What is Independence: Definition and 354 Discussions

Hello MHB, I got one question. If we got this vector V=(3,a,1),U=(a,3,2) and W=(4,a,2) why is it linear independence if determinant is not equal to zero? (I am not interested to solve the problem, I just want to know why it is) Regards, |\pi\rangle

Use Stokes's theorem to show that line integral of ##\vec{F}(\vec{r})## over an curve ##L##, given by ##\int_L \vec{F}(\vec{r}) d\vec{r}##, depends only on the start and endpoint of ##L##, but not on the trajectory of ##L## between those two points. Hint: Consider two different curves, ##L##...

Here is the question: Here is a link to the question: Linear algebra help please please? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Here is the question: Here is a link to the question: Differential Equations...Linear independence question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Is the set $$ \{cos(x), cos(2x)\} $$ linearly independent?Homework Equations Definition: Linear Independence A set of functions is linearly dependent on a ≤ x ≤ b if there exists constants not all zero such that a linear combination of the functions in the set are equal to...

How can two different regularization schemes give the same physical results? This seems unbelievable. If you impose the same renormalization conditions, then in all regularization schemes, the cutoff, or dimension, or the heavy masses get absorbed into constants in such a way that the...

V1 = (1,2,3,4) V2 = (0,1,0,-1) V3 = (1,3,3,3) a) I already expressed them a linearly dependent set in R4 b) Express each vector in part (a) as a linear combination of the other two linear combo is just {c1v1 + c2v2...cnvn} right? But I don't get where to start to prove this

Homework Statement suppose v(t) , u(t) are two linearly independent solution of the 2nd DE. (6t^2-t-1)y''+t^2e^ty'-(3t^3-t-1)y=t^2e^t-3t^3+1 satisfying the condition v(0)=u(0)=1 , prove that u'(0) ≠ v'(0) Homework Equations The Attempt at a Solution I've tried to use Wronskian...

Consider a plane P in ℝ^{3}. Is it necessarily the case that any vector outside this plane cannot be expressed as a linear combination of finitely many vectors on this plane? I would think yes; if you tried to parametrize the plane P with two parameters, could we somehow show that there are...

Homework Statement Suppose that A, B and C are not linearly independent. Then show how the a_i can be computed, up to a common factor, from the scalar products of these vectors with each other Homework Equations a_1A + a_2B + a_3C = 0 a_1=a_2=a_3=0 Hint - Suppose that there are non-zero...

I cannot visualize the geometry for either of these ideas. Is it the case that two vectors can be linearly independent or dependent of each other? In which case, what is the dependency or independency based on? What are these two vectors independent or dependent of with respect to each other?

Homework Statement Suppose v_1,v_2,v_3,...v_n are vectors such that v_1 does not equal the zero vector and v_2 not in span{v_1}, v_3 not in span{v_1,v_2}, v_n not in span{v_1,v_2,...v_(n-1)} show that v_1,v_2,v_3,...,V_n are linearly independent. Homework Equations linear independence...

The problem is attached. I don't know why he called all 4 vectors V1, I guess it was a typo. Anyways, part I) This is not linearly independent as the determinant of the matrix containing those 4 vectors is 0 I am having trouble with part II) I think I know the answer, but I don't...

Homework Statement Suppose that S = {v1, v2, v3} is linearly independent and w1 = v2 w2 = v1 + v3 and w3 = v1 + v2 + v3 Determine whether the set T = {w1,w2,w3} is linearly independent or linearly dependent. Homework Equations Let c1, c2, c3=scalars c1w1+c2w2+c3w3=0...

The problem is attached. I just wanted to see if the way I proved my statement is correct. My answer: No, because there exists more columns than rows, thus at least one free variable always exists, thus these vectors are linearly dependent.

Hello, I understand that if we have three functions f, g, and h, they are linearly independent <=> the only c1, c2, and c3 that satisfy (c1)f+(c2)g+(c3)h=0 are c1=c2=c3=0. In order to solve for these c1, c2, and c3, we want three equations in the three unknowns. To do this we can...

Hi Physicsforums I am re-learning classical mechanics and having a tough time dealing with a certain line from Thornton/Marion. On page 269 (5th ed), a little after introducing Hamiltonian dynamics and the canonical conjugate equations of motion, the author says: "the qk and the pk are...

