What is Independence: Definition and 354 Discussions

Homework Statement Under what conditions on the numbers a and b are the vectors (1,a), (1,b) linearly independent in R2? Homework Equations The Attempt at a Solution x(1,a)+y(1,b)=(0,0) (x,ax)+(y,by)=(0,0) (x+y, ax+by)=(0,0) x+y=0, ax+by=0 x=-y, ax=-by unsure where to go from...

Homework Statement Let U be the subspace of P3(ℝ) spanned by E={x^3,x^3-x^2,x^3+x^2,x^3-1} find a linearly independent subset F of E spanning U. Homework Equations E={x^3,x^3-x^2,x^3+x^2,x^3-1} The Attempt at a Solution a(x^3)+b(x^3-x^2)+c(x^3+x^2)+d(x^3-1)=0x^3+0x^2+0x+0...

Homework Statement Show if S = {v1,v2,v3} is independent or dependent . . . Homework Equations (0,0,0,0) = k1(a,b,c,d) + k2(e,f,g,h) + k3(i,j,k,l) where {a,b,c,d,e,f,g,h,i,j,k,l \in ℝ} The Attempt at a Solution im trying to tell if i can say that this set of 3 vectors in R4 is...

Homework Statement Show that the set {1,x,x^2,...,x^n,...} is linearly independent in Q[x]. The Attempt at a Solution Since an infinite set of vectors is linearly independent if each finite subset is also linearly independent, I think I need to show that every subset of...

If we take the derivative of n functions that are linearly independent to each other and we write it down like c1f1(x) + c2f2(x) +...+ cnfn(x)=0, then would the linear independence be preserved if we differentiate the equation with respect to x?

Hello, I'm currently self-studying "An Introduction to Convex Polytopes" and I'm having some trouble understanding the different characterizations of affine independence. I understand that for an n-family (x_{1},...,x_{n}) of points from R^{d}, it is affinely independent if a linear combination...

Restrict attention to vectors in ℝ^m where m is a natural number. Let σ be the vector of ones. Let V be the set of vectors whose largest entry is 1 and whose smallest entry is 0. When is it (generically) the case that the set of vectors {σ, v_1, v_2, ..., v_n} is linearly independent...

Are photons in the laser output beam mutually independent and can the multi-photon output be described as the tensor product of individual photon states?

Hi friends, sorry that i have posted so many threads recently regarding complex analysis. i am trying hard to understand as much as possible.anyway i was wondering if anyone had any good geometric interpretation for the equivalence between a differential being exact and it being path...

I am trying to establish whether the following is true (my intuition tells me it is), more importantly if it is true, I need to establish a proof. If $X_1, X_2$ and $X_3$ are pairwise independent random variables, then if $Y=X_2+X_3$, is $X_1$ independent to $Y$? (One can think of an...

what is the basic difference between mutual exclusiveness and independence? actually i got this as a difference between fundamental principle of addition and fundamental principle of multiplication. it is quite urgent.

A server sends 3 digits (0 and 1) to a computer. The probability of sending 0 is p. I have to check under which conditions the events: A={at least 2 of 3 digits is 0} B={all the digits are the same} are independent. I thought that they will be independent if the first 2 digits are...

Explain why the method of decomposition when applied to the solution set of a homogeneous linear system always yields a linearly independent set of vectors whose span is the set of solutions... Can someone explain this it seems reasonable but I can't seem to prove it to myself

Homework Statement there is the vector space F(R) = {f | f:R -> R } show that {1, sin^2(x), sin(2x)} is linearly independent Homework Equations a(1) + b(sin^2(x)) + c(sin(2x)) = 0, where the ONLY solution is a=b=c=0, for the set to be implied linearly independent.The Attempt at a Solution...

Homework Statement Let v_1,v_2 be any two solutions of the differential equation y''+ay'+by=0 such that \frac {v_2}{v_1} is not constant, and let f(x) be any solution of the differential equation as well. Use the properties of the Wronskian to prove that constants c_1,c_2 exist such that: c_1...

Homework Statement The Hamilton-operator is given as \hat{H} and describes the movement of a free rigid object that has the moments of inertia I_{i} Under what circumstances is <\Psi|\hat{L_{1}}|\Psi> time-independent? Homework Equations...

