Linearly Definition and 217 Threads

Let ${}\left\{X, Y, Z, W\right\}$ be an independent set in $\Bbb{R}^n$, is the following set independent? ${}\left\{X+Y, Y+Z, Z+W, W+X\right\}$ My textbook says it isn't, but I'm not sure why. Let $\lambda_1, \lambda_2, \lambda_3, \lambda_4$ be scalars, then...

Question: If \textbf{v}_1,...,\textbf{v}_4 are in \mathbb{R}^4 and \{\textbf{v}_1, \textbf{v}_2, \textbf{v}_3\} is linearly dependent, is \{\textbf{v}_1, \textbf{v}_2, \textbf{v}_3, \textbf{v}_4\} also linearly dependent? My Solution: http://s29.postimg.org/4wvwjlkqd/Linearly_Independent_Sets.png

Hi, so I am given this problem: Using the Wronskian, show that 1, x, x^2,..., x^(n-1) for n>1 are linearly independent. The wronskian is not zero for at least one value in the interval so it is linearly independent, I just do not know how to show it properly.Thank you! :D

How the system of equations ##y=-2x+1## ##y=x+1## ##y=2x+1## are linearly dependent. In wiki its written for the above system of equations "one equation linearly dependent on the others"

Does an emf create across the body of a magnet when the magnet is linearly moving across its own field. If ABCD is a rectangular magnet of thickness t, AB is the north pole and CD is the south pole. If the magnet moves in the direction of t, that is perpendicular to ABCD plane, will there be an...

Hey everyone I just had a quick thought that was bothering me. For a set to be a basis it must be linearly independent and span the vector space. I've seen cases however of only two vectors forming a basis for R4 I don't see how two vectors could span 4 space or am I missing something. Thanks!

http://tinypic.com/r/euht06/8 Hi everyone! It was the question given to us in today's test & I was wondering if anyone can tell me what is the correct answer. The teacher will post the answers by the end of this week but I am feeling anxious to know the answer. Thanks in advance. P.S. Sorry...

Homework Statement Determine if the given set of functions is linearly independent or linearly dependent.Homework Equations $$S=x~sin~x, ~ x~cos~x$$The Attempt at a Solution My first instinct was to use the Wronskian. $$W[y_1(x), y_2(x)]=\begin{vmatrix} x~sin~x & x~cos~x\\ x~cos~x+sin~x &...

Homework Statement For which value of k are the vectors linearly dependent? v1=[1,0,1] v2=[1,1,2] v3=[1,2,k] <--Originally written as column vectors Homework EquationsThe Attempt at a Solution None of the vectors look like multiples of each other to me...so time to row reduce. Since each...

Homework Statement Hey everyone. I'm in an introductory differential equations class, and I think this homework problem has got me stumped. The functions y1(x), y2(x), ... , yn(x) are linearly independent on an interval I. c1y1(x)+c2y2(x)+...+cnyn(x)=0 for all x in I, implies that...

I have attached both the question and my attempt at solution. My problem is reducing the equation further to get the value of a/L

I have a linear time-varying linearly perturbed ODE of the form: \dot{x} = [A(t)+B(t)]x where A(t) is a bounded lower-triangular matrix with negative functions on the main diagonal, i.e. 0>a^0\ge a_{ii}(t). The matrix B(t) is bounded, so that ||B(t)|| \le \beta. The question is...

Homework Statement Given 2 1 1 0 0 0 1 1 0 0 0 3(i) Show that the rows of A are linearly independent. (ii) Show that the nonzero rows of any matrix in row echelon form are linearly independent.The Attempt at a Solution i) REF gives 1 0 0 | 0 0 1 0 | 0 0 0 1 | 0 0 0 0 | 0 x1 = x2 = x3 =x4 =...

I was doing a question in Taylor book (example 3.3) where a sticky putty is thrown at a stationary wheel. To solve it we use conservation of angular momentum. What I am confused about is that the wheel is initially at rest and has no angular momentum initially. But when the putty is thrown at it...

