 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 and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 24 problems in chapter 5.8 have been answered, more than 45353 students have viewed full stepbystep solutions from this chapter. Chapter 5.8 includes 24 full stepbystep solutions.

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

Column space C (A) =
space of all combinations of the columns of A.

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

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.

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.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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 .

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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).

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 ().

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

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