Independence Definition and 346 Threads

So I'm watching this demo posted in another problem by @PeroK by Prof. Walter Lewin and I can't help but see that in the part where he is demonstrating the independence of acceleration cylinder length that the (presumably) shorter aluminum cylinder is edging out the win...Not intended, but...

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

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...

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...

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

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}##?

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...

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.

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...

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...

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}...

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 ...

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...

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...

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...

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...

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...

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) -...

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...

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...

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...

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...

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...

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...

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...

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 $\text{dim}X^* = \text{dim}X=n$...

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...

(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...

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...

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...

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...

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...

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)...

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 +...

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.

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...

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...

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...

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...

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...

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{"""} &...

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...

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.

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...

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 $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...

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...

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?

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...

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...

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: $...