What is Independence: Definition and 354 Discussions

What could we say if a matrix is invertible? Could we say that it can span and is linearly independent?

Hello everyone. I was going through my Linear Algebra Done Right textbook that threw me off. I hope this forum is appropriate for my inquiry; while there is no problem I'm trying to solve here, I don't know whether just asking for clarification would belong to the homework forum instead. If...

Homework Statement I have five sets of three functions that I need to test for linear independence. I know that I could write them as a linear combination of scalar multiples of the functions, set it equal to 0 and see if I can solve for the scalar multiples. To clarify I mean: Functions...

Homework Statement "if u,v,w are linearly independent then au+bv+cw=0 for some a,b,c in R" first of all is this correct? and how do i proof if this is correct? should i show like this, i know 0u+0v+0w=0 so that imply that statement is true? because 0 is in R is this...

Homework Statement \psi(x,t) = \psi_1 e^{-i E_1 t/2} + \psi_2 e^{-iE_2 t/2} under what conditions is the probability density time independent? Homework Equations |\Psi(x,t)|^2 = \psi(x,t)* \psi(x,t) The Attempt at a Solution i found a statement in pg 71 of prof Richard...

How can I prove given an arbitrary set of vectors v1 and v2, given they are linearly independent, that their sum (v1 + v2) is also linearly independent?

Hasn't this been disproved by wave function collapses?

Homework Statement Hi I seem to remember that if you have a homogenous ODE y'' + p(t)y' + q(t)y = 0 which have the solutions y1 and y2. Where we are told that y1(t) \neq 0 then y1 and y2 are linear independent. I found the simular claim on sosmath.com but are they simply...

Homework Statement If a set is linear independent, proove that every of its non empty subset is linear independent. i'm sorry, but I'm not sure is my sentence correct or not, Homework Equations n/a The Attempt at a Solution let {v1,v2,v3,...,vn} be linear independent set. so...

How do you know when a matrix (or equivocally a system of equations) is linearly independent? How do you know that it's linearly dependent? For example, given this matrix, [ 1 1 2 1] [-2 1 4 0] [ 0 3 2 2] How do we know if this matrix is linearly independent or dependent...

Why did Britain lose the war over America's independence? Yesterday was independance day and it got me wondering why Britain losed the war for American independance. I've never really heard the full story behind it, at the time the British empire had the most powerful military ever known so it...

Thanks to all those who have served and are serving. Thank you for protecting our freedoms and our rights to live as human beings. Let us all sit back, have a beer, and blow **** up.

Homework Statement Let (A_n : n\in \mathbb{N}) be a sequence of events in a probability space. Show that the events A_n are independent if and only if the \sigma-algebras \sigma(A_n)=\{\emptyset, A_n, A_n^c, \Omega\} are independent. Homework Equations For \sigma-algebras \mathcal{A}_i...

Homework Statement Homework Equations If X and Y are statistically independent, then f(x,y) = g(x)h(y) where g(x) = \int f(x,y) dy h(y) = \int f(x,y) dx The Attempt at a Solution (a) g(x) = \int f(x,y) dy = \int_{y=0}^{1-x} 6x\, dy \Rightarrow g(x)=6x(1-x) and...

Hello Forum, since my GR tutor can't help me with some issues arising I thought it is time to register here. I am very confused about the phrase "coordinate independence". Especially regarding the Lie Derivative and the Commutator of two vector fields. 1) The Lie Derivative is said...

Homework Statement I have a table of paired measurements: IQ and brainsize of a person. Question: is there a significant connection between brainsize and IQ?Homework Equations / The Attempt at a SolutionThe only test in my course notes that checks indepedence of continuous variables is a...

Homework Statement Show that the internal energy of a material whose equation of state has the form p = f(V), T is independent of the volume and the pressure. That is \left(\frac{\partial U}{\partial V}\right)_{T} = 0 \left(\frac{\partial U}{\partial p}\right)_{T} = 0 Homework...

If x1, x2,..., xn span \mathbb{R}^n, then they are linearly independent. This is true since n-1 vectors can't span R^n. How can this be written in a more meaningful way?

(Note: this isn't a homework question, I'm reviewing and I think the textbook is wrong.) I'm working through the Gram-Schmidt process in my textbook, and at the end of every chapter it starts the problem set with a series of true or false questions. One statement is: -Every orthogonal set...

Homework Statement A is a 3x3 matrix with distinct eigenvalues lambda(1), lambda(2), lambda(3) and corresponding eigenvectors u1,u2, u3. Suppose you already know that {u1, u2} is linearly independent. Prove that {u1, u2, u3} is linearly independent. Homework Equations ?? The...

Homework Statement Define linear functionals lk on Pn by lk(p) = p(tk) Show that the lk are linearly independent. Homework Equations q1(t)= (t-t2)(t-t3)...(t-tn) The Attempt at a Solution I really don't know where to start. I do have the following equation: If the lks are...

:smile: What do you guys think of the new non-background independent Chamseddine and Connes paper?

Homework Statement Given S = (1+x2, x +x3 And augment S to form a Basis S' of P3The Attempt at a Solution 0 + 0x + 0x2 + 0x3 = a(1+x2)+b(x +x3) = a + ax2 + bx + bx3

This is a question about a problem (not homework) from Ed Thorp's book, http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/ElementaryProbability.pdf . Problem 13 on page 85 outlines a proof that betting systems designed to make an unfavorable game favorable cannot work when there...

