 5.6.1E: Find the first four terms of each of the recursively defined sequen...
 5.6.2E: Find the first four terms of each of the recursively defined sequen...
 5.6.3E: Find the first four terms of each of the recursively defined sequen...
 5.6.4E: Find the first four terms of each of the recursively defined sequen...
 5.6.5E: Find the first four terms of each of the recursively defined sequen...
 5.6.6E: Find the first four terms of each of the recursively defined sequen...
 5.6.7E: Find the first four terms of each of the recursively defined sequen...
 5.6.8E: Find the first four terms of each of the recursively defined sequen...
 5.6.9E: Let a0, a1, a2, ... be defined by the formula an = 3n + 1, for all ...
 5.6.10E: Let b0, b1, b2, ... be defined by the formula bn = 4n, for all inte...
 5.6.11E: Let c0, c1, c2, ... be defined by the formula cn = 2n –1 for all in...
 5.6.12E: Let s0, s1, s2, ... be defined by the formula for all integers n ? ...
 5.6.13E: Let t0, t1, t2, ... be defined by the formula tn = 2 + n for all in...
 5.6.14E: Let d0, d1, d2, ... be defined by the formula dn = 3n – 2nfor all i...
 5.6.15E: For the sequence of Catalan numbers defined in Example prove that f...
 5.6.16E: Use the recurrence relation and values for the Tower of Hanoi seque...
 5.6.17E: Tower of Hanoi with Adjacency Requirement: Suppose that in addition...
 5.6.18E: Tower of Hanoi with Adjacency Requirement: Suppose the same situati...
 5.6.19E: FourPole Tower of Hanoi: Suppose that the Tower of Hanoi problem h...
 5.6.20E: Tower of Hanoi Poles in a Circle: Suppose that instead of being lin...
 5.6.21E: Double Tower of Hanoi: In this variation of the Tower of Hanoi ther...
 5.6.22E: Fibonacci Variation: A single pair of rabbits (male and female) is ...
 5.6.23E: Fibonacci Variation: A single pair of rabbits (male and female) is ...
 5.6.24E: Use the recurrence relation and values for F0, F1, F2, ... given in...
 5.6.25E: The Fibonacci sequence satisfies the recurrence relation Fk = Fk1 ...
 5.6.26E: Prove that Fk = 3Fk–3 + 2Fk–4 for all integers k ? 4.
 5.6.27E: Prove that for all integers k ? 1.
 5.6.28E: Prove that F F F = 2FFfor all integers k ?1.
 5.6.29E: Prove that for all integers k ? 1.
 5.6.30E: Use mathematical induction to prove that for all integers n ? 0, .
 5.6.31E: Use strong mathematical induction to prove that Fn<2n for all integ...
 5.6.32E: Let F0, F1, F2, … be the Fibonacci sequence defined in Section 5.5....
 5.6.33E: It turns out that the Fibonacci sequence satisfies the following ex...
 5.6.34E: (For students who have studied calculus) Find assuming that the lim...
 5.6.35E: (For students who have studied calculus) Prove that exists.
 5.6.36E: (For students who have studied calculus) Define x0, x1, x2, ... as ...
 5.6.37E: Compound Interest: Suppose a certain amount of money is deposited i...
 5.6.38E: Compound Interest: Suppose a certain amount of money is deposited i...
 5.6.39E: With each step you take when climbing a staircase, you can move up ...
 5.6.41E: Use the recursive definition of summation, together with mathematic...
 5.6.42E: Use the recursive definition of product, together with mathematical...
 5.6.43E: Use the recursive definition of product, together with mathematical...
 5.6.44E: The triangle inequality for absolute value states that for all real...
Solutions for Chapter 5.6: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.6
Get Full SolutionsSince 43 problems in chapter 5.6 have been answered, more than 45399 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.6 includes 43 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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).

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).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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

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 .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

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.