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Matrices

Diagonal Matrix Representation, Standard Basis Vectors

A linear mapping T : R2 -> R3 is defined on the standard basis vectors via T(e1) = (1, 0, 1) and T(e2) = (1, 0, -1) 1. Calculate T(3,3) 2. Find the dimension of the range of T and the dimension of the kernel of T. 3. Find the matrix representation of T relative to the standard bases in R2, R3. 4. Find bases {v1, v2}

Diagonal Matrix Representation : Linear Mapping, Basis and Kernels

A linear mapping T: R3 → R2 is defined on the standard basis vectors via: T (e1) = (0, 0), T (e2) = (1, 1), T (e3) = (1, -1) i. Calculate T(4,-1,3) ii. Find the dimension of the range of T and the dimension of the kernel of T. iii. Find the matrix representation of T relative to the standard bases in R3, R2. iv.

Invertibility of Matrices

Can you please explain to me why the columns of the nxn matrix A span R^n when A is invertible? I feel that if matrix A has columns that span R^n, then the inverse of A should likewise share that same characteristic, the spanning. But I'm not sure if that is a sufficient relationship. Can you give an example of matrix A spanning

Parametric/simultaneous equations and matrices.

See attached file for full problem description with diagrams and equations --- Parametric equations and matrices. The diagram below shows a line defined by the parametric equations , which crosses the x- and y-axes at the points (a, 0) and (0, b), respectively. The region marked A, is bounded by this line, the x- axes, th

Duality Principal and Multiplying Matrices of Differing Size

See the attached file for the full solution. 1) Please explain in your own words the duality principle. 2) The biggest problem I have with matrices is the multiplication. I get them right but I believe the confusion comes from the way it is set up. To be more clear the way it is set up as far as the rows and columns. If it is

Nilpotent of a Matrix

Linear Algebra Matrices (XII) Nilpotent of a Matrix Show that the matrix A = [8, 10 , -16; 32 , 0 , 0; 24 ,

Generating seed values

I am a licensed land surveyor in Illinois and Montana, and I write surveying software (I've been out of college for 20+ years). Currently, I am programming a 3D Conformal Coordinate Transformation, also known as the seven-parameter similarity transformation. I have the book "Adjustment Computations" by Wolf & Ghilani. Section 1

Gaussian Elimination and Operation Count

Consider the problem Ax=b where A is a tridiagonal matrix. What is the operation count for the forward elimination and the back substitution steps of Gauss elimination in this case? Count add/sub and mult/div operations separately, then give the overall order of the total operations needed. (Use O(n^p) notation).

1. This question is concerned with subgroups of the group S5 of symmetries (or permutations) on the set {1,2,3,4,5}, a group with 120 elements. (a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6 and give examples of each of these. Explain why this group does not have cyclic subgroups of any order. (b) By considering the symmetry groups of appropriate geometric figures, give examples of : (iv) a subgroup of order 4 that is not cyclic; (v) a subgroup of order 6 that is not cyclic; (vi) a subgroup of order 8. (c) By considering those permutations that fix one element, or, otherwise, give an example of a subgroup of order 24 and another of order 12. [You need not list all the elements of these groups, but you should explain clearly which elements constitute each subgroup.] (d) List the potential orders of subgroups of S5 (other than S5 itself), according to Lagrange's theorem, in addition to those already considered in this question. Give an example of a subgroup of one of these orders. 2. (a) Which of the following sets are groups under the specified binary operation? In each case, justify your answer. (ix) Z, the set of integers, under operations * defined by a*b = a + 2b (x) R*, the set of non-zero real numbers, under the operation × defined by x×y = 5xy (xi) The set {3,6,9,12} under multiplication modulo 15. (xii) The set of matrices {(1, p;0,1)/pЄZ} under matrix multiplication. (b) G is a group of real functions with domain and co-domain the non-negative real numbers, i.e. functions [0,∞) → [0,∞). The group operation in G is function composition. If one of the elements of G is the squaring function, f, defined by f(x) = x2 , explain why G must be an infinite group. 4. (a) Define the motion of conjugacy as it applies in a general group. Prove that the inverses of a pair of conjugate elements are also conjugate. Prove that conjugate elements have the same order. The remainder of this question concerns the group G, whose Cayley table is as follows: e a b c d f g h i j k l ----|-------------------------------------------------------------------------------------------------- e | e a b c d f g h i j k l a | a b e d f c h i g k l j b | b e a f c d i g h l j k c | c f d e b a j l k g i h d | d c f a e b k j l h g i f | f d c b a e l k j i h g g | g h i j k l e a b c d f h | h i g k l j a b e d f c i | i g h l j k b e a f c d j | j l k g i h c f d e b a k | k j l h g i d c f a e b l | l k j i h g f d c b a e (b) Determine the inverse and the order of each of the elements of G. (c) Simplify each of the following: (i) acb (ii) bca (iii) ajb (iv) bja (v) gcg (d) Given that the only element conjugate to g is g itself ( you need not prove this), determine the conjugacy classes of G. (e) Find H, a normal subgroup of G having three elements. Identify the elements of the quotient group G/H and determine its isomorphism type.

