 4.8.27E: Exercise refer to the following definition.Definition: The least co...
 4.8.1E: Find the value of z when each of the algorithm segments in 1 and 2 ...
 4.8.2E: Find the value of z when each of the algorithm segments in 1 and 2 ...
 4.8.3E: Find the value of z when each of the algorithm segments in 1 and 2 ...
 4.8.4E: Find the values of a after execution of the loops
 4.8.5E: Find the values of e after execution of the loop
 4.8.6E: Make a trace table to trace the action of Algorithm 4.8.1 for the i...
 4.8.7E: Make a trace table to trace the action of Algorithm 4.8.1 for the i...
 4.8.8E: The following algorithm segment makes change; given an amount of mo...
 4.8.9E: Find the greatest common divisor of each of the pairs of integers. ...
 4.8.10E: Find the greatest common divisor of each of the pairs of integers. ...
 4.8.11E: Find the greatest common divisor of each of the pairs of integers. ...
 4.8.12E: Find the greatest common divisor of each of the pairs of integers. ...
 4.8.13E: Use the Euclidean algorithm to calculate the greatest common diviso...
 4.8.14E: Use the Euclidean algorithm to calculate the greatest common diviso...
 4.8.15E: Use the Euclidean algorithm to calculate the greatest common diviso...
 4.8.16E: Use the Euclidean algorithm to calculate the greatest common diviso...
 4.8.17E: Make a trace table to trace the action of Algorithm 4.8.2 for the i...
 4.8.18E: Make a trace table to trace the action of Algorithm 4.8.2 for the i...
 4.8.19E: Prove that for all positive integers a and b, ab if, and only if, ...
 4.8.20E: a. Prove that if a and b are integers, not both zero, and d = gcd(a...
 4.8.21E: Complete the proof of Lemma by proving the following: If a and b ar...
 4.8.22E: a. Prove: If a and d are positive integers and q and r are integers...
 4.8.23E: Rewrite the steps of Algorithm 4.8.2 for a computer language with a...
 4.8.24E: An alternative to the Euclidean algorithm uses subtraction rather t...
 4.8.25E: Exercise refer to the following definition.Definition: The least co...
 4.8.26E: Exercise refer to the following definition.Definition: The least co...
 4.8.28E: Exercise refer to the following definition.Definition: The least co...
 4.8.29E: Exercise refer to the following definition.Definition: The least co...
Solutions for Chapter 4.8: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.8
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Chapter 4.8 includes 29 full stepbystep solutions. Since 29 problems in chapter 4.8 have been answered, more than 49083 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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!

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).