Let $V$ be a finite dimensional vector space. Let $T$ be a linear transformation on $V$ with eigenvalue $0$. A vector $v \in V$ is said to have rank $r > 0$ w.r.t eigenvalue $0$ if $T^rv=0$ but $T^{r-1}v\neq 0$. Let $x,y \in V$ be linearly independent and have ranks $r_1$ and $r_2$ w.r.t...

Edit: I think I may have posted this in the wrong section, sorry about that. Note that this isn't a homework problem though, I"m not enrolled in this class, I was just reading over some of this stuff and trying some problems since I"m majoring in physics. I have a textbook "discussion" problem...

Hi How to be mathematically correct about assumption of independence about arrivals of clients at a bank? Physically I understand that there is no possible dependence between2 sequent arrivals of clients but anyway when I make this assumption I want to be correct according literature...

A supplier sends boxes of screws to a factory: 90% of the boxes contain 1% defective, 9% contain 10% defective, and 1% contain 100% defective (eg wrong size). i) What percentage of screws supplied are defective? ii) Two screws are chosen from a randomly selected box. What is the probability...

Dear all, I'm currently reading papers of statistical modelling. I encountered with the concepts of uncorrelatedness and independence. I understand the definitioins, but I am wondering what is the real effects they can make in statistical analysis? For example, I have a dataset and I use...

Homework Statement This is from Serge Lang's "Linear Algebra, 3rd Edition", page 15. Consider the vector space of all functions of a variable t. Show that the following pairs of functions are linearly independent: (a) 1,t (b) t, t2 (c) t, 4 Homework Equations...

This is from my text, "Linear Algebra" by Serge Lang, pg 11: -The two functions et, e2t are linearly independent. To prove this, suppose that there are numbers a, b such that: aet + be2t=0 (for all values of t). Differentiate this relation. We obtain aet + 2be2t = 0. Subtract...

Is it possible that 2 sigma-algebras could be independent under one measure but not independent under another? Many thanks.

Homework Statement I have attempted the questions below but am not sure if I am applying the method correctly to show linear dependence/independence. a)Show that the vectors e1=[1 1 0]T, e2=[1 0 1]T, e3=[0 1 1]T are linearly independent b) Show that the vectors e1=[1 1 0]T, e2=[1 0 -1]T...

What would be the best way to show that functions f(x)=1, g(x)=sin(x) and h(x)=cos(x) are linearly independent elements of the vector space \mathbb{R}^{\mathbb{R}}? I know that the linear independence means that an expression like \alpha \mathbb{x}_1 + \beta \mathbb{x}_2 + \gamma \mathbb{x}_3...

Question : Let A be a 7 × 4 matrix. Show that the set of rows of A is linearly dependent. Answer: The row vectors of a matrix are linearly independent if and only if the rank of the matrix is equal to the number of rows in the matrix. Since rank (A) = 4 , and the number of rows in the...

Q: Is there a set of four vectors in R3, any three of which form a linearly independent set? Prove. Okay so i know what linearly independent is, i have 3 vectors which are linearly independent but I can't find a fourth vector to satisfy the need of the questions like: vectors: v1 =...

Homework Statement Hello! I've been trying to prove a problem. I attach the problem. I refer to the book " Advanced calculus" written by watson fulks. You can find the below information in p. 405~p.417 I want to prove C Exercise problem number 2. Homework Equations The...

Homework Statement Let S be a basis for an n-dimensional vector space V. Show that if v1,v2,...,vr form a linearly independent set of the vectors in V, then the coordinate vectors (v1)s, (v2)s,...,(vr)s form a linearly independent set in the Rn, and conversely. Homework Equations...

Hi guys, I've been working on a question which is as follows: For which real values of c will the set $\{1+cx, 1+cx^2, x-x^2\}$ be a basis for $P_2$? I'm coming up with the answer as no values of c, but am I really wrong? I've only checked linear independence, because it would imply that it...

Homework Statement Let E= A \cup \bar{B} and F= \bar{D} \cup C Assuming that A,B,C,D are independent show that F and E are independent Homework Equations By definition A and by are independent if and only P(AB)=P(A)P(B). The Attempt at a Solution I tried to use set theory to...

Homework Statement Two variables, X and Y have a joint density f(x,y) which is constant (1/∏) in the circular region x2+y2 <= 1 and is zero outside that region The question is: Are X and Y independent? Homework Equations Well, I know that for two random variable to be independent...

