What is Independence: Definition and 354 Discussions

Independence is a condition of a person, nation, country, or state in which its residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the status of a dependent territory.

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  1. A

    I The third central moment of a sum of two independent random variables

    Is it true that when X and Y are independent, E ({X+Y}3) = E (X3)+E(Y3)?
  2. G

    I Independence of Trace-Partition function

    I am trying to calculate the partition function of the system of two completely decoupled systems. Probability-wise, the decoupled nature means that the PDF is the product of the PDF of each subsystem. I just wanted to be sure that it would translate into: $$ H = \sum_{k_i...
  3. uxioq99

    Time Independence of the Momentum Uncertainty for a Free Particle Wave

    Mine is a simple question, so I shall keep development at a minimum. If a particle is moving in the absence of a potential (##V(x) = 0##), then ##\frac{\langle\hat p \rangle}{dt} = \langle -\frac{\partial V}{\partial x}\rangle=0## will require that the momentum expectation value remains...
  4. VVS2000

    A Independence of generalized coordinates and generalized velocities

    How can I make sense of this and further how to think of this in the context of phase space diagrams?
  5. Salmone

    I Linear independence of three vectors

    If I've got three vectors ##\vec{a}##, ##\vec{b}## and ##\vec{c}## and ##\vec{a}##, ##\vec{b}## are linearly independent and ##\vec{c}## is linearly independent from ##\vec{a}##, is ##\vec{c}## also linearly independent from ##\vec{b}##?
  6. Lynch101

    B Statistical Independence in Quantum Mechanics

    Very basic question here, about statistical independence in quantum mechanical experiments. The quote from PD below is what prompted the question. When we talk about "some kind of pre-existing correlation" are talking about a simple correlation in the sense of the correlation of sunglasses and...
  7. isaacdl

    A Proving $g(u,v)≠0$ with Linear Independence

    I'm trying to prove that there exist always a vector w whose contraction with a lightlike vector u (g(u,u)=0) it's always different from zero: $g(u,v)≠0$I know how to do this with coordinates, but in a free cordinate scheme I'm totally lost. Any help? PD: Both vectors are linearly independent.
  8. rudransh verma

    B Independence of vertical and horizontal motion

    It says when the ball is shot the can is released and they both hit each other at the same height ie they travel same distance down. But that is only possible when the ball starts it’s downward journey the same time as the can starts it’s own. Shooting a ball upward direction will give it some...
  9. dextercioby

    I One more talk about the independence of Einstein's SR axioms

    Sorry if this is discussed here previously, but I just stumbled upon an article from 1911 which I would like to bring forth to you. Preamble: it is generally thought that Einstein's (refined) two axioms of SR (1. The laws of physics are invariant upon shifting from one IRF to another. 2. The...
  10. 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}...
  11. F

    I Proving linear independence of two functions in a vector space

    Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...
  12. P

    Other How to develop independence as a PhD student?

    Hello everyone! So, just out of curiosity, how does one learn to come up with their own questions to answer and come up with ideas for new experiment/science as a graduate student? I ask because in my experience, graduate students get assigned a project from their professor (which hopefully the...
  13. P

    I Are these events independent?

    Hello everyone. Let us consider 3 events A,B,C such that: $$P((A \cap B )\cup C)=P(A)*P(B)*P(C)$$ Notice that the second term is a union and not an intersection. Are they independent? And what if the assumption was: $$P(A \cap( B \cup C))=P(A)*P(B)*P(C)$$? I know that the independence condition...
  14. entropy1

    I (In)dependence in entanglement experiments

    If an entanglement experiment, whereby an entangled pair of particles is measured at both ends, is independent of the next entanglement experiment with another pair of entangled particles, how can there be a correlation? It seems that each independent run does not influence the next run, but...
  15. GlassBones

    Isomorphisms preserve linear independence

    Homework Statement Let ##T:V \rightarrow W## be an ismorphism. Let ##\{v_1, ..., v_k\}## be a subset of V. Prove that ##\{v_1, ..., v_k\}## is a linearly independent set if and only if ##\{T(v_1), ... , T(v_2)\}## is a linearly independent set. Homework EquationsThe Attempt at a Solution...
  16. S

    Exponential Order Statistics and Independence

    Homework Statement Consider the exponential probability density function with location parameter ##\theta## : $$f(x|\theta) = e^{-(x-\theta)}I_{(\theta,\infty)}(x)$$ Let ##X_{(1)}, X_{(2)},...,X_{(n)}## denote the order statistics. Let ##Y_i = X_{(n)} - X_{(i)}##. Show that ##X_{(1)}## is...
  17. QuasarBoy543298

    Hamiltonian mechanics - the independence of p and q

    in the Lagrangian mechanics, we assumed that the Lagrangian is a function of space coordinates, time and the derivative of those space coordinates by time (velocity) L(q,dq/dt,t). to derive the Hamiltonian we used the Legendre transformation on L with respect to dq/dt and got H = p*(dq/dt) -...
  18. M

