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Matrices

Hessian Matrix : Maximizing Profit

Question: The profit maximizing input choice A competitive firm's profit function can be written as π := p * q - w * L - r * k where p is the competitive price of the product and w and r the per unit cost of the two inputs labour (L) and capital (k). the firm takes p,w, and r as given and chooses L and k to maximize

Coding Theory : Vectors and Generator Matrices

Please see the attached file for the fully formatted problems. 1(i) Explain what is meant by (a) a linear code over Fq, (b) the weight w(u) of a vector u and the distance d(u, v) between vectors u and v. (c) Define the weight tu(C) and the minimal distance d(C) of a code C. Prove that w(C) = d(C) if C is linear. (ii) Give

LaPlace Transforms for Circuit Schematics

Write the LaPlace-transformed loop equations for these two circuits by inspection. Use matrix notation. Include initial conditions. See attached file for full problem description.

Linear Combinations of Matrices

4 0 1 -1 0 2 A = B = C = -2 -2 2 3 1 4 Which of the following 2 X 2 matrices are linear combinations of A, B, or C 6 -8 0 0 6 0 -1 5

Matrices, Vector Spaces and Subspaces

Give a demonstration as to why or why not the given objects are vector subspaces of M22 a) all 2 X 2 matrices with integer entries A vector space is a set that is closed under finite vector addition and scalar multiplication. It is not a vector space, since V is NOT closed under finite scalar multiplication. For insta

Products of Diagonal Matrices

The n × n matrix A = [aij] is called a diagonal matrix if aij = 0 when i != j. Show that the product of two n × n diagonal matrices is again a diagonal matrix. Give a simple rule for determining this product.

Addition and Scalar Multiplication of Matrices

Given the set of objects and the operations of addition and scalar multiplication defined in each example below do the following: Determine which sets are vector spaces under the given operations For those sets that fail give the axiom(s) that fail to hold 1. The set of all triples of real numbers(x, y, z) with the op

Find the determinant of 2x2 matrices

Q: Find the determinant of these 2x2 matrices. (a) mat[4 3; -2 9] (b) mat[5 -7; 4 12] Please see the Word document for a cleaner version of the problem.

Graphs : Adjacency Matrix, Order and Valency

Let A be the adjacency matrix of a regular graph of order v and valency k. Let J be the all-ones matrix of the same order. Show that A*J = J*A = K*J "Definition 2.1 A graph r with adjacency matrix A = A(r) is called regular if there exists a natural number k such that AJ = JA = kJ. The number k is called valency of r."

Linear Binary Codes

You can not use the definition Hermitian inner product to solve this problem. You need to use the definition of weight. We need to generalize the equation. 1-Define the product, x*y , of two binary vectors of the same length to be the vector whose ith component is the product of the ith components of x and y. Show tha

Binary Hamming Code Vectors

See attached file for full problem description. 7. The parity check matrix of a binary [n, k, d] = [15, 11, 3] Hamming code whose columns are in numeric order is a) Find the corresponding generator matrix. Use the method of permuting and unpermuting the columns for the [7, 4, 3] binary Hamming code. b) Encode the

Binary repetition code

See attached file for full problem description. 6. Let be the binary repetition code with parameter [n, k] = [12, 4]. a) Give a generator matrix for that is in standard form. What does standard form means? b) Give a parity check matrix for that is in standard form. c) What is the minimum distance of ? d)

Binary linear block code

See attached file for full problem description. 3. Let be a binary linear block code with parameter [n, k] = [6, 3] which has as its generator matrix. Assume that the first three bits of each codeword are the information bits. a) Find a generator matrix for that is in standard form. What does standard form mea

Orthogonality of a Generator Matrix of a Binary Block Code

Please see the attached file for the fully formatted problems. 4. Let be the generator matrix of a binary block code . a) Show that each row of G is orthogonal to itself and to each of the other rows of G. b) Show that each codeword in is orthogonal to itself and to every other codeword in . Note: Use part a)

Binary Hamming code

1-Define the product, , of two binary vectors of the same length to be the vector whose ith component is the product of the ith components of and . Show that wt ( + ) =wt ( ) +wt ( ) - 2wt ( ) . wt means weight. 2. Suppose that a binary Hamming code is modified by adding an additional check bit to each codeword. This ad

Parity check matrix problems

1.- Find a parity check matrix for the [12,4] repetition code. Can you expalin what does mean repetition code and the parity check matrix? 2.- Show that the complement of each codeword in the [12,4] repetition code is again a codeword. Note: do not do this problem by checking all the sums of codewords. Please Can yo

Matrix Functions

1.Consider the 2 functions f1(t) and f2(t); 1. f1(t) = { a1.e^ -2t for t>=0 and = { 0 for t<0 } f2(t) = { a2.e^ -t + a3.e^ -2t for t>=0 and = { 0 for t<0 } Find a1,a2 and a3 such that f1(t) and f2(t) are orthonormal on the interval 0 to infinity. 2. If Q is an n x n symmetric matrix and a1,a2 are such that 0 < a1I <

Solving Systems of Equations with Three Variables

1. Production Turley Tailor Inc. makes long-sleeve, short sleeve,and sleeveless blouses. A long-sleeve blouse requires 1.5 hours of cutting and 1.2 hours of sewing. A short-sleeve blouse requires 1 hour of cutting and .9 hour of sewing. A sleeveless blouse requires .5 hour of cutting and .6 hour of sewing. There are 380 hours of

Matrices Values of Determinants

These are problems from the text that were advised to study for the next exam. (see attachment for equations) 1) Determine the values of r for which det(A-rI) = 0 2) Verify that X(t) is a fundamental matrix for the given system and compute X-1(t). Use the result, x' = Ax, x(t0) = x0 to find the solution to the g

Vectors, matrices, and polynomials

5) Use cofactor expansion to find the determinant | 4 -1 1 6 | | 0 0 -3 3 | | 4 1 0 14| | 4 1 3 2 | 6)Find the characteristic equation, eigenvalues, and corresponding eigenvectors for A = 1 2 2 1 7) Answer True or False for the following statements: (True means always true; False means sometimes false). Ju

Vectors and Matrices : Operations and Transformations

1) Let u=(2,3,0), and v=(-1,2,-2). Find a) ||u + v|| b) ||u|| + || v || c) Find two vectors in R³ with norm 1 orthogonal to be both u and v d) Find norm of vector u / || u || 2) For which values of t are vectors u = (6, 5, t), and v = (1,t,t) orthogonal? 3) a) Find the standard matrix [T] for the linear tra