Homework Statement Say we have the plane, x+2y+4z=8 (part of a larger problem) Homework Equations The Attempt at a Solution The vectors (8,0,0) and (0,0,2) both lie in the plane. They are linearly independent. But (0,4,0) lies in the plane and is not a linear combination of the first two...

i.e., does the set of functions of the form, \{ x^{\frac{n}{m}}\}_{n=0}^{\infty} for some fixed m produce a linearly independent set? Either way, can you give a brief argument why or why not? Just curious :)

Homework Statement Consider a harmonic, electromagnetic plane wave traveling along a line from the origin to the point (3,5,6). It is linearly polarized and its electric field lies in a plane perpendicular to the direction of travel of the wave. The wavelength of the wave is 2.0mm and it has a...

Homework Statement I need to find the Kinetic energy of a bar rotating about its center of mass. I know the bar as length 3b and it's center of mass is located at 2b, the bar density changes linearly along it's length. Homework Equations T=1/2 W^2 I The Attempt at a Solution...

Homework Statement A cart is acclerating to the right with a=3m/s^2. Fluid is in hydrostatic state. Find the force on the back wall. Cart goes .8m into the page. In the image dotted line is free surface when cart is stationary. Homework Equations \vec{\nabla P}=\rho (\vec{g} -...

Why column 1 is M^2*v? How can we know? Please see attached. Many thanks.

Homework Statement Check if the following set of vectors are linearly dependent or independent: A) V1= \stackrel{1}{1} V2= \stackrel{1}{3} B) V1= \stackrel{\stackrel{1}{2}}{3} V2= \stackrel{\stackrel{2}{1}}{3} C) V1= \stackrel{1}{3} V2= \stackrel{2}{1} V3= \stackrel{-1}{2} Homework...

Consider the thought experiment in the diagram below. According to the authors, the vector will rotate even in the rest frame of the moving train relative to the body of the train. Is it possible to consider the spin axis of the gyroscope represented by the vector as a simple rod and...

I know eigenvectors corresponding to different eigenvalues are linearly independent but what about a set ${e_{1},...,e_{n}}$ of eigenvectors corresponding to different eigenvalues?

1. Prove that if A is symmetric and B is skew-symmetric, then {A,B} is a linearly independent set. I am going to need some help to solve this. Not sure how to begin. Homework Equations The Attempt at a Solution

Say I have a matrix ##A## that has linearly independent columns. Then clearly ##A^T## has lin. indep. rows. So what can we say about ##A^TA##? Specifically, is there anything we can say about the rows/columns of ##A^TA##? I'm thinking there has to be some sort of relation but I don't know what...

Hi, i would like to have a hint for the following problem: Let $$v_1, v_2 \&\ v_3 $$ in a vector space V over a field F such that$$ v_1+v_2+v_3=0$$, Show that $\{v_1,v_2\}$ spans the same subspace as $\{v_2,v_3\}$ Thanks in advance

The question is: are x and ix linearly dependent or independent? My first guess is that they should be linearly dependent since i is a constant. But when you apply the definition of linear independence i.e. when you solve ax+ibx=0 (where x≠0), you get a=-ib which shows that the only...

i know S = { $$1 , x, x^2$$} is linearly dependent set for p2. where $$(a_0, a_1, a_2) = (0,0,0) $$ I wanted to use the Wronskian on { $$1 , x, x^2, x^3$$} , but as I understand, it only proves linear independence and not the converse.

Homework Statement Let a1, a2, ... an be vectors in Rn and assume that they are mutually perpendicular and none of them equals 0. Prove that they are linearly independent. Homework Equations The Attempt at a Solution Consider βiai + βjaj ≠ 0 for all i, j => βiai + βjaj +...

Given two vectors x(t) = (e^t te^t)^T y(t) = (1 t)^T a) Show that x and y are linearly dependent at each point in the interval [0, 1] b) Show that x and y are linearly independent on [0, 1] I compute det([x y]) = 0, so they are linearly dependent how about part b. Isn't a)...

Homework Statement For what value(s) of a are the following vectors v1=[1,2,-1], v2=[0,1,3], and v3=[a,4,5] linearly dependent? Homework Equations Since linear dependence means that anyone of the vectors can be expressed as a linear combination of the other vectors: sv1v1+s2v2=v3 where s1and...

Hi there, I have a homework where I have to do this: Prove that square matrix is invertible if the columns of the matrix are linearly independent. There is also a hint: You can help with the following statement: Linear transformation L: U->V is bijection when a vector space basis N for U...

I have a partial linearly distributed load on a cantilever and would to determine the bending moment distribution along the beam. I usually do this by taking cuts along different sections of the beam and finding the expressions for shear force and bending moment. However when I do this...