What's wrong with the following: \frac {d \left< E_{k} | \Psi \right>}{dt} = \frac{\partial \left< E_{k} \right|}{\partial t} \left| \Psi \right> + \left< E_{k} \right| \frac{\partial \left| \Psi \right>}{\partial t} = \frac{i}{\hbar} \left< E_{k} | H | \Psi \right> - \frac{i}{\hbar} \left<...

Homework Statement I was trying to prove a theorem from Axler's Linear Algebra text and my proof is different from the one in the book, and I'm wondering if someone can check whether or not my proof works, since I'm just starting to write proofs. Theorem 2.6 (pg. 25): In a finite-dimensional...

Hi everyone, General relativity gives us the definition of the distance between two "events" d=\int \sqrt{g_{\alpha\beta}dx^\alpha dx^\beta}. I don't think this will complicate things but let's say its between two balls. Now like I said, with GR we have the definition of the distance...

63 years Israel was created as the only democratic country with western values in Middle East. In the same time, 60% of Palestinian people left their land and they are not allowed to return until now. If those refugees are allowed to return back, then Israel will not be a Jews state anymore, and...

Under what conditions is Faraday’s Law independent of Ampere’s Law? I want to say that this is so only in the static case, however this isn't right (or at least there's more to it). What am I missing?

Does anyone know a good way to check if a given set of vectors (assume we just know we have a set, not their values) is linearly dependent or linearly independent without a calculator? Ex: Given a set of n-dimensional vectors, say, vector1, vector2, and vector3, how would one determine if these...

Give an example to show that if not assuming independence of X1, X2, ..., Xn it is possible to show that Var(1/n * sum from k = 1 to n of Xk) >> \sigma^2/n

Homework Statement Suppose {V1, V2, ..., Vp} form a linearly independent set of vectors. Show that any subset of this collection of vectors is also linearly independent. Is it necessarily true that is the vectors are dependent, that any subset is also dependent? Homework Equations The...

Homework Statement [PLAIN]http://img220.imageshack.us/img220/7427/diff5.jpg The Attempt at a Solution Done (a). How do I go about (b) and (c)?

Homework Statement Let V be a vector space and \{v_1,...,v_{n+1} \} \subset V a set of linearly independent vectors of V . Show directly: (Don't just quote a theorem!) (a) The set \{v_1,...,v_{n} \} is linearly independent. (b) v_{n+1} \not \in span \{v_1,...,v_{n} \} Homework...

Homework Statement Critique my understanding. Homework Equations From the omniscient Wikipedia: The Attempt at a Solution So if I had (0 0 1)T, (0 2 -2)T, (1 -2 1)T, and (4 2 3)T, then I'd check whether at least one of them can be written as a linear combination of the others by looking...

I'm trying to finish these linear independence proofs: 3. Let S = {v1, v2, v3} be a linearly independent subset of V and let T = {v1 + v2, v2 + v3, v1 + v3}. (a) Show that if char F is not 2, then T is linearly independent. (b) Show that if char F = 2, then T is not linearly independent. 4...

Let V be the vector space of all real-valued continuous functions. t, e^t, sin(t) are in V. Is t, e^t, sin(t) in V linearly independent? My answer is yes. However, how can I prove it which is that which do I have to show or can I just say the def of linear independent?

1) Let u and v be nonzero vectors in a vector space V. show that u and v are linearly dependent if and only if there is a scalar k such that v = ku. Equivalently, u and v are linearly independent if and only if neither vector is a multiple of the other. 2) Let S = {v1, v2, ..., vk} be a set of...

Homework Statement 1. Consider three linearly independent vectors v1, v2, v3 in Rn. Are the vectors v1, v1+v2, v1+v2+v3 linearly independent as well? 2. Consider a subspace V of Rn. Is the orthogonal complement of V a subspace of Rn as well? 3. Consider the line L spanned by [1 2...

1. Homework Statement There are two proofs: Let X and Y be two matrices such that the product XY is defined. Show that if the columns of Y are linearly dependent, then so are the columns of the matrix XY. Let X and Y be two matrices such that the product XY is defined. Show that if...

I tend to think very independently, often coming up with unconventional, sometimes unorthodox, ways of solving problems. I do not like to listen to authority such as having to code up software a certain way or following strict guidelines/formats. Do you think the software...

