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Multiple Choice and some to show work/matrices

This has multiple choice answers as well as some that need some written work. The attachment shows exactly what it needs. Thanks for the time. Please Answer the following questions, some are multiple choice, others require some work. 1. Write the augmented matrix for the given system: 2. Use the system in pro

Determinant Functions and n-Linear Functions

Each of the following expressions defines a function D on the set of matrices over the field of real numbers. In which of these cases is D a 3-linear function? Justify your conclusions. a) D(A) = A11 + A22 + A33 b) D(A) = A11 A22 A33 Recall: Let K be a commutative ring with identity, n a positive integer, and let D

Similar matrices and trace proof

(See attached file for full problem description with symbols) --- If A and B are matrices over the field F, show that the Now show that similar matrices have the same trace. Recall: Let A and B are matrices over the field F. We say that B is similar to A over F if there is an invertible matrix P over F such that . -

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.

Triangles and matrices

Researchers at the National Interagency Fire Center in Boise, Idaho coordinate many of the firefighting efforts necessary to battle wildfires in the western United States. In an effort to dispatch firefighters for containment, scientists and meteorologists attempt to forecast the direction of the fires. Some of this data can be

Union, Intersection, Composition and Symmetric Different

Let R1 and R2 be relations on a set A. represented by the matrices: M R1 0 1 0 M R2 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 find the matrices that represent ( show all work) a) R1 union R2 b) R1 intersection R2 c) R2 º R1 (composition) d) R1 º R1 (co

Sovling Eigenvalues Problem

For an n x n matrix A, show that if one or more of the eigenvalues is zero, A has no inverse. Also show that if, A does have an inverse, the eigenvalues of A^-1 are the reciprocals of the eigenvalue A.

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 and Matrices: Matrix Operations

1. Take the following three row vectors: A = (1, 3), B = (7, 9), C = (7, 2) 1. Find the column vectors V = AT, W = BT, X = CT 2. Create the Matrix D such that A is the first row, B is the second row, and C is the third row 3. Create the Matrix E such that V is the first column, W is the second column, X is the third column 4

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

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