    A Numerics: Wronskian and linear independence

    Hi PF! I'm solving a differential eigen-value problem in weak form, so I have trial functions. If the Wronskian of trial functions is small but not zero, is linear independence an issue? I have analytic trial functions but am numerically integrating.
  19. thariya

    A Independence of Operator expectation values

    Hi! I want to know under what conditions the operator expectation values of a product of operators can be expressed as a product of their individual expectation values. Specifically, under what conditions does the following relation hold for quantum operators (For my specific purpose, these are...
  20. J

    MHB Linear Map, T^p(x)=0, Show Linear Independence

    Problem: Suppose V is a complex vector space of dimension n, and T is a linear map from V to V. Suppose $x \in V$, and p is a positive integer such that $T^p(x)=0$ but $T^{p-1}(x)\ne0$. Show that $x, Tx, T^2x, ... , T^{p-1}x$ are linearly independent.During class my professor said it was "a...
  21. Pushoam

    Checking the linear independence of elements of 2 X 2 matrices

    Homework Statement Homework Equations 3. The Attempt at a Solution [/B] ## |3 \rangle = |1 \rangle - 2 ~ |2 \rangle ## So, they are not linearly independent. One way to find the coefficients is : ## |3 \rangle = a~ |1 \rangle +b~ |2 \rangle ## ...(1) And solve (1) to get the values of a...
  22. shahbaznihal

    A On metric and connection independence

    Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast. I will be very thankful if...
  23. M

    Linear algebra, field morphisms and linear independence

    Homework Statement Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K. Homework Equations f1, ..., fn are field morphisms => Ker (fi) = 0 (injective) The Attempt at a Solution I...
  24. Q

    Independence of Position and Velocity in Lagrangian Mechanics

    In Lagrangian mechanics, both q(t) and dq/dt are treated as independent parameters. Similarly, in Hamiltonian mechanics q and p are treated as independent. How is this justified, considering you can derive the generalized velocity from the q(t) by just taking a time derivative. Does it have...
  25. barcodeIIIII

    Proof: Time independence of the entropy under unitary time evolution

    Homework Statement The unitary time evolution of the density operator is given by $$\rho(t)=\textrm{exp}(-\frac{i}{\hbar}Ht)\,\rho_0 \,\textrm{exp}(\frac{i}{\hbar}Ht)$$ General definition of entropy is $$S=-k_B\,Tr\,\{\rho(t) ln \rho(t)\}$$ Proof: $$\frac{dS}{dt}=0$$ Homework Equations I am not...
  26. L

    MHB Question about proof of the linear independence of a dual basis

    This is from Kreyszig's Introductory Functional Analysis Theorem 2.9-1. Let $X$ be an n-dimensional vector space and $E=\{e_1, \cdots, e_n \}$ a basis for $X$. Then $F = \{f_1, \cdots, f_n\}$ given by (6) is a basis for the algebraic dual $X^*$ of $X$, and $\text{dim}X^* = \text{dim}X=n$...
  27. J

    A Time independence of a Noether charge in QFT?

    In classical field theories, I believe I understood how to derive a Noether charge that corresponds to a symmetry of action. And there is no problem in understanding its time independence. But in quantum field theory, it looks like the two different approaches, 1) Canonical quantization...
  28. Clifford Engle Wirt

    B Conditional Probability, Independence, and Dependence

    (Mentor note: link removed as not essential to the question.) The problem is: what is relevance anyhow? My questions are these: did I get the math right in the following? Is there a better, more acceptable way to lay out the sample space Ω and the two events F and E? Apart from the math...
  29. LarryS

    I Measures of Linear Independence?

    My formal education in Linear Algebra was lacking, so I have been studying that subject lately, especially the subject of Linear Independence. I'm looking for functions that would qualify as measures of linear independence. Specifically, given a real-valued vector space V of finite dimension...
  30. S

    Independence of Random Variables

    Homework Statement Given ##f_{X,Y}(x,y)=2e^{-x}e^{-y}\ ;\ 0<x<y\ ;\ y>0##, The following theorem given in my book (Larsen and Marx) doesn't appear to hold. Homework Equations Definition ##X## and ##Y## are independent if for every interval ##A## and ##B##, ##P(X\in A \land Y\in B) = P(X\in...
  31. Drakkith

    Linear Independence of Two Functions

    Homework Statement Use definition (1) to determine if the functions ##y_1## and ##y_2## are linearly dependent on the interval (0,1). ##y_1(t)=cos(t)sin(t)## ##y_2(t)=sin(t)## Homework Equations (1) A pair of functions is said to be linearly independent on the interval ##I## if and only if...
  32. S

    Proving Independence of Fano's Geometry Axiom 4

    Homework Statement In Fano's Geometry, we have the following axioms a. There exists at least one line b. Every line has exactly three points on it c. Not all points are on the same line d. For two distinct points, there exists exactly one line on both of them e. Each two lines have at least one...
  33. redtree