Homework Statement There is a vector space with real entries of all 2x2 matrices. You have to find what values of \alpha\inℝ make the set Z = \{ \begin{pmatrix} 1 & 2\\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 3 & 7\\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 2 & 6\\ \alpha & 0...

Homework Statement Determine whether the following vectors are linearly independent in P3 "not sure what P3 stands for maybe polynomial of third degree?" 1,x2,x2-2 let p1(x)=1 p2(X)=x2 p3=x2-2 c1p1(x)+c2p2(x)+c3p3(x)=z where z=0x2+0x+0 I then create a matrix using the above...

According to special relativity, velocity cannot be added to another velocity linearly. But I was thinking, what about kinetic energy? K.E. = mv2/2. As velocity increases, the object's mass also increases. The way I see it, the increase in mass "compensates" for the less-than-expected increase...

Hi, Homework Statement Given a vector space V and its basis B = {v1, v2, ..., vn}, I was asked to prove that: a group of vectors {u1,...,uk} in V is linearly dependent if and only if {[u1]B,...[uk]B} is linearly dependent. Homework Equations The Attempt at a Solution I proved...

I've a small wind turbine that is rated at about 24V. The turbine produces AC voltage but it's got an internal recifier converting this to DC. I need to create an embedded system that measures the output of this turbine displaying AC voltage, current and power output. I did some research...

Forgive me ahead of time, I don't really know how to use LaTeX, (it's on my to do list). Homework Statement Given the vector space C([0,pi]) of continuous, real valued functions on the given interval, as well as the inner product <f,g>=integral(f(t)*g(t))dt from 0 to pi: a) Prove the set...

Hello everybody I have to show that this set of vectors a = (e-t, e-it, et, eit ) is linearly independent. My attempt : f(x) = k1 * e-t + k2 * e-it + k3 * et + k4 * eit f '(x) = k1 * -e-t + k2 * -ie-it + k3 * et + k4 * ieit f ''(x) = k1 * e-t + k2 * -e-it + k3 * et + k4 * -eit f(0) =...

Hi, I want to whether there is a function (/matrix) such that it can generate a m-dimensional vector such that this generated vector will always be linearly independent of the set of vectors the function has already generated. My problem can be written in pseudocode format as follow. I...

I don't know how to write out matrices nicely on this forum, but suppose you have some matrices:[1 0 3] [2 0 4] [0 0 5] This would, by definition, be linearly dependent, spanning a plane in r3..is this correct? Since c1=0, c2=anything, c3=0 where c1v1+c2v2+c3v3=0 With this: [1 0 3 5] [3 0...

Homework Statement Prove that if you have 3 linearly independent equations in 3 variables, then there exists only 1 solution to the system. Homework Equations linear independence implies none of the equations can be expressed as a linear combination of the other equations. The...

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.

Question Source : Elements of Engineering Electromagnetics 6th edition by Rao. Page 202 problem3.30 Problem: Three sinusoidally time-varying polarized vector fields are given at a point by F1 = 3^(1/2) * ax * cos(wt +30) F2 = az * cos(wt+30) F3 = [ 0.5ax + 3^(1/2)ay + 0.5*3^(1/2)az ] *...

A thin wire of length L and uniformly density ρ is bent into a circular loop with center at O.The moment of inertia of it about a tangential axis lying in the plane of loop is. Ans : Mass M is not given,but ρ is given. So M=ρL3->(1) (L3 means L cube,no idea how to post it in that manner!). For...

I had kind of a general question. Say I have a second order, homogeneous ODE. Say I use one of the general techniques to generate a complementary solution for my ODE and I end up with something of the form y = C1(solution1) + C2(solution2) Am I gauranteed that these two solutions will be...

I'm trying to find possible errors in this lab experiment that I did, and one question I am thinking about is "Does ice melt linearly?" I assumed that it does so in my calculations, but now I'm not so sure. I measured the mass of melted ice (water) over a period of time and assumed that it...

Homework Statement Suppose that T:W -> W is a linear transformation such that Tm+1 = 0 but Tm ≠ 0. Suppose that {w1, ... , wp} is basis for Tm(W) and Tm(uk) = wk, for 1 ≤ k ≤ p. Prove that {Ti(uk) : 0 ≤ i ≤ m, 1 ≤ j ≤ p} is a linearly independent set.Homework Equations The Attempt at a Solution...