 9.2.1: Let f be a homomorphism from a group 3G, # 4 to a group 3H, +4. Pro...
 9.2.2: Let f be a homorphism from the group 3Z, +4 to the group 3Z, +4 giv...
 9.2.3: The function f defined by f(x) = x # 8 2 is a homomorphism from 3Z,...
 9.2.4: The function f defined by f(x) = x # 8 4 is a homomorphism from 3Z1...
 9.2.5: A function f: Z Z S Z is defined by f(x, y) = x + y. a. Prove that ...
 9.2.6: Let F be the set of all functions f: R S R. For f, g [ F, let f + g...
 9.2.7: Let S = 50, 4, 86. Then 3S, +12 4 is a subgroup of 3Z12, +12 4. Fin...
 9.2.8: Let S = 5i,(2, 3)6. Then 3S, +4 is a subgroup of the symmetric grou...
 9.2.9: Consider the canonical paritycheck matrix h = F 1 1 1 0 1 1 1 0 1 ...
 9.2.10: Consider the canonical paritycheck matrix h = I 1 1 0 1 0 1 1 1 0 ...
 9.2.11: Give an example of a canonical paritycheck matrix that will genera...
 9.2.12: Give a canonical paritycheck matrix for a singleerror correcting ...
 9.2.13: Let H be an n r binary matrix mapping :Zn 2, +2; to :Zr 2, +2; by t...
 9.2.14: Which of the following are perfect codes? Which are singleerror co...
 9.2.15: Complete the coset leader/syndrome table of Example 28.
 9.2.16: Use your table from Exercise 15 and the matrix H from Example 28 to...
 9.2.17: For Exercises 17 and 18, use a canonical paritycheck matrix for th...
 9.2.18: For Exercises 17 and 18, use a canonical paritycheck matrix for th...
Solutions for Chapter 9.2: Coding Theory
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 9.2: Coding Theory
Get Full SolutionsSince 18 problems in chapter 9.2: Coding Theory have been answered, more than 4201 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.2: Coding Theory includes 18 full stepbystep solutions. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Mathematical Structures for Computer Science was written by Patricia and is associated to the ISBN: 9781429215107.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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