Linearly Definition and 227 Threads

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...

Hello everyone, this nxn matrix arises in my numerical scheme for solving a diffusion PDE. M = \left(\begin{array}{cccccccccc}1-\frac{Dk}{Vh} & \frac{Dk}{Vh} & 0 & 0 & & & \ldots & & & 0 \\[6pt] \frac{Dk}{h^2} & 1-2\frac{Dk}{h^2} & \frac{Dk}{h^2} & 0 & & & & & & \\[6pt]0 &...

I have a large real symmetric square matrix (with millions of rows/columns). How can I identify the sets of rows that are linearly dependent? More generally, can I determine linear independence of rows with a continuous function where, say, the function is 1.0 for a row that is linearly...

The numbers are subscripts. U1 + U2 + U3 = V1 + V2 + V3 U1 + U2 = V2 I have tried solving for each V in terms of U, but this isn't working out too well.

An example of what I mean: Suppose you had a blueprint for a chemical rocket. You build one, and it has mass m and provides thrust x. Suppose you scale the whole blueprint up by 1% and build another. The volume (and therefore the mass) of each part in the rocket has increased by a factor of...

Hello everyone! This is my first posting. According to Maxwell, an accelerating charge emits a EM wave. All the books I have referred to, talk about the frequency of oscillating charge. How can we determine the frequency of EM wave emitted by a charge that is accelerating linearly? Thank you...

Hi I am trying to do this problem. Verify that \( y_1=x^3 \) and \(y_2=|x|^3 \) are linearly independent solutions of the diff. equation \( x^2y''-4xy'+6y=0\) on the interval \((-\infty,\infty) \). Show that \( W(y_1,y_2)=0 \) for every real number x. I could actually show the above by...

question in attachment. please help!

Hi guys,if i mount a dc motor on a glider that slides in 2 directions without friction, AND, instead of having it drive a mechanism, have a circular plate with an eccentrically drilled hole in it (not centre) mounted on the motor shaft though this hole, i should observe the glider sliding back...

Homework Statement Let S be a linearly independent subset of a Hilbert space. Prove that span(S) is a subspace, that is a linear manifold and a closed set, if and only if S is finite. Homework Equations The Attempt at a Solution Assuming S is finite means that S is a closed set...

Homework Statement Show that if {a, b, c} is a linearly independent set of vectors, then so are {a, b}, {a, c}, {b, c}, {a}, {b}, and {c}. Homework Equations None. The Attempt at a Solution Well I was just thinking that if {a, b, c} is a linearly independent set of vectors, then...

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...

hallo I am trying to calculate the probability to obtain 2 sets of linearly independent vectors from a set of binary vectors of length k. For example: k = 4, and therefore I have 2^k = 16 vectors to select from. I want to randomly select 7 vectors (no repetition). What is the...

Homework Statement Calculate the total charge embodied in a solid with charge density that decreases linearly with height from a value of λ at the bottom to 0 at the top. Solve for a rectangular prism and a sphere. Homework Equations ∫∫∫ρdxdydz ∫∫∫pr^2sinθdrdθd∅ The Attempt at a Solution...

Homework Statement Let U be the subspace of P3(ℝ) spanned by E={x^3,x^3-x^2,x^3+x^2,x^3-1} find a linearly independent subset F of E spanning U. Homework Equations E={x^3,x^3-x^2,x^3+x^2,x^3-1} The Attempt at a Solution a(x^3)+b(x^3-x^2)+c(x^3+x^2)+d(x^3-1)=0x^3+0x^2+0x+0...

Homework Statement Let U be the subspace of R5 spanned by the vectors E={(1,1,0,0,1),(1,1,0,1,1),(0,1,1,1,1),(2,1,-1,0,1)}. Find a linearly independent subset F of E with Span(E)=U Homework Equations The Attempt at a Solution I figured out that E is linearly dependent and that...

Homework Statement If the rows of A are linearly dependent, prove that the rows of AB are also linearly dependent.The Attempt at a Solution A = \begin{pmatrix}a&-a\\b&-b\end{pmatrix} the rows are linearly dependent because a - a = 0 and b - b = 0. B =...

Finite-dimensional V and W are linearly isomorphic vector spaces over a field. Prove that if \{v_{1},...,v_{n}\} is a basis for V, \{T(v_{1}),...,T(v_{n})\} is a basis for W. My attempt at a proof: Let T:V\rightarrow W be an isomorphism and \{v_{1},...,v_{n}\} be a basis for V. Since T is an...

Homework Statement Using the fact that a set S is linearly dependent if and only if at least one of the vectors, vj, can be expressed as a linear combination of the remaining vectors, obtain necessary and sufficient conditions for a set {u,v} of 2 vectors to be linearly independent. Determine...

Registered events X in time interval t are distributed linearly n = n0 + bt. Find probability density function, then t = 10, n0 = 5 and b =2. Find average amount of registered events per day and Mean squared error. Find the probability to register an event per 5th and 6th days. What is the...

Homework Statement Let V be a real vector space and {b_1,b_2,b_3,b_4} a linearly independent set of vectors in V Is the set \left \{ b_1,b_2,b_3,b_1+b_4,b_2+b_4 \right \} The Attempt at a Solution \alpha_1b_1+\alpha_2b_2+\alpha_3b_3+\alpha_4\left \{ b_1+b_4 \right \}+\alpha_5\left \{...