Homework Statement Let u,v,w be three linearly independent vectors in ℝ7. Determine a value of k, k= , so that the set S={u-3v,v-5w,w-ku} is linearly dependent. Homework Equations The Attempt at a Solution I don't really know why knowing that we're in ℝ7 will help. I know a...

for a system of particles why is centre of mass independent of the origin?

Homework Statement I have a set of Vector v_1,v_2,v_3,v_4 in \mathbb{R}^4 and need to show that E = v_1,v_2,v_3,v_4 is an ordered basis for \mathbb{R}^4 The Attempt at a Solution I know that for this being the case v = c_1 \cdot v_1 + \cdots + c_4\cdot v_4 where v \in...

Can a non square matrix have linearly independent columns? I can't take the determinant so I can't tell.

Edit: There was a mistake in the question, see below for right question.

Homework Statement One satellite is scheduled to be launched form Cape Canaveral in Florida, and another launching is scheduled for Vandenberg AFB in California. Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral...

Hi I just want to confirm something. I have A,B\in M_{nm}(\mathbb{C}) and I want to prove they are linearly independent. Since they are over the complex field, do I have to prove: (a_1+ib_1)A+(a_2+ib_2)B=0 \ \Leftrightarrow \ a_1=a_2=b_1=b_2=0 ? Thanks.

Homework Statement Consider the two lines, given in the paramentric form L1: x = (0, 1 ,2) + s(1, 0, 2) L2: x = (4, 2, c) + t(-2, 0, d) where c and d are constants. a) For what value of d are the lines parallel? b) With the value of d above, for what value(s) of c (if any) are the...

If u,v andw are three linearly independent vectors of some vectorial space V, show that u + v , u-v and u -2v + w are also linearly independent. Okay, first of all, i know that: \lambda_{1} \times u + \lambda_{2} \times v + \lambda_{3} \times w = (0,0,0) admits only the solution that...

Homework Statement If the set {v1,v2,v3} of vectors in R^(m) is linearly dependent, then argue that the set {v1,v2,v3,v4} is also linearly dependent for every choice of v4 in R^(m). Homework Equations Definitions would be more relevant so... Linearly Independent: If the only solution...

What is the best way for the US (or any other country) to generate 50% of its energy from renewable (or at least source that will last for 500 years) source in the next 20 years? My personal favorite is PV in the US Southwest with water->hydrogen as a storage medium for over night and for...

suppose v1,...,vk are nonzero vectors with the property that vi.vj=0 whenever i is not equal to j. Prove that {v1,...,vk} is linearly independent.

Let V be a linear space and u, v, w \in V. Show that if {u, v, w} is linearly independent then so is the set {u, u+v, u+v+w}

Why electromagnatic waves do not require any medium for their propagation while other waves always require meduim for their propagation...?

Homework Statement We are ask to find the probability density of psi(x,t). I know that psi have an exp term but i don't understand how by squaring psi make the exp term disappear. Homework Equations Psi = sqrt(2/L)sin(n*pi*x / L)e^(-2*pi* i(E/h)t The Attempt at a Solution I...

In the expression ln(-s^2-i\epsilon) , s^2 and \epsilon are positive (this expression can result from for example a loop diagram where s^2 is a Mandelstam variable). In mathematics, the branch cut of ln() is usually taken to be the negative real axis, so that the value above the negative axis...

This is not really a homework problem but it relates to a number of problems, so I thought this would be the most appropriate place to post it. Homework Statement The basic question is about how we define linear dependence in a vector space. For a vector space over some field \mathbb{F}, we...

Independence Problem: Please Help! I have been trying to figure out the proof to this problem for the past couple of days and still don't have an answer. The question is as follows: Let (Q,F,P) be a probability triple such that Q is countable. Prove that it is impossible for there to exist a...

Could GR's "background independence" be a theoretical artifact? ==quote from Rovelli "Unfinished Revolution" (2006) page 2== ...Others, on the other hand, and in particular some hard–core particle physicists, do not accept the lesson of GR. They read GR as a field theory that can be...

Homework Statement If V1...V4 are linearly independent vectors in R4, then {V1, V2, V3} are also linearly independent. True or False. The Attempt at a Solution My solution involved reducing the problem down the 3 vectors in R3. Then show a counter example of this in R3 although I...

Homework Statement Show that if vectors v1 , . . . , vk in a vector space V have the properties that v1 does not = 0, and each vi is not in the span of the preceding ones, then the vectors are linearly independent. Conversely, show that if v1 , . . . , vk is an ordered list of linearly...

Show that the set of functions SIN(nx) where n=1,2,3... is linearly independent ?

Has there been any attempts to create time-independent model of physical world? Thanks!

Homework Statement Let A and B be vector spaces, T:A->B be a linear transformation. Give examples of: (a) T, where a(1),... a(n) are linearly independent vectors in A, but T(a(1)),...T(a(n)) are not. (b) T, where T(a(1)),...T(a(n)) span the range of T, but a(1),... a(n) do not span A...

Homework Statement Prove that if u and v are given non-zero vectors in the arbitrary inner-product space V, and are such that <u,v>=0, then {u,v} is a linearly independent subset of V. Homework EquationsThe Attempt at a Solution I have no idea where to start. It's hard to prove because the...

Homework Statement Show that two planar vectors a and b are linearly independent if and only if they are not parallel. The Attempt at a Solution I know that, if they are not parallel, they will meet and cross in a line. What else should I know before proving this question?