- 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 parity-check matrix h = F 1 1 1 0 1 1 1 0 1 ...
- 9.2.10: Consider the canonical parity-check matrix h = I 1 1 0 1 0 1 1 1 0 ...
- 9.2.11: Give an example of a canonical parity-check matrix that will genera...
- 9.2.12: Give a canonical parity-check matrix for a single-error 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 single-error 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 parity-check matrix for th...
- 9.2.18: For Exercises 17 and 18, use a canonical parity-check matrix for th...
Solutions for Chapter 9.2: Coding Theory
Full solutions for Mathematical Structures for Computer Science | 7th Edition
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.
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.
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.
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 .
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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