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Linear Algebra

Homomorphisms

? Let G be a group and let a,b be two elements of G. The conjugate of b by a is, by definition, the element . The centralizer of a, denoted by s the set of all elements g in G such that ga=ag. i) Find all possible conjugates f the permutation ii) Find the centralizer p in . iii) Prove that for any element a in a g

Solbing System of Linear Equations

1. Solve by substitution or elimination method: 3x - 2y = 8 -12x + 8y = 32 2. Solve by substitution or elimination method: 7x - 5y = 14 -4x + y = 27 3. Solve by substitution or elimination method: -4x + 3y = 5 12x - 9y = -15 4. A university boo

Solving Systems of Linear Equations

1. Why do intersecting lines represent a unique solution? Give examples to support your answer. 2. What is the significance of the name 'linear equation' to its graphical representation? 3. The solutions of line m are (3, 9), (5, 13), (15, 33), (34, 71), (678, 1359), and (1234, 2471). The solutions of line n are (3, -9)

Equations

1.Can you show that, given two equations y = m1x + c1 and y = m2x + c2 where c1 and c2 are different, there is no solution if m1 = m2. Interpret this result graphically. Also show that if c1 = c2 then there will be at least one solution no matter what m1 and m2 are. Interpret this result on a graph. 2.In your reading you have

Matrix : Convergence, Pseudoinverse and Single Value Decomposition

Only problems #3 &4-a,(without using any software). 3 . Ax = b we consider the iterative scheme .... where the matrix Q is nonsingular. (a) If ... for some subordinate matrix norm, show that the sequence produced by the above scheme converges to the solution of the system for any initial vector x(0). 4. Given singular

Solving Simultaneous Linear Equations

3. (a) Solve the following systems of equations i) x + 2 y - z = 2 -3 x - y + z = -3 - x+ 3 y - z = 1 ii) 4 x+ -3 y+ z = - 1 -3 x+ y+ -5 z = 0 -5 x -4 z = 0 iii) x1 + x2 + x3 = 3 -3 x1 -17 x2 + x3 + 2 x 4 = 1 4 x -7 x2 + 8 x3 -5 x4 = 1 -5 x2 -2 x3 + x4 = 1 (b) Find the values of k for which

Linear Transformation and Matrix

Please help me solve the following linear algebra questions involving linear transformation and matrices. (see attached) ? Let and let . Define a map by sending a vector to . a) let and be the standard basis vectors of V. let , and be the standard basis vectors of W. Find the matrix of T with respect to

Systems of Equations : Matrices

Please give step by step instructions and name each step like triangular form, augmented matrix etc so I know when and what to do and can understand it. We are not using calculator so the steps need to be shown to the solution. 1) Solve the system using elementary row operations on the equations of the augmented matrix. Fol

Linear Trend Question

The following linear trend equation was developed for the annual sales of the Jordan Manufacturing Company, Y1 = 500 + 60X (in $ of dollars). By how much per year and per month are sales increasing?

The Linear Diophantine Equation

Find the general solution ( if solution exist) of each of the following linear Diophantine equations: (a) 2x + 3y = 4 (d) 23x + 29y = 25 (b) 17x + 19y = 23 (e) 10x - 8y = 42 (c) 15x + 51y

Fermat Numbers

The Fermat numbers are numbers of the form 2 ^2n + 1 = &#934;n . Prove that if n < m , then Φn │ϕ m - 2. The Fermat numbers are numbers of the form 2 ^2n + 1 = (Phi)n . Prove that if n < m , then (Phi)n │(Phi)m - 2.

Perturbed Linear System

Consider the perturbed linear system x' = (A + eB(t))x, x is an element of R^n, where A is a constant matrix, B is a bounded continuous matrix valued function, and e is a small parameter. Assume that all eigenvalues of A have non-zero real part. 1) Show that the only bounded solution of the system is 0. 2) If A ha

Systems of Equations : Five Word Problems

There are hens + rabbits. The heads = 50 the feet = 134. How many hens & how many rabbits ? Flies + spiders sum 42 heads and 276 feet. How many of each class? J received $1000 and bought 9 packs of whole milk & skim milk that totalled $960 - How many packs bought of each kind? A number is composed of two integers and its sum

Transcendental Equation, Positive-Definite and Orthonormal

Solve the eigenvalue problem as follows: Let U = ... be a two-component vector whose first component is a twice differentiable function u(x), and whose second component is a real number u1 Consider the corresponding vector space H with inner product Let S C H be the subspace .... and let .... The above eigenvalue

Differential Operators: Eigenvalues and Eigenfunctions

Please see the attached file for the fully formatted problems. Let L = with boundary conditions u(0) = 0, u'(O) = u(1) ,so that the domain of L is S = {u Lu is square integrable; u(0) = 0, u'(O) = u(1)}. (a) For the above differential operator FIND S* for the adjoint with respect to (v,u) =S 1-->0 v-bar u dx and compare S

Eignevalues and Eigenvectors: Example Problem

Please see the attached file for the fully formatted problems. The Fourier transform, call it F, is a linear one-to-one operator from the space of square-integrable functions onto itself. (In fact, we also know that F is an "isometric" mapping, but we will not need this feature in this problem). Indeed, Note that here x an

Linear algebra

Determine the characteristic values of the given matrix and find the corresponding vectors: [ 2 -2 1 ] [ 1 -1 1 ] [ -3 2 -2 ]

Linear Algebra : Wronskian

Compute the Wronskian of the given set of functions, then determine whether the function is linearly dependent or linearly independent: x^2 - x, x^2 + x, x^2, all x

Linear Algebra And Differential Equations: Real Vector Space

Determine if the given set constitutes a real vector space. The operations of "multiplication by a number" and "addition" are understood to be the usual operations associated with the elements of the set: The set of all elements of R^3 with first component 0

Change of Basis: Eigenvectors

For the problem, refer to the linear transformation T: R^3 --> R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Write the change of basis matrix K from the basis F of R^3 which consists of the eigenvectors of T to the standard basis E for R^3.

Ellipsoid to canonical form

Problem attached. (a) Find the shortest and the largest distance from the origin to the surface of the ellipsoid. (b) Find the principal axes of the ellipsoid.