rise of liquid at the back of a linearly accelerating tank....

    Homework Statement why the vertical rise of the surface is given by the formula as shown in the notes ? shouldn't it = Δs = (g +az ) Δx / ax ? Homework EquationsThe Attempt at a Solution
  40. B

    Proving a set of vectors is linearly independent

    Hey all, student100's brother here, he got me to create an account to ask my question here... I'm taking an introduction to linear algebra class and we had a test problem to prove a set of vectors is linear independent: ##V = v_1, \dots, v_n \in \Bbb R^n## such that each element of the set is...
  41. J

    Is thrust linearly related to propeller-pitch?

    Simple question, Given a constant speed, is the thrust of a propeller linearly related to it's pitch?
  42. G

    Quantum Mechanics, Cart filling, + friction as it moves

    Homework Statement A cart roles down the track with an initial velocity vo. Because of falling rain, water starts filling the cart such that its mass increases linearly with time. The rain that has fallen on the track cause the wagon to experience a frictional force characterized with a...
  43. R

    Proof: Max number of Linearly Independent Vectors

    Homework Statement Prove that a set of linearly independent vectors in Rn can have maximum n elements. So how would you prove that the maximum number of independent vectors in Rn is n?I can understand why in my head but not sure how to give a mathematical proof. I understand it in terms of the...
  44. ELiT.Maxwell

    Spring (with mass) kinetic energy -- velocity assumption

    why we assume that velocity decreases linearly in a spring (i.e. if one end is fixed, then velocity of a particle (of spring) at x from fixed =vx/l where v is the velocity of the free end) and why does it hold good too when the spring (linear mass density) is non uniform... EDIT: spring has...
  45. Dorian Black

    Faraday's Law for a linearly rising magnetic field

    Hi, Imagine a conductive wire bent to the shape of a loop without its ends meeting. A magnet is moved with respect to the loop such that the magnetic field crossing it (perpendicularly) is linearly increasing with time (Φ=kt) where k is a constant. The induced emf is the rate of change of...
  46. K

    Time Dilation & Mass: Does 2x Mass Mean 2x Slower Time?

    I was wondering if time on a planet which is exactly twice the mass of the Earth would pass exactly twice as slow relative to Earth time? So after a year on this big earth, two years would have passed on earth? Thanks
  47. P

    Find two linearly independent eigenvectors for eigenvalue 1

    Homework Statement A linear transformation with Matrix A = ## \begin{pmatrix} 5&4&2\\ 4&5&2\\ 2&2&2 \end{pmatrix} ## has eigenvalues 1 and 10. Find two linearly independent eigenvectors corresponding to the eigenvalue 1. Homework Equations 3. The Attempt at a Solution [/B] I know from the...
  48. E

    Resulting system of equations is not linearly independent

    Homework Statement Solve 2x''+3x'+40x = 40y+3y' Homework Equations y = 0.05sin(10t) The Attempt at a Solution I used the annihilator method to find the answer of x(t) = Acos(10t)+Bsin(10t)+Ce-0.75tcos(sqrt(311)/4t)+De-0.75tsin(sqrt(311)/4t) where A, B, C and D are constants. The initial...
  49. H

    Is E-field in pn-junctions changing linearly with bias?

    Hi guys, there are a lot of textbooks and articles describe the electric field in depletion region at thermal equilibium, but very few of them tell what happens to the field under bias. I am wondering if the electric field in the depletion region simply changes linearly with reverse bias? Or if...
  50. B

    Linearly Increasing Potential Well. Help

    [Note: no template because this post was moved from the QM subforum] I was working on problem #41 and was confused about what the wave function would look like from the time x = 0 to when E=V0. (See image in...
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