 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: 4. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Since 32 problems in chapter 5.4 have been answered, more than 57800 students have viewed full stepbystep solutions from this chapter.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

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.

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.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

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

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

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 ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

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·