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# Matrices

### Matrix relative to a basis for a linear transformation.

--- Let be the linear transformation defined by . a) If is the standard ordered basis for and is the standard ordered basis for what is the matrix of T relative to the pair b) If and , where , , , , what is the matrix T relative to the pair --- See attached file for full problem description.

### Prove the uniqueness of I, the n x n identity matrix.

Prove the uniqueness of I, the nxn identity matrix.

### Discrete Mathematics : Integer Algorithms, GCD, Solving Congruences and Diagonal Matrices

4. Describe an algorithm that takes as input a list of n integers and produces as output the largest difference between consecutive integers in the list. Integers 28. What is the greatest common divisors of these pairs of integers? a) 22 * 33 * 55, 25 * 33 * 52 b) 2 * 3 * 5 * 7 * 11 * 13, 211 * 39 * 11 * 1714 c) 17, 17

### Discrete Math Definitions : Algorithm, Searching algorithm

On the following terms could you please give my an English text description - in your own words. Thanks. 1. Algorithm: 2. Searching algorithm: 3. Greedy algorithm: 4. Composite: 5. Prime: 6. Relatively prime integers: 7. Matrix: 8. Matrix addition: 9. Symmetric: 10. Fundamental Theorem of Arithmetic: 11. Euclidean A

### Vectors : Area of Parallelogram, Perpendicular Vectors , Angles Between Vectors, Orthogonal Vectors and Determinants

1. For vectors v and w in , show that v - w and v + w are perpendicular if and only if . 2. Let u = (-3, 1, 2), v = (4, 0, -8), and w = (6, -1, -4) be vectors in . Find the components of the vector x that satisfies 2u - v + x = 7x + w. 3. Find a non-zero u vector such that satisfies the following. a. u has the same d

### Matrices and Their Use in Coding and Encription

The use of coding has become particularly significant in recent years. One way to encrypt or code a message uses matrices and their properties. We start with a message coded into matrix form, called A. Multiply A by another matrix B to get AB and send the message. a) What would we need to decode the message at the other end t

### Systems of Differential Equations : Fundamental Matrix, Linearly Independent Solutions and Vectors,

Please see the attached file for the fully formatted problems. I need: (c) on #4 (c), (d) and check (b) on #6 (e) on #7 For this to help me with the test coming up I will need all work and answers,

### Diagonal Matrix Representation : Linear Mapping, Basis and Kernels

A linear mapping T: R3 &#8594; 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

### 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

### Symplectic Matrix, Determinant, Characteristic Polynomial and Eigenvalues

If ' is a symplectic matrix show that det ' = 1. Also prove that....where P(...) = det(,,,) is the characteristic polynomial of '. Finally, if ... is an eigenvalue with multiplicity k of ' then the same is true for...

### 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)/p&#1028;Z} 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,&#8734;) &#8594; [0,&#8734;). 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

### Gaussian Elimination and Back Substitution

X + 2y + z = 0 -3x + 3y + 2z = -7 4x - 2y - 3z = 2

### Matrix Operations : Matrix Addition

Given A = 1 2 -2 3 4 5 B = 2 0 1 3 -2 5 C = -4 -6 1 2 3 0 a.) Find A+B and B+A b.) Find A+B+C

### 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

### 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

### Gauss-Jordan Method

2x+3y-2z=1 x-2y-3z=-9 5x+4y-4z=2

### Writing a System of Equations in Matrix Form

Please write the given system in matrix form. *(Please see attachment for complete problem, including system and outline of the form)

### Angles and matrices

Project 3 2. Lance Armstrong won the 2003 Tour de France. The wheel on his bike had a 63 inch diameter. His average speed was 40 km / hour. What was the angular speed of the wheel in revolutions per hour? 3. A construction company is making picnic pavilions where the roof will be supported by two sets of beams. The bea

### Matrices : Vector Equations

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) Compute u + v and u - 2v u = -1 ; v = -3 2 -1 2) Display the following vector

### Matrix Reduction

I am to use matrix reduction on this problem. I can solve systems by matrixs with out any problem but I can't seem to set up the two equations for this problem so that I can even turn them into a matrix and solve. A company produces desks on both the east and west coast. The east coast plant, fixed costs are 16000 per year an

### Evaluate: The Inverse of a Matrix

Matrix row 1 = [0 1] row 2 = [1 0] Please find the inverse of the matrix. Make sure to show all steps and work involved.

### Find the Inverse of a Matrix

Please solve for the following: Matrix row 1 = [1 .5] row 2 = [0 .5] Task: Find the inverse.

### Multiply matrices

Multiply matrix a by matrix b matrix a: row 1 = [0 1 2] row 2 = [-1 4 .5] row 3 = [1 3 0] matrix b: row 1 = [3 -1 5] row 2 = [0 2 2] row 3 = [4 -6 0]

### Multiply Matrices

Multiply Matrix[4 -1] over [2 .5] by matrix [3] over [2]