    I Independence of variables in Convolution

    Given a convolution: \begin{equation} \begin{split} g(x) * h(x) &\doteq \int_{-\infty}^{\infty} g(z) h(x-z) dz \end{split} \end{equation} Do ##z## and ##x## have to be independent? For example, can one set ##x=z+y## such that: \begin{equation} \begin{split} \int_{-\infty}^{\infty} g(z)...
  34. Mr Davis 97

    T/F Question of linear independence

    Homework Statement T/F: Let ##T: V \rightarrow W##. If ##\{v_1,v_2,...,v_k \}## is a linearly independent set, then ##\{T(v_1), T(v_2),..., T(v_k) \}## is linearly independent. Homework EquationsThe Attempt at a Solution This seems to be true, because we know that ##a_1v_1 + a_2v_2 + \cdots +...
  35. Dimani4

    I Classical Newtonian Physics: Space & Time Independence

    Hi folks, Tell me please why in classical Newtonian physics one can say that the space and time are independent? But we have equations of motion which clearly show this dependence (x=Vt; x=x0+1/2at^2+v0t). Thank you.
  36. A

    Question about linear independence

    Homework Statement Homework EquationsThe Attempt at a Solution if there exists a set with 3 vectors, and all of them are linear independent, then by definition no linear combination of the 3 vectors can equal to 0. I believe that's an accurate definition right? So in this case, the answer...
  37. Mr Davis 97

    Linear independence of polynomials of different degree

    Homework Statement Let S be a set of nonzero polynomials. Prove that if no two have the same degree, then S is linearly independent. Homework EquationsThe Attempt at a Solution We will proceed by contraposition. Assume that S is a linearly dependent set. Thus there exists a linear dependence...
  38. Mr Davis 97

    Linear Independence of a Set of Vectors

    Homework Statement Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. Homework EquationsThe Attempt at a Solution I think that that it would be easier to prove the logically equivalent statement: Prove that a set S of vectors...
  39. nomadreid

    I Independence of the conditions

    Working through a paper about whose rigor I have my doubts, but I am always glad to be corrected. In the paper I find the following: "We now investigate the random variable q. There are the following restrictions on q: 1) The variable q must characterize a stochastic process in the test...
  40. Mr Davis 97

    I Linear independence of functions

    Is there a difference between the linear independence of ##\{x,e^x\}## and ##\{ex,e^x\}##? It can be shown that both only have the trivial solution when represented as a linear combination equal to zero. However, the definition of linear independence is: "Two functions are linearly independent...
  41. Mathman2013

    Doing Chi-square independence test in SPSS?

    Homework Statement Hi there I have problem I hope some can help me solve. My H0, there is no connection between people who received the new and old medication? And them getting well or not well. I suppose to test the following data using Chi-square test in SPSS. \pmatrix{\\\textrm{"""} &...
  42. K

    Math proof: Linear Independence

    Homework Statement How can I show that if a vector (in a vector space V) cannot be written as a linear combination of a linearly independent set of vectors (also in space V) then that vector is linearly independent to the set? Homework Equations To really prove this rigorously it would make...
  43. C

    I Question On Linear Independence

    We were going over linear independents in class and my professor said that if y1 and y2 are linearly independent then the ratio of y2/y1 is not a constant, but he never explained why it is not a constant.
  44. A

    MHB Proving Linear Independence

    In a problem I am working on, it is given that $V_1, V_2, ... , V_n$ are mutually perpendicular vectors in a space defined with a certain scalar product. I need to prove or disprove that $V_i$ are linearly independence regardless of any definition of scalar product. I think the solution should...
  45. weezy

    Proof of independence of position and velocity

    A particle's position is given by $$r_i=r_i(q_1,q_2,...,q_n,t)$$ So velocity: $$v_i=\frac{dr_i}{dt} = \sum_k \frac{\partial r_i}{\partial q_k}\dot q_k + \frac{\partial r_i}{\partial t} $$ In my book it's given $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...
  46. F

    I Vector components, scalars & coordinate independence

    This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##). In my more...
  47. Nick tringali

    Is a force field like the one in Independence Day possible in real life?

    When the US fighter aircraft fire on the alien spacecraft , the pilots learn that the spacecraft is protected by some type of an invisible force field. Does our current understanding of how the universe works allow for such a thing?
  48. D

    Path Independence: Solving a Physics Problem

    Homework Statement Hi,I am trying to understand the proof attached. My problem is shown by a red arrow.Can someone explain those 2 steps? And please answer it as simply as you can...Since I haven't done multivariable calculus..It is a physics course.Thanks Homework EquationsThe Attempt at a...
  49. S

    I Independence of speed of light and velocity of source

    I've been attempting to learn special relativity, but I've encountered a stumbling block. I understand that the speed of light is independent of the speed of the source of the light (similar to how sound waves travel at a speed that is independent of the speed of the energy source of those...
  50. G

    MHB How can the Wronskian be used to determine linear independence?

    I'm asked to check whether $\left\{1, e^{ax}, e^{bx}\right\}$ is linearly independent over $\mathbb{R}$ if $a \ne b$, and compute the dimension of the subspace spanned by it. Google said the easiest way to do this is something called the Wronskian. Is this how you do it? The matrix is: $...