What is Independence: Definition and 354 Discussions

Just in time for the 4th of July. http://www.dailymail.co.uk/news/article-1197046/Rare-valuable-copy-American-Declaration-Independence--Surrey.html

CAN somebody, please write down a formula defining the linear independence of the following functions?? { e^x,e^{2x}}

I'm having trouble with one of the rules of probability P(A n B) = P(A)P(B) which holds if events A and B are independent The following problem illustrates my confusion. I've defined Events A and B below, are these events dependent? Per the solution in the book P(A \cap B) = P(A)P(B)...

How can you show that an arbitrary n \times n matrix has n linearly independent eigenvectors? What if all you know about the matrix is that it's the product of a positive-definite matrix and a semi-positive-definite matrix?

Any relationship between mass, length and time in general relativity can be considered using tensors of Einstein's Field Equations which are independent of the coordinate system used and of the origin of that coordinate system Is there a formal name for such coordinate independence and origin...

Can anybody explain to me what string theorists mean when they are talking about background independence in string theory? I understand this concept in the context of general relativity and loop quantum gravity; there it means invariance with respect to active, local diffeomosphisms. In...

Homework Statement If linearly dependent, write one matrix as a linear combination of the rest. \left[\begin{array}{cc} 1&1 \\ 2&1 \end{array}\right] \left[\begin{array}{cc} 1&0 \\ 0&2 \end{array}\right] \left[\begin{array}{cc} 0&3 \\ 2&1 \end{array}\right] \left[\begin{array}{cc} 4&6 \\ 8&6...

So, I have read that the SZ effect is virtually independent of redshift. I follow the argument that the factor of (1+z)^-4 in the surface brightness cancels the (1+z)^4 factor in the photon energy density at the cluster (three factors from space being smaller in each dimension, one from the...

Homework Statement x1, x2, and x3 are linearly dependent. Show that x1 and x2 are linearly independent. Homework Equations After reduction using gaussian elimination, x1, x2, and x3 are proven to be linearly dependent because x1 and x2 are defined by x3 (being the free variable) as...

prove that vectors v_1,..,v_n on a vectorinc space V over field F are linearly dependant if and only if there is an index 1<=i<=n so v_i is a lenear combination of the previus vectors by its index v_1,..,v_{i-1} ?? i got a prove but i can't fully understand it: suppose v_i is a lenear...

The question is: Show that the given integral is independent of the path. F(x,y) = (2xy)dx + (x^2)dy So i take the integral of 2xy w.r.t x and it gives: x^2*y + g(y) now I take the partial derivative of that function w.r.t to y and i get: x^2 + g'(y) I set it equal to...

HEY...so here I go! Abstract: To determine the level of independence between an introductory solid state course to an introductory quantum mechanics course. The deal is I’m going in my third year of a Bsc with honours (yes Canada lol) and specialization in Physics-Mathematics and I’m...

I attempted the proof but I don't know how to complete it.. Let u,v,w be linearly independent vectors and x is in <u,v,w>. Then there are unique a,b,y such that x=au+bv+yw

Linear independence!? Homework Statement Let {p, q} be linearly independent polynomials. Show that {p, q, pq} is linearly independent if and only if deg(p)>=1 and deg(q)>=1. The Attempt at a Solution I am pretty sure the statement to prove is incorrect. If we use deg(p) = -1 and...

Homework Statement Let u and v be two nonzero vectors in R^2. If there is no c E R such that u = cv, show that {u, Bv} is a basis of R^2 and that R^2 is a direct sum of the subspaces generated by U = <u> and V = <v> respectively. Homework Equations Clearly, u and v are linearly...

Determine whether the set {[1,2,-1,6], [3,8,9,10],[2,-1,2,-2]} is linearly independent. 3. The Attempt at a Solution I construct A = \left[\begin{array}{ccccc} 1 & 2 & -1 & 6 \\ 3 & 8 & 9 & 10 \\ 2 & -1 & 2 & -2 \end{array}\right] The row echelon form is A =...

how do i prove that A and (B intersection C) are mutually independent? first of all how do i even read that question, is it read: A union (B intersection C) ??