So i have 3 vectors: a= [1 1 1] b= [2 L 0] c= [L 2 3] How do I calculate the L in order to make these vecotrs linearly dependent? How does ß depend from L if v= [ß 0 -1] and v is in span(a b c)? Thank you!

If I create a matrix whose columns are the vectors, and then I row-reduce it and there's a zero row, are the vectors lineraly dependent? why?

Homework Statement Let F be a subset of the complex numbers. Let V be a vector space over F, and suppose α, β and γ are linearly independent vectors in V. Prove that (α+β), (β+γ) and (γ+α) are linearly independent. Homework Equations None. The Attempt at a Solution None. Thanks for your time.

Homework Statement "In each of the given cases, decide whether the specified elements of the given vector space V (i) are linearly independent, (ii) span V, and (iii) form a basis. Show all reasoning. V is the space of all infinite sequences (a0, a1, a2, ...) of real numbers v1 =...

Can circularly polarized light interfere with linearly polarized light?

Given matrices in a vectorspace, how do you go about determining if they are independent or not? Since elements in a given vectorspace (like matrices) are vector elements of the space, I think we'd solve this the same way as we've solved for vectors in R1 -- c1u1 + c2u2 + c3u3 = 0. But I'm...

if i have a 4x3 matrix, this means there are more equations than unknowns and so there are no solutions to the system. does this mean that the row vectors are linearly dependent?

hey i have the set s = {(t,1,1),(1,t,1),(1,1,t)} and i want to find for which values of t this set is linearly independent. For a set of vectors containing all numbers i setup c1v1 + c2v2 .. +cnvn = 0 and I need the only solution to be c1=c2=c3..=cn=0 for linear independence. so then put...

Is a linear equation y'+P(x)y=Q(x) not linear if P(x) and Q(x) are not linearly dependent function? Does linearly dependent mean a constant multiplied by P(x) will equal Q(x)? Thank you.

If I'm given a set of vectors {-4; 3; -10} = v1 {2; -2; -3+k} = v2 {2; -6; 14} = v3 I want to find that they are linearly independent if and only if k != something to solve this is simple but a huge tedious pain (although not nearly as tedious as trying to find a solution to this...

Homework Statement A half wave plate and a quarter wave plate are placed between a Polariser and an analyser .All of these are parallel to each other and perpendicular to the direction of propagation of unpolarised incident light.The optic-axis of the half-wave plate makes an angle 300 with...

Hi, Assume that the real positive numbers x_1,x_2,...,x_n are linearly dependent over the rational numbers, i.e. there are q_1,...,q_n in Q such that x_1*q_1+...+x_n*q_n=0. Is there an algorithm to calculate the coefficients q_i? Is there an algorithm to even check if the x_i's are linearly...

Hey guys was wondering if anyone knew what the go is with linearly dependent solutions to test for exactness, by that I mean I have the differential equation (2x + y^2)dx + 4xydy = 0 (M,N) So i test for exactness and \partialM/\partialy = 2y \partialN/\partialx = 4y So I...

Homework Statement I need to argue this properly Let's say I have a matrix A and rref(A) is given as \begin{bmatrix} 1 & 0&-1 \\ 0& 1 & -1 \end{bmatrix} Since I have a pivot in every row, why isn't this linearly independent? Don't give me other arguments like "because there...

Homework Statement Here is a really simple lin.alg problem that for some reason I'm having trouble doing. Assume that \left\{ v_i \right\} is a set of linearly independent vectors. Take w to be a non-zero vector that can be written as a linear combination of the v_i . Show that \left\{ v_i...

So i have a problem in front of me Let A be a m x n matrix whose rows are linearly independent. Prove that there exists a vector p such taht Ap = e_1 where e_1 =( 1, 0 , 0, 0, 0,0 ,0 ... 0)T i don't even know where to begin

Homework Statement The vectors a, b are linearly independent. For what values of t are = t^2a + b and d = (2t-3)(a-b) linearly independent. also another similar question If the vectors a, b , c are linearly independent, show that a-2b-c, 2a+b, and a+b+c are also linearly...

Homework Statement Basically, the title says it all, I need to figure out whether these functions are linearly independtend on (-infinity, infinity) Homework Equations Wronskian (the determinant of the matrix composed of the functions in the first row, first derivative in the second...

Homework Statement Let u1 = (2; 1; 1; 1) and u2 = (4; 2; 2;-1).I need to extend the linearly independent set u1 and u2 to obtain a basis of R^4. Homework Equations The Attempt at a Solution u1 and u2 are linearly independent since both vectors are non-zero and none is a multiple...

Homework Statement I need to prove that, if {u;v;w} is a linearly independent set in a vector space, then the set {2u + v + w; u + 2v + w; u + v + 2w} is also linearly independent. Homework Equations ... The Attempt at a Solution if {u;v;w} is a linearly independent set=>...

Homework Statement V is a subspace of Rn and S={v1,...,vk} is a set of linearly independent vector in V. I have to prove that any list of linearly independent vectors can be extended to a basis for V. Homework Equations None that I can think of. The Attempt at a Solution So to be...

Let t€R be fixed. Show that {u,v}CR^2 with u=(cost,sint), v=(-sint,cost) is a linearly inpedendent set.

Homework Statement Suppose that a matrix A has real entries (which we always assume) and a complex (non-real) eigenvalue  \lambda= a + ib, with  b not equal to 0. Let W = U + iV be the corresponding complex eigenvector, having real and imaginary parts U and V , respectively. Show that U...