Independence Definition and 346 Threads

Let V be a vector space over a field F, v_{1}, \cdots, v_{n} \in V and \alpha_{1}, \cdots, \alpha_{n} \in F. Further on, let the set \left\{v_{1}, \cdots, v_{n}\right\} be linearly independent, and b be a vector defined with b=\sum_{i=1}^n \alpha_{i}v_{i}. One has to find necessary and...

I need to check the proof of the proposition below we got for homework, thanks in advance! Proposition. Let V be a vector space over a field F, and S = \left\{a_{1}, \cdots, a_{k}\right\}\subset V, k\geq 2. If the set S is linearly dependent, and a_{1} \neq 0, and if we assume there is an...

It would be nice if we had a transcript of the very interesting KITP video of a couple of days ago (20 October) especially the remarks by David Gross about the "background independence" of string theory (by which something is meant that seems to be different from what has been usual in QG...

I'm stuck on a question in linear algebra, it reads "Show that the subset S={cos mx, sin nx: m between 0 and infinity, n between 1 and infinity} is linearly independent. I really just don't know where to start. I've seen a similar question which was just sin (nx) and the lecturer integrated...

LINEAR ALGEBRA: 3 vecotrs in R^4 (with 6 variables) -- Are they linearly independent? For which values of the constants a, b, c, d, e, anf f are the following vectors linearly independent? Justify your answer...

I'm trying to get the idea of independent events well grounded in my mind, but I'm having some difficulty. First of all, what would a venn diagram for independent events look like? I know you have the two events intersecting, and the intersection is equal to the product of the probabilities...

My teacher gave us an intuitive idea of what it means for two vectors in \mathbb{R}^2 to be linearly independent (they aren't multiples of each other) and for three vectors in \mathbb{R}^3 (they aren't on the same plane). Now the book has generalized the idea of linear independence to n...

Is emergent spacetime the same as saying background independence?

if X and Y are events which are independent of each other, but neither are independent with A, is this equality true for conditional probabilities: P( X, Y | A) = P(X|A) * P(Y|A) if not, how do you solve for P(A | X,Y) given that you only know P (A) and P(X|A) and P(Y|A)? The reason I came...

REF: Smolin; Perimeter Institute; -- “The case for background independence” What does background independence mean. Are statistical probabilities needed to compare separated events. If the background that physics works on varies relative to distant locations in GR does that mean GR is...

"I think energy independence is going too far" - Shell Oil http://www.msnbc.msn.com/id/13296235/page/5/ I think it is clear that this is not who we need making the calls for what's good for the US. Energy needs to be taken out of the hands of big oil...

Here's a simple question that I can't seem to get: "Suppose for some v T^{m-1}v\neq 0 and T^mv=0. Prove that (v,Tv,...,T^{m-1}v) is linearly independent." I know that m\leq \dim V and v,Tv,...,T^{m-1}v are all nonzero.

Here's an interesting way to look at CR I feel is often overlooked: Let: z = x + i y z^{\ast} = x - i y One common form for the CR condition is to say that if some function f is analytic then it does not depend on z^{\ast}\;. That is, \frac{\partial f}{\partial z^{\ast}} = 0 But...

I am trying to understand how the Cauchy-Goursat theorem of complex analysis differs from the usual conditions for independence of path in real vector calculus. My complex analysis textbook emphasizes that the Cauchy-Goursat theorem is true even if the function we are integrating does not...

Is it possible to prove 2 vectors are linearly independent with just the following information?: A is an nxn matrix. V1 and V2 are non-zero vectors in Rn such that A*V1=V1 and A*V2 = 2*V2. Is this enough information, or is more needed to prove the LI of the 2 vectors?

Hi, I don't know how to do the following proof: If (v1, ...vn) are linearly independent in V, then so is the list (v1-v2, v2-v3, ...vn-1 -vn, vn). I can do the proof if I replace 'linearly independent' with 'spans V' ...so what connection am I missing? Thanks much!

Hello, there is this question in the book: --------- Consider the vector space of functions defined for t>0. Show that the following pairs of functions are linearly independent. (a) t, 1/t --------- So if they are linearly independent then there are a,b in R, such that at + b/t = 0 So if we...

Hello, my book has this question, and no examples (very) similar to it, so I am wondering if I did it correct :smile: --------- The following circuit operates if and only if there is a path of functional devices from left to right. The probability that deach device functions is as shown...

Hi I just need some help on understanding some general notation in this quesiton: Prove if {x_1,x_2,..,x_m} is linearly independent then so is {x_1,x_2,...,x_i-1, x_i+1,...,x_m} for every i in {1,2,...,m}. I don't really understand what the difference between {x_1,x_2,...,x_i-1...

If a,b,c are vectors in an R-vector space then their sums a+b, a+c, b+c are also linearly independent. If R is replaced by Z_2 then this fails, because there's the nontrivial solution to x(a+b)+y(a+c)+z(b+c)=0 where x=y=z=0 or x=y=z=1 right?

Is there a linear algebra theorem or fact that says something like For a linear transformation T:Rn -> Rm and its standard m x n matrix A: (a) If the columns of A span Rn the transformation is onto. (b) If the columns of A are linearly independent the transformation is one-to-one. Is...