The man keeps going! I know nothing about this topic other than what is available in mainstream media. Was wondering if those of you in the know could comment on the possibility and viability of these goals and policies? Obama aims for oil independence

Please refresh my memory; if a finite set S is L.I., then does this imply the existence of a set T of the same size (i.e. |T| = |S|) so that the elements T are pairwise orthogonal?

Homework Statement Suppose X is a discrete random variable with probability mass function pX(x)=1/5, if x=-2,-1,0,1,2 pX(x)=0, otherwise Let Y=X2. Are X and Y independent? Prove using definitions and theorems. Homework Equations The Attempt at a Solution The random variables X and Y...

Homework Statement F(r) = r/!r!^3 (Sorry but the ! is supposed to imply that its scalar) I found the curl using the cartesian coordinate definition of curl. It came out to be zero. Now the question is, is F path independent? Its silly, becuase if the curl is zero then it does imply that...

Homework Statement Let X be a random vairable which can only take three values: -1,0,1 and they each have the same probability. Let Y also be a random vairable defined by Y = X2. Show that i) X and Y are not independent ii) X and Y are uncorrelated Homework Equations To show that two...

Homework Statement A=[-2 -7 -1] B=[-2 -4 -3] C=[0 6 -4] Determine whether or not the three vectors listed above are linearly independent or linearly dependent. I have determined that they are linearly DEPENDENT. If they are linearly dependent, determine a non-trivial linear...

Hi all, I've recently been reading about string field theory (note: I'm a novice). As I understand, the string field is an infinite collection of classical fields. But I'm uncertain as to why this formulation leads to background independence? Thanks all.

In probability is there a difference between sets that are disjoint and sets that are independent.

Can you help me with this, or at least give me an idea how to proceed: Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0, but T^m(v) is equal...

Show that a roller coaster with a circular vertical loop. The difference in your apparent weight at the top of the circular loop and the bottom of the circular loop is 6 g's-that is, six times your weight. Ignore friction. Show also that as long as your speed is above the minimum needed, this...

Homework Statement For each n \in \mathbb{N}, let f_n(x) = e^{nx} for x \in \mathbb{R}. Prove that f_1, ... , f_n are linearly independent vectors in {\cal F}(\mathbb{R}, \mathbb{R}) Homework Equations The Attempt at a Solution I know that the simple way to prove this for n=2...

When are to functions y1 = f1(x) and y2 = f2(x) independent? It would apper never, because, we can always write x = f1-1 (y1), and therefore y2 is a function of y1. Every function is dependent of any other function. Generally, dy1/dy2 != 0 for arbitrary functions y1 and y2. Is this reasoning...

My professor says that a linearly independent subset of a vector space automatically spans the vector space, and that a subset of a vector space that spans the vector space is automatically linearly independent. I don't understand why either of these is true.

Homework Statement If f is a harmonic function, that is del^2(f)=0, show that the line integral: (integral)f_y dx - f_x dy is independent of path in any simple region D. The Attempt at a Solution I tried to rewrite the given integral as integral of Q dx - P dy, since path...

Homework Statement Let v, x_1, x_2, x_3 be elements in R^4, and suppose that there are distinct real numbers c1, c2, and c3 such that v = c_1*x_1 + c_2*x_2 + c_2*x_2. Prove that x_1, x_2, and x_3 are independent. The Attempt at a Solution Let A=[x_1 x_2 x_3]. Then Col(A) = span{x_1...

Thought this book was really interesting and folks here might be interested in it. Basically the author claims that the idea of making America "energy independent" is neither reasonable, possible, nor deisrable, and that the whole concept of it is based on a bunch of myths and falsehoods...