Hopefully north and south will learn to get along. http://www.bbc.co.uk/news/world-africa-12317927 http://www.bbc.co.uk/news/world-africa-12115013

Q. Consider the following statements about events A, B, and C. - p(A) = 2/3 - p(B) = 1/2 - B c A - Events A and C are independent - Events B and C are mutually exclusive Given that B is a subset of A is what is P(A n B). B is completely contained in A so any point in B is also in A...

If (x,y,z) and (x,y) is algebraically dependent, is then (x,z) algebraically dependent? Does by the way anyone have a good web-source for information about transcendence degrees?

Is there any way to prove that entropy change is independent from the path? dS=dQ/T dQ=dE+PdV d(PV)=PdV+VdP dS=nCd[ln(T)]+nR[ln(T)]-VdP/T i go this far, but it is not very different from dS=nCd[ln(T)]+PdV/T

Duke has been "Velcro Dog" for over a year since we adopted him from a shelter. Tonight was the first time that he has spent the night in the bedroom without me. He is always stuck to me like a burr. Last night, my wife and I spent the evening (New Year's Eve) playing cribbage, dominoes...

Homework Statement Given a set of polynomials in x: x^{r_1}, x^{r_2},...,x^{r_n} where r_i \neq r_j for all i \neq j (in other words, the powers are distinct), where the functions are defined on an interval (a,b) where 0 < a < x < b (specifically, x \neq 0), I'd like to show that this...

Homework Statement Give that u and v are LI and that u and w are LI and that v and w are LI, is the set {u,v,w} LI ? Prove or disprove. The Attempt at a Solution I know that this can be done by providing a counterexample. But I wanted to know if there is a way to prove it generally? That...

Homework Statement http://uploadpie.com/fHoAj Homework Equations The Attempt at a Solution [PLAIN][PLAIN]http://uploadpie.com/fCgEI

I don't this this is an overly complicated proof but it is one I have never seen or done before. Let f be a polynomial with atleast two non-zero terms having different degrees. Prove that the set {f(x),xf'(x)} is linearly independent in P Proof: With out loss of generality we can...

Homework Statement You are given 4 vectors in R^4 which are linearly independent. Do they always span R^4?Homework Equations The Attempt at a Solution Intuitively, I think the answer is yes. I know if I want to show they span R^4, I need to use the general terms, but all I can think of is the...

Homework Statement 0<=x<=1 0<=,y <=1 event A: y<=x event B: y<=1-x Are events A,B independent? Also, if B: x<=1/4 , are A,B independent? The Attempt at a Solution if independent, P(A|B) = P(A) P(A|B) = P(A)P(B) / P(A)P(B)+(1-P(A))P(B) ... ? For B: x<=1/4 they are obviously...

Let the joint PDF of (X,Y) be of the form: f(x,y) = 1/8x(x-y), 0<=x<=2, |y|<=x f(x,y) = 0 elsewhere Find P(X+Y<=2). The answer that my teacher gave was P(X+Y<=2)=∫01dx ∫-xx 1/8x(x-y)dy + ∫12dx ∫-x2-x 1/8x(x-y)dy I do not understand how my teacher could separate the integral like...

Homework Statement Let X~Bernoulli(θ) and Y~Geometric(θ), with X and Y independent. Let Z=X+Y. What is the probability function of Z? Homework Equations The Attempt at a Solution I am getting PX(1) = θ PX(0) = 1-θ PX(x) = 0 otherwise pY(y) = θ(1-θ)^y for y >= 0...

Homework Statement [PLAIN]http://uploadpie.com/nsXSv Homework Equations The Attempt at a Solution I have no idea how to start. To be linearly independent, c1u1+c2u2+...+cnun = 0 has only trivial solution. But I don't know how can I use the given information to prove that

Homework Statement Let {X, Y, Z} be linearly independent in Rn. If {X, Y, Z, W} is linearly dependent, show that W \epsilon span{X, Y, Z}. NB: You must SHOW this. Homework Equations The Attempt at a Solution For W to belong to the span of {X,Y,Z}, W = aX + bY + cZ where a, b, c...

Can i use the invertability of a matrix as an alternative way of determining the linear independence of a set? Thank you.

I'm studying the Poisson Process (rate R) and I'm hung up on the issue of dependence. This seems like and easy question but I have no background in probability whatsoever. By definition, the number of events in disjunction time intervals are independent. Okay. Fine. But say we have an...