What is Independence: Definition and 354 Discussions

Hello, I am studying independence of events and I came across a formula that I don't understand. It is rather technical, not very interesting, but I feel stuck and it stays in my mind. Could you explain the following : If ## (A_i)_{i\in I}## is a sequence of independent events on...

What are the prerequisites for learning 'Stable Domination and Independence in Algebraically Closed Valued Fields (ACVFs)' ?

I need help with the following situation--- I need to justify the independence of all four axioms using computation and/or narrative. This is what I have so far, but would appreciate any ideas and help you may have to offer. Axiom 1: Each game is played by two distinct teams. Axiom 2: There...

Hi pf, I am having trouble with understanding some of the steps involved in a mathematical proof that a normalized wavefunction stays normalized as time evolves. I am new to QM and this derivation is in fact from "An introduction to QM" by Griffiths. Here is the proof: I am fine with most of the...

i am asked to determine whether 3 vectors which have 5 dimensions (x,y,z,w,u) are linearly dependent or independent in R^3. it doesn't make any sense. should i ignore w and u dimensions and take x,y,z only? because if i dont, all answers would be same, doesn't matter in r^3 or R^4 etc. the...

Hello Forum, I have a lot of questions concerning Wave propagation, but I will not pose them all into one thread, because of my experience, that part of them will be overlooked. So let's get started: 1. Phase shift, E-B- Independence Consider a plane wave generated by a distant dipole...

Homework Statement There's no reason to give you the problem from scratch. I just want to show that 5 trigonometric functions are linearly independent to prove what the problem wants. These 5 functions are sin2xcos2x. sin2x, cos2x, sin2x and cos2x. Homework Equations...

Homework Statement Consider two random variables X and Y with joint PMF given by: PXY(k,L) = 1/(2k+l), for k,l = 1,2,3,... A) Show that X and Y are independent and find the marginal PMFs of X and Y B) Find P(X2 + Y2 ≤ 10) Homework Equations P(A)∩P(B)/P(B) = P(A|B) P(A|B) = P(A) if independent...

This question mostly pertains to how looking at affine independence entirely in terms of linear independence between different families of vectors. I understand there are quite a few questions already online pertaining to the affine/linear independence relationship, but I'm not quite able to...

Homework Statement Prove that if ({A_1, A_2, ..., A_k}) is a linearly independent subset of M_nxn(F), then (A_1^T,A_2^T,...,A_k^T) is also linearly independent. Homework EquationsThe Attempt at a Solution Have: a_1A_1^T+a_2A_2^T+...+a_kA_k^T=0 implies a_1A_1+a_2A_2+...+a_kA_k=0 So...

I was just wondering why the strength of the electric fields of insulating surface charges like sheets and shells aren't dependent on the distance from the charge according to Gauss's Law?

If someone can check this, it would be appreciated. (Maybe it can submitted for a POTW afterwards.) Thank-you. PROBLEM Prove that if $H$ and $K$ are torsion-free groups of finite rank $m$ and $n$ respectively, then $G = H \oplus K$ is of rank $m + n$. SOLUTION Let $h_1, ..., h_m$ and $k_1...

I have been reading these notes: http://isites.harvard.edu/fs/docs/icb.topic455971.files/l10.pdf in which they claim that if two spacetime events are coincident in one frame of reference then they are coincident in all inertial frames of reference, thus spacetime events are absolute i.e. they...

Homework Statement Assume vectors ##a,b,c\in V_{\mathbb{R}}## to be linearly independent. Determine whether vectors ##a+b , b+c , a+c## are linearly independent. Homework EquationsThe Attempt at a Solution We say the vectors are linearly independent when ##k_1a + k_2b +k_3c = 0## only when...

Hi All, I am trying to understand the directional independence of LT. In contrast, we all know that Doppler Effect is dependent on the direction of motion. I have tried to find any reasoning or explanation and could not find one so far. May be I did not use correct terms in my searches. If...

Homework Statement Given the system of vectors \cos x, \cos (x+2), \sin (x-5). Determine whether the system is linearly independent. Homework EquationsThe Attempt at a Solution If it were linearly dependent, there would exist a non-trivial linear combination, such that: k_1\cos x + k_2\cos...

http://postimg.org/image/pb0eu3ap3/ So I've got this graph, and magnitude, M, depends on the number of points, N,... but it shouldn't. http://postimg.org/image/pb0eu3ap3/ The line of best fit is... M = a*N ^b It's been a while since I studied, but knowing their relationship would it be...