Why is the given line integral independent of path in the entire xy-plane? Int((y2 + 2xy)dx + (x2 + 2xy)dy)[/i]

I remember reading about charge independence; about how the energy levels of mirror nuclei (correcting for differences in the colomb term) are identical… I think this suggests that the force between any two nucleons is the same, so the attraction of neutron-proton=proton-proton=neutron-neutron...

[SOLVED] Independence of Motion I have a problem with a diagram that goes along with independence of motion. When air resistance is negligible, objects are supposed to fall at the same rate. In the diagram (a flash photograph), it shows 2 balls falling, one straight down, and the other...

the vectors: v= [-5, -8, 7], u= [2, 4, (-17+k)] and w= [2, 7, 1] are linearly independent if and only if k does not equal ___? - note that the vectors are supposed to be setup vertically with only one column and 3 rows. det[v, u, w] The Attempt at a Solution - I tried...

I'm kinda confused about whether the vectors in a linear span has to be independent. It makes sense intuitively. For example say v and u spans a plane. Then v and u has to be linearly independent. Otherwise they would lie in a line. Can anyone give me an example where vectors span a space and...

Can I ask for some help? Suppose that {v1,v2...vn} is a linearly independent set of vectors and A is a singular matrix. Prove or disprove: The set {Av1, Av2, ...Avn} is linearly independent.

Homework Statement I have three vectors in R^(2x2): (1 0 , 0 1) (That is "1 0" horizontal first line, and "0 1" horizontal second line), (0 1, 0 0) and (0 0, 1 0). I have to determine if they are linear independent or not. I know how to do it in R^(2x1), but not in R^(2x2). What's the...

I can't seem to figure this one out: Question: Let D be a nonempty subset of a vector space V over a field F. Let B be a finite linearly independet subset of span D having n elements. Prove there exists a subset D' of D also having n elements such that span[(D-D') U B] = span(D)...

I need some help with examples. Especially number 2. 1) Name a subset which is closed under vector addition and additive inverses but is not a subspace of R squared. I think I got this one. {(x,y) st x,y are elements of integers} because this isn't closed under scalar multiplication...

Homework Statement where C is the contour given with direction marked by increasing y, and where -2≤y≤2 , compute itgeral(z^2-2z+1)dz. With the condition x=5; Firstly I solved the auestion with the classical way ; taking z= 5 + it where -2≤t≤2; we take the i*integral((5+it)^2-2(5+it)...

I think I am missing a key info below. I have listed the problem statement, how I am approaching and why I think I am missing something. Please advise why I am wrong. Thanks Asif ============ Problem statement: Let T: U->V be an isomorphism. Let U1, U2,...,Un be linearly...

Homework Statement Using the wronskian (determinant basically), determine if e^x, sin(x), cos(x) are linearly independent Homework Equations I used this: | e^{x} sin(x) \:cos(x)| |e^{x} cos(x) -sin(x)| |e^{x} -sin(x) -cos(x)| But pretend that's just a 3x3 matrix and you take...

Homework Statement Proof 1: Show that S= {v1, v2, ... vp} is a linearly independent set iff Ax = 0 has only the trivial solution, where the columns of A are composed of the vectors in S. Be sure to state the relationship of the vector x to the vectors in S 2. The attempt at a solution As far...

Homework Statement Let V and W be vector spaces, Let T: V --> W be linear, and let {w1, w2,..., wk} be linearly independent subset of R(T). Prove that if S = {v1,v2,...vk} is chosen so that T(vi) = wi, for i = 1, 2,...,k, then S is linearly independentHomework Equations The Attempt at a...

Will a set of vectors stay linearly independent after a change of basis? If it's not always true then is it likely or would you need a really contrived situation?

In all descriptions of the http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley" I know of, the two arms of the interferometer are at right angles to each other. Does anyone know of an experiment which, just for completeness, tried other angles? If not, does anyone know of a reference with...

Hi, I'm just learning about Maxwell's equations in high-school and was playing around with them. Supposedly they are 4 independent and self-sufficient equations that when connected with the Lorentz force law will predict classical electrodynamics in its entirety. But then, it appears to me...