 5.8.1E: Which of the following are secondorder linear homogeneous recurren...
 5.8.2E: Which of the following are secondorder linear homogeneous recurren...
 5.8.3E: Let a0, a1, a2, . . . be the sequence defined by the explicit formu...
 5.8.4E: Let b0, b1, b2, . . . be the sequence defined by the explicit formu...
 5.8.5E: Let a0, a1, a2, . . . be the sequence defined by the explicit formu...
 5.8.6E: Let b0, b1, b2, . . . be the sequence defined by the explicit formu...
 5.8.7E: Solve the system of equations in Example 5.8.4 to obtain Reference:...
 5.8.8E: In each of 8–10: (a) suppose a sequence of the form . where t ? 0, ...
 5.8.9E: In each of 8–10: (a) suppose a sequence of the form . where t ? 0, ...
 5.8.10E: In each of 8–10: (a) suppose a sequence of the form . where t ? 0, ...
 5.8.11E: In each of 11–16 suppose a sequence satisfies the given recurrence ...
 5.8.12E: In each of 11–16 suppose a sequence satisfies the given recurrence ...
 5.8.13E: In each of 11–16 suppose a sequence satisfies the given recurrence ...
 5.8.14E: In each of 11–16 suppose a sequence satisfies the given recurrence ...
 5.8.15E: In each of 11–16 suppose a sequence satisfies the given recurrence ...
 5.8.16E: In each of 11–16 suppose a sequence satisfies the given recurrence ...
 5.8.17E: Find an explicit formula for the sequence of exercise 39 in Section...
 5.8.18E: Suppose that the sequences both satisfy the same secondorder linea...
 5.8.19E: Show that if r, s, a0, and a1 are numbers with r ? s, then there ex...
 5.8.20E: Show that if r is a nonzero real number, k and m are distinct integ...
 5.8.21E: Prove Theorem 5.8.5 for the case where the values of C and D are de...
 5.8.22E: Exercises 22 and 23 are intended for students who are familiar with...
 5.8.23E: Exercises 22 and 23 are intended for students who are familiar with...
 5.8.24E: The numbers that appear in the explicit formula for the Fibonacci s...
Solutions for Chapter 5.8: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.8
Get Full SolutionsDiscrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. This expansive textbook survival guide covers the following chapters and their solutions. Since 24 problems in chapter 5.8 have been answered, more than 24797 students have viewed full stepbystep solutions from this chapter. Chapter 5.8 includes 24 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Iterative method.
A sequence of steps intended to approach the desired solution.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.