Suppose we take the three Newton’s Laws as axioms. Existence of inertial reference frames F = ma F(A on B) = -F(B on A) Also suppose also we are considering purely classical mechanical processes on point particles (no heat transfer, etc.). It is clear to me that the conservation of momentum...

Hello I'm taking linear algebra and have a couple of questions about linear independence, spanning, and basis Let me start of by sharing what I think I understand. -If I have a matrix with several vectors and I reduce it to row echelon form and I get a pivot in every column then I can assume...

I have four axioms and I am stuck trying to prove the independence of these axioms. Axiom 1: Each game is played by two distinct teams. Axiom 2: There are at least four teams. Axiom 3: Exactly six games are played. Axiom 4: Each distinct team played once against the same team. I've justified...

Are a field and its complex conjugate independent? It seems like they're not, as one is the complex conjugate of the other, so if you have one, you know the other. However, it seems in path integrals, you integrate over the field and its conjugate, so they can take on values that are not the...

Hey Guys! I am working on a math project and I am stumped. I'm not sure weather I should use the chi-squared test or another test with my set of data. I am testing weather there is a relationship between crime rates and the unemployment rate of cities. Could someone please help? I'm not testing...

Problem: True or False? If $x$ and $y$ are linearly independent, and if $\{\textbf{x}, \textbf{y}, \textbf{z}\}$ is linearly dependent, then $\textbf{z}$ is in Span $\{\textbf{x},\textbf{y}\}$ Solution: $\textbf{True}$. If $a\textbf{x} + b\textbf{y} = \textbf{0}$ is true and if $a\textbf{x} +...

Hi, so I am given this problem: Using the Wronskian, show that 1, x, x^2,..., x^(n-1) for n>1 are linearly independent. The wronskian is not zero for at least one value in the interval so it is linearly independent, I just do not know how to show it properly.Thank you! :D

Hello I'm french so sorry for the mistake. If we have a manifold and a point p with a card (U, x) defined on on an open set U which contain p, of the manifold, we can defined the tangent space in p by the following equivalence relation : if we have 2 parametered curve : dfinded from...

So I was reading my textbook and I confused myself about a theorem Where if S={v,v2,...,vr} and in ℝn then if r>n, then it is linearly dependent It doesn't make sense to me because if we look at 2 vectors in ℝ3 (lets say u and v) we have u=(u1,u2,u3) and v=(v1,v2,v3) So i do...

Hi, I've been wondering recently what types of systems/processes can give rise to independence. E.g. 'a' and 'b' are independent only given that the system constraints exist. I'm coming from biology so by independence I don't necessarily mean the strict mathematical version but something like...

There's a question in charles curtis linear algebra book which states: Let ##f1, f2, f3## be functions in ##\mathscr{F}(R)##. a. For a set of real numbers ##x_{1},x_{2},x_{3}##, let ##(f_{i}(x_{j}))## be the ##3-by-3## matrix whose (i,j) entry is ##(f_{i}(x_{j}))##, for ##1\leq i,j \leq 3##...

my book says that the surface integral of a gradient field is path independent so long as the gradient field is continuous. This seems fishy to me. I'm envisioning a continuous gradient filed where z=grad f(x,y) and the object traced out looks like a mountain range. The equation for such a...

Apriori -- before taking any of the postulates of special relativity into account -- we might say that the lorentz transformations between two frames K and K', where K' is moving w. speed v along the x-axis of K, is given by $$\vec{x}' = F(\vec x, t)$$ and $$t' = G(\vec x, t).$$ Now, i want...

Homework Statement Determine all values of the constant k for which the given set of vectors is linearly independent in \mathbb R^4. {(1, 1, 0, −1), (1, k, 1, 1), (4, 1, k, 1), (−1, 1, 1, k)} Homework Equations The Attempt at a Solution So far I set up a coefficient matrix...

There is a famous example that electron couldn't absorb the whole incoming photon without emitting another one. Instead of the normal way, I try to prove it simply by argument ( which might be wrong ). There are four constraints in the process, one from energy conservation, three from momentum...

On page 13 of Griffith's "Introduction to Quantum Mechanics 2nd ed" David goes into a long (relatively speaking) proof of why a normalized pair of quantum state vectors will not at some time later become "un-normalized". It seems like just putting the Psi's in a braket the e^(-it) "time...

Hi, I came across this statement while I was reading an article on resolution of vector in arbitrary basis. "v = αa + βb + γc ---(1) , where a, b and c are three independent vectors. we observe that the coefficient α cannot involve any overlap of v with either b or c ; β cannot involve any...