I put together two questions : a) suppose there is a point mass with mass M..if it is moving, then from a certain oberver, the total energy is higher, via E=Mc^2...hence, following the generaly relativity qualitatively, where the energy density defines the curvature, the gravitation should be...

How do I prove the linear independence of the standard basis vectors? My book is helpful by giving the definition of linear independence and a couple examples, but never once shows how to prove that they are linearly independent. I know that the list of standard basis vectors is linearly...

Observation Independence -- what and why? So, heuristically, observation independence is the condition that separated measurements are statistically independent. I have lots of questions about this! (Actually, I have lots of questions about the theoretical foundation of using statistics at...

Prove this questions using ration ideal in intuitive way. Prove this implications and explain the results: (a) A _|_ B => not A _|_ not B, onde _|_ means that events A and B are independent. (b)[ P(A|C) >= P(B|C) ] and [ P(A|not C) >= P(B|not C) ] ==> P(A) > P(B)

Hi, can someone help me with the following question? Q. Show that if \left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to } \right\} is linearly independent and \mathop {v_{k + 1} }\limits^ \to \notin span\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to...

Why the "need" for background independence in String It seems to me in any theory the first cause is still missing regardless of whether it is background dependent or independent so what does it matter ? What is wrong with having a background against which nature is acted out ?

In the list of talks on the program for Strings 06, how many titles do you expect will mention background independence? Looking at the Strings 05 program, you see this year the number is ZERO http://www.fields.utoronto.ca/programs/scientific/04-05/string-theory/strings2005/speakers.html...

What's meant by - background independence/ dependence -landscape ? thank you

How do you know if this: | 0_-8_5| |3_-7_4 | |-1_5_-4| | 1_-3_2| a linearly independent set? The answer at the back of the book say that it is independent, but obvious there are free variable in this matrix , thus imply a nontrival solution for AX=0, so it must be depend. Let...

Can someone help me out with the following question? Use coordinate vectors to determine whether or not the given set is linearly independent. If it is linearly dependent, express one of the vectors as a linear combination of the others. The set S, is \left\{ {2 + x - 3\sin x + \cos x,x +...

Given a set of m real functions of n variables, what is a necessary and sufficient condition for the functions to be functionally independent ? A set a functions f_i(x_1,...x_n)\quad i=1,...m are functionally independent, if the only function \phi(u_1,...u_m) such that \phi(f_1,...f_m)=0 is...

Hi, If I transform a set of linearly independent vectors by a one-to-one linear transformation, is the transformed set also linearly independent?

Hey all, I need to show whether or not the following statement is true: For v_1,...,v_n\in Z^m, the set \{v_1,...,v_n\} is linearly independent over Z \Leftrightarrow it is linearly independent over R. The reverse direction is true of course, but I'm having some trouble showing whether or...

Hello, I have a homework problem I need some help with. Given 3 matrices: A = [1,2,1,2]^t B = [2,3,-1,0]^t C = [1,0,1,0]^t I need to use test of independence to find out whether they are independent. So, the matrix I ended up with is this 1 2 1 0 0 1 2 0 0...

Can anyone tell me why friction does not depend on the area of contact??

Why is the formula for period of simple harmonic motion independent of amplitude?

If the curl of a vector field is zero, then we can that the vector field is path independent. But there are cases where this is not true, I was wondering how? Whats the explanation for this? Thanks in advance for any help. - harsh

How do I determine this: Problem: The vectors: v1, v2, ... , vn, n >= 4 and are linearly independent. Determine if the following vectors are also linearly independent. a) the vectors v1 - v2, v2 + v3, v3 + v1 b) the vectors v1 - v2, 2(v2 - v3), 3(v3 - v4), ..., n(vn - v1) c) the...

Hi...Can you please check if my proof is correct? Exercise: A1,A2,...An are independently events. Prove that : P(A1[union]A2[union]...[union]An) = 1-Πi[element-of]I(1-P(Ai)) note for this (Πi[element-of]I(1-P(Ai)) I={1,2,...n) P([intersect]Ai)= Π P(Ai) for 3 events A1,A2,A3 means...

Of course, the majority opinion among knowledgeable physicists seems to be “Yes”. However, there are at least two cogent arguments that Background Independence (hereafter BI) should be regarded as a bug, rather than a feature. First, local conservation of energy. In particular, GR, the...

It is said that a flaw in string theory is that it is background dependent, whilst loop quantum gravity isn't background dependent. I don't see why this would actually be a flaw, because from what I understand is that background dependence means that the theory relies on a background of...

Tensors, a reason of great schism in physics? Yes, this application of that aristotelian principle, you have described so well, where the third is excluded and as so uncertainty is the reason why there won`t be a quantum gravitation theory, and a reason why physical laws must not be expressed...

I read from a website that the electric field is independent of the separation between the plates if the separation is negligibly small. But how to derive this?

Hello. I want to ask questions... I hope you can guide me in showing the proof. 1. Let a, b and c be vectors in a vector space such that {a, b} is linearly independent. Show that if c does not belong to span {a, b} , then {a, b, c} is linearly independent. I know that is {a,b}...

A PF poster recently asked about the Background Independence feature that distiguishes classical 1915 General Relativity from Stringy theories. She suggested that the explanation be at the "basic Xray tech" level. That probably means starting with a short exerpt from Smolin article in...