Modern Algebra Group theory Symmetric Groups Permu

Associative & Commutative Rule

Prove that addition modulo n, written +n is: 1) associative 2)comutative there are two ways to prove these properties. each way requires a definition or two: 1) for n&#8805;2, 0&#8804;a, b&#8804;n+1 a+n(written as a power in a corner downside, but dont know how to put it tho) b={condition 1 - a+b if a+b<n;condition 2 - a

Determinants, Cofactors and Permutations

Q1. Suppose An is the n by n tridiagonal matrix with 1's everywhere on the three diagonals... Let Dn be the determinant of An; we want to find it. (a) Expand in cofactors along the first row of An to show that Dn = Dn-1 - Dn-2 (b) Starting from D1 = 1 and D2 = 0 find D3, D4, ..., D8. By noticing how these numbers cycle a

Systems of Equations : Real World Situations and Determinants

1. In real-world situations, what is the advantage of using the Method of Substitution to solve a system of equations rather than using the Method of Addition? 2. When solving a 3x3 determinant, we broke the determinant down into a sequence of 2x2 determinants, remembering to alternate the signs of the leading coefficients in

Equations, point of intesection, plane of intersection

14) Solve the following system of equations x1 - 3 x2 =5 -x1 + x2 + 5 x3= 2 x2 + x3 =0 15) Determine if the system is consistent. Do not completely solve x1 + 3 x3 =2 x2-3x4 =3 -2x2+ 3x3 + 2 x4= 1 7 x4= -5 17) Do the three lines x1 -4 x2 = 1, 2 x1 -x2 = -3, - x1 -3 x2 =4 , have a common point of intersection? Exp

Matrices

Need to see problems on attachment done to further grasp concepts I am missing. Please do not leave out any details in solving the problems. If I ask questions please answer them in detail and please make everything simple and clear to understand. My goal is to apply what you solve to other problems I want to work on on my ow

Matricies

Need to see problems on attachment done to further grasp concepts I am missing. Please do not leave out any details in solving the problems. If I ask questions please answer them in detail and please make everything simple and clear to understand. My goal is to apply what you solve to other problems I want to work on on my ow

Matrices : Gauss-Jordan Elimination

65. A father, when dying, gave to his sons 30 barrels, of which 10 were full of wine, 10 were half full, and the last 10 were empty. Divide the wine and flasks so that there will be equal division among the three sons of both wine and barrels. Find all the solutions of the problem. (from Alcuin) 4, 5 Find all solutions of the

Matrices: Gaussian Elimination, Calculation Time and Cramer's Rule

Questions: a) How many multiplications are necessary to find the determinants of matrices which are 2x2, 3x3, 4x4? b) The number of multiplications for an nxn matrix may be found in terms of the number for an (n-1)x(n-1) matrix. FIND THIS FORMULA and use it to obtain the number of multiplications for a 10x10 matrix. c) Fo

System of Equations: Matrix Form and Eigenvalues

Consider the attached system of equations. (a) Write the system in the given matrix form {see attachment} (b) Determine the eigenvalues of A in terms of the parameter {see attachment} (c) The qualitative nature of solutions depends on .... (d) Sketch a typical phase portrait... Please see attachment for complete set of

Finding Adjoint Operators

Problem: Find the adjoint operator and its domain for: a) . (Assume is continuously differentiable and is continuous on the interval . This is the answer, I just need to see the steps to arrive there: , with boundary conditions . b) Again, here is the answer, but I need to see the solution method:

Matrix Proof

Prove that ||x^(k) - x|| <= (||T||^k)(||x^(0) - x||) and ||x^(k) - x|| <= (||T||^k/(1-||T||))(||x^(1)-x^(0)||), where T is an n x n matrix with ||T|| < 1 and x^(k)=Tx^(k-1)+c, k=1,2,..., with x^(0) arbitrary, c belonging to R^n, and x=Tx+c.

Matrix Norm

The frobenius norm (which I know is not a natural norm)is defined for an n x n matrix A by ||A||_f = (sum i=1 to n, sum j=1 to n, |a_ij|^2)^1/2 Please show that ||.||_f is a matrix norm. That is, satisfy the five axioms. NOte: _ is subscript.

Least Square Problems

Let P = A(A^TA)^-1A^T, where A is an mxn matrix of rank n. (1) Show that P^2 = P. (2) Prove P^k = P for K = 1,2,.... (3) Show that P is symmetric. ____________________________________________________ Let A&#1028;R^(mxn) and let r be a solution to the least square problem Ax=b. Show that a vector y&#1028;R^n

Isomorphism Example Problem

If G ={a + b*sqrt2 | a,b rational} and H = {matrix a 2b, b a | a,b rational}, H is a 2 x 2 matrix - a 2b b a show that G and H are isomorphic under addition. Prove that G and H are closed under multiplication. I know I need to define the function map first, but I don't know what it is in this problem, let alone prove