It seems to me that if a row is able to be zeroed out through Gaussian reduction that the determinate of that matrix would equal zero. Therefore, we know that at least one of equations/vectors that constructed the matrix was formed from the other two rows. That is -- that equation is dependent...

Hello, I have a 16x16 contingency table, and I would like to perform a test in order to reject the hypothesis of independence between the variable X (columns) and the other variable Y (rows). Unfortunately many cells in the table contain zeros. The zeros are due to the fact that the sample...

Homework Statement Given that { u1, u2, u3, u4, u5, u6 } are linearly independent vectors in R16, and that w is a vector in R16 such that w ∉ span{ u1, u2, u3, u4, u5, u6 }. a) Is the set { 0, u1, u5 } is linearly independent? b) the set { u1, u2, u3, u4, u5, u6, w } is linearly...

Homework Statement A vector field $$ \vec{u}=(u_1,u_2,u_3) $$ satisfies the equations; $$ \Omega\hat{z} \times \vec{u}=-\nabla p , \nabla \bullet \vec{u}=0$$ where p is a scalar variable, \Omega is a scalar constant. Show that \vec{u} is independant of z. Hint ; how can we remove p from...

After four years of postdoc I'm wondering how one develops further. I have set up a research line that is independent from my supervisors, but I'm using the lab resources. That means my supervisors are claiming the project and are closing me in. I've obtained funding for a PhD student, which was...

Homework Statement Let x1 = (1, 2, -1, 1), x2 = (-1, -1, -1, -1), x3 = (1, 1, 1, 0), x4 = (-2, -1 -4 -1) Show that x1, x3 and x4 are linearly independent Homework Equations The Attempt at a Solution Now I used the equation: ax1+bx2+cx3+dx4=0 Hence forth the augmented...

What is the difference between path independence and a conservative vector field? It seems like both definitions given in the textbook mean the same. What is the difference between the two? According to wikipedia, path independence is a consequence of a conservative vector field, but...

Say I have a matrix ##A## that has linearly independent columns. Then clearly ##A^T## has lin. indep. rows. So what can we say about ##A^TA##? Specifically, is there anything we can say about the rows/columns of ##A^TA##? I'm thinking there has to be some sort of relation but I don't know what...

The question in my book is posed as: Find an example in which P(AB) < P(A)P(B). The answer is given as a coin toss letting A be the event of obtaining heads and B being the even of obtaining tails. I'm confused as to how that's working. My understanding is no doubt 'off', but I thought it was...

Homework Statement Homework Equations The Attempt at a Solution For part (a): a*1 + b*√2 + c*√3 = 0 assume a, b, c not all zero a + b√2 = -c√3 a2 + 2b2 + 2ab√2 = 3c2 a2 + 2b2 - 3c2 = -2ab√2 (a2 + 2b2 - 3c2)/(-2ab) = √2 which is not possible since we take a, b, c to...

The probability that john will go to UCLU is estimated at 1/5; the probability that he will go to some other university is 1/3. The probability that his sister Mary will go to UCLU is 1/4. Calculate the probabilities that: a) John and Mary both go to UCLU; b) John will not go to university...

Hello! I've read thousand of explanations about how q and q-dot are considered independent in the Lagrangian treatment of mechanics but I just can't get it. I would really appreciate if someone could explain how is this so and (I've seen something about an a-priori independence but I couldn't...

Hi I know this may be a silly question but i am doubting myself on how i did this question: Suppose X and Y are independent, with E(X) = 5 and E(Y) = 6. For each of the following variables Z, either compute E(Z) or explain why we cannot determine E(Z) from the available information: Z =...

Given two vectors x(t) = (e^t te^t)^T y(t) = (1 t)^T a) Show that x and y are linearly dependent at each point in the interval [0, 1] b) Show that x and y are linearly independent on [0, 1] I compute det([x y]) = 0, so they are linearly dependent how about part b. Isn't a)...

Help needed. Let A1, A2 and B be events with P(B)>0. Events A1 and A2 are said to be conditionally independent given B if P(A1nA2|B)=P(A1|B)P(A2|B). Prove or disprove the following statement: Suppose 0<P(B)<1. If events A1 and A2 are conditionally independent then A1 and A2 are also...

I don't know whether this has much general interest. I am interested by communication horizons in cosmology, or causal horizons if you prefer. This paper claims to study several cases including how far apart two quasars would have to be to have been out of contact ever since the end of the...