 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 9582 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 and is associated to the ISBN: 9781429215107.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).