 5.4.25E: Imagine a situation in which eight people, numbered consecutively 1...
 5.4.1E: Suppose aj, a2, a3, ... is a sequence defined as follows:a1 = 1, a2...
 5.4.2E: Suppose b1, b2, b3, ... is a sequence defined as follows:b1 = 4, b2...
 5.4.3E: Suppose that c0, c1, c2, ... is a sequence defined as follows:c0 = ...
 5.4.4E: Suppose that d1, d2, d3, ... is a sequence defined as follows: dk =...
 5.4.5E: Suppose that e0, e1, e2, ... is a sequence defined as follows:e0 = ...
 5.4.6E: Suppose that f0, f1, f2, ... is a sequence defined as follows:f0 = ...
 5.4.7E: Suppose that g1, g2, g3, ... is a sequence defined as follows:g1 = ...
 5.4.8E: Suppose that h0, h1, h2, ... is a sequence defined as follows:h0 = ...
 5.4.9E: Define a sequence a1, a2, a3, ... as follows: a1 = 1, a2 = 3, and a...
 5.4.10E: The problem that was used to introduce ordinary mathematical induct...
 5.4.11E: You begin solving a jigsaw puzzle by finding two pieces that match ...
 5.4.12E: The sides of a circular track contain a sequence of cans of gasolin...
 5.4.13E: Use strong mathematical induction to prove the existence part of th...
 5.4.14E: Any product of two or more integers is a result of successive multi...
 5.4.15E: Any sum of two or more integers is a result of successive additions...
 5.4.16E: Use strong mathematical induction to prove that for any integer n ?...
 5.4.17E: Compute 41, 42, 43, 44, 45, 46, 47, and 48. Make a conjecture about...
 5.4.18E: Compute 90, 91,92, 93, 94, and 95. Make a conjecture about the unit...
 5.4.19E: Find the mistake in the following “proof” that purports to show tha...
 5.4.20E: Use the wellordering principle for the integers to prove Theorem 4...
 5.4.21E: Use the wellordering principle for the integers to prove the exist...
 5.4.22E: a. The Archimedean property for the rational numbers states that fo...
 5.4.23E: Use the results of exercise and the wellordering principle for the...
 5.4.24E: Use the wellordering principle to prove that given any integer n ?...
 5.4.26E: Suppose P (n) is a property such that1. P(0), P(1), P(2) are all tr...
 5.4.27E: Prove that if a statement can be proved by strong mathematical indu...
 5.4.28E: Give examples to illustrate the proof of Theorem 1.Theorem 1Existen...
 5.4.29E: It is a fact that every integer n ? 1 can be written in theform whe...
 5.4.30E: Use mathematical induction to prove the existence part of the quoti...
 5.4.31E: Prove that if a statement can be proved by ordinary mathematical in...
 5.4.32E: Use the principle of ordinary mathematical induction to prove the w...
Solutions for Chapter 5.4: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.4
Get Full SolutionsChapter 5.4 includes 32 full stepbystep solutions. 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. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. Since 32 problems in chapter 5.4 have been answered, more than 24621 students have viewed full stepbystep solutions from this chapter.

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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.

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.

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·