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. 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. 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...

30. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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...