Forgive me for not writing in latex, but I searched this site for 10 minutes looking for a latex reference and could not find anything on matrices. Also, excuse for the excessive amount of info. Homework Statement Determine whether this list of 3 polynomials in P1: p1 = 1+3x p2 = 1+2x...

1. Homework Statement If set A={u,v,w} ⊂ R^n is linearly independent, is B={u-v, u+w, v+w}⊂ R^n linearly independent? 2. Homework Equations 3. The Attempt at a Solution Since A is linearly independent, there exist no all non-zero scalars a1, a2, a3 such that a1*u+a2*v+a3*w=0...

Any one please tell me about the term linear independence?and when we say that the function is linear independent

Homework Statement So the dimension is R4. V1=[3 1 1 2], V2=[-2 -1 2 2] and V3=[2 1 2 1] Homework Equations The Attempt at a Solution The only way I know of to test for convergence is to make a matrix out of the row vectors of the vectors above (with the row vectors becoming the...

Homework Statement Let A be an m x n matrix of rank n. Suppose v_1, v_2, ..., v_k \in \mathbb{R}^n and \{v_1, v_2, ..., v_k\} is linearly independent. Prove that \{Av_1, Av_2, ..., Av_k\} is likewise linearly independent. Homework Equations The Attempt at a Solution It says I...

Homework Statement Let Yi = (Z1 + ... + Zi)/(Z1 + ... + Zi+1) for i = 1,...,n and Yn+1 = Z1 + ... + Zn+1 where Zi ~ independent gamma(pi) for i = 1,...,n+1. Prove that the Yi's are mutually statistically independent.Homework Equations U ~ Dirichlet(p1,...,pn;pn+1) iff U = Z/T where Z is the...

Homework Statement Test the set of {1, ln(2x), ln(x^2)} for linear independence in F, the set of all functions. If it is linearly dependent, express one of the functions as a linear combination of the others. Homework Equations N/A The Attempt at a Solution I know if [ a(1)...

I know this isn't quite advanced probability, but I'm not sure if I have this right. I want to show that conditional independence of $X$ and $Y$ given $Z$ does not imply independence of $X$ and $Y$ (and vice versa). So I used coin tosses where: $X=\{$ first coin tails $\}$ $Y=\{$ second coin...

Homework Statement If, in a matrix, there is a column of all zeros, does this mean the given vector/matrix is linearly dependent? An example would be: [1 2 0 4] [2 3 0 1] [5 2 0 7] A few questions to clear up some possible misconceptions: 1) The matrix above is a 4-dimensional vector...

Homework Statement The Attempt at a Solution I don't think I'm really understanding this problem. Let me tell you what I know: A set is linearly independent if a_1 A_1 +...+a_n A_n = \vec0 for a_1,...,a_n \in R forces a_1 = ...=a_n = 0. If f,g,h take any of the x_i \in S, then one of the...

Homework Statement Well it isn't so much the problem as it is the notation used within the problem. But here is the question: Determine whether or not \overline{w} and \overline{v} are linearly independent in R4/U Homework Equations If v \in V then \overline{v} = v + U The...

Hello, I am stuck with the following question. 1. Suppose T ind. C |Z, does it follow that T ind. C ? 2. Suppose T ind. C , does it follow that T ind. C |Z? I think both don't follow, but I don't know how to show it Thanks in advance

Homework Statement If we have a normed vector space, and a sequence of vectors \{\mathbf{v}_k\}_{k=1}^{N} in the normed vector space. If there exists a constant B>0 such that the following holds for all scalar coefficients c_1,c_2\cdots c_N B\sum\limits_{k=1}^N |c_k|^2 \leq...

Homework Statement http://s2.ipicture.ru/uploads/20120117/ReWSCD1f.jpg The attempt at a solution \frac{\partial P}{\partial y}=\frac{2y}{x^3} \frac{\partial Q}{\partial x}=\frac{2y}{x^3} \frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y} According to my notes: Both functions are...

Homework Statement Let E' and E'' be linearly independent sets of vectors in V. Show that E' \cap E'' is linearly independent. The Attempt at a SolutionTo show a contradiction, let E' \cap E'' be linearly dependent. Also let A be all of the vectors in E' \cap E''. Thus, A \subseteq E' and A...

Homework Statement Determine whether the commutativity of (V,+) is independent from the remaining vector space axioms. Homework Equations N/A The Attempt at a Solution I am having a really hard time with this problem. Off the top of my head I could not think of any way to prove...