 5.1.1E: Write the first four terms of the sequences defined by the formulas...
 5.1.2E: Write the first four terms of the sequences defined by the formulas...
 5.1.3E: Write the first four terms of the sequences defined by the formulas...
 5.1.4E: Write the first four terms of the sequences defined by the formulas...
 5.1.5E: Write the first four terms of the sequences defined by the formulas...
 5.1.6E: Write the first four terms of the sequences defined by the formulas...
 5.1.7E: Write the first four terms of the sequences defined by the formulas...
 5.1.8E: Compute the first fifteen terms of each of the sequences in 8 and 9...
 5.1.9E: Compute the first fifteen terms of each of the sequences in 8 and 9...
 5.1.10E: Write the first four terms of the sequences defined by the formulas...
 5.1.11E: Find explicit formulas for sequences of the form a1, a2, a3, ... wi...
 5.1.12E: Find explicit formulas for sequences of the form a1, a2, a3, ... wi...
 5.1.13E: Find explicit formulas for sequences of the form a1, a2, a3, ... wi...
 5.1.14E: Find explicit formulas for sequences of the form a1, a2, a3, ... wi...
 5.1.15E: Find explicit formulas for sequences of the form a1, a2, a3, ... wi...
 5.1.16E: Find explicit formulas for sequences of the form a1, a2, a3, ... wi...
 5.1.17E: Consider the sequence defined by for all integers n ? 0. Find an al...
 5.1.18E: Let a0 = 2, a1 = 3, a2 = –2, a3 = 1, a4 = 0, a5 = –1, and a6 = –2. ...
 5.1.19E: Compute the summations and products.Exercise
 5.1.20E: Compute the summations and products.Exercise
 5.1.21E: Compute the summations and products.Exercise
 5.1.22E: Compute the summations and products.Exercise
 5.1.23E: Compute the summations and products.Exercise
 5.1.24E: Compute the summations and products.Exercise
 5.1.25E: Compute the summations and products.Exercise
 5.1.26E: Compute the summations and products.Exercise
 5.1.27E: Compute the summations and products.Exercise
 5.1.28E: Compute the summations and products.Exercise
 5.1.29E: Write the summations in expanded form.Exercise
 5.1.30E: Write the summations in expanded form.Exercise
 5.1.31E: Write the summations in expanded form.Exercise
 5.1.32E: Write the summations in expanded form.Exercise
 5.1.33E: Evaluate the summations and products in the indicated values of the...
 5.1.34E: Evaluate the summations and products in the indicated values of the...
 5.1.35E: Evaluate the summations and products in the indicated values of the...
 5.1.36E: Evaluate the summations and products in the indicated values of the...
 5.1.37E: Rewrite by separating off the final term.Exercise
 5.1.38E: Rewrite by separating off the final term.Exercise
 5.1.39E: Rewrite by separating off the final term.Exercise
 5.1.40E: Write a single summation.Exercise
 5.1.41E: Write a single summation.Exercise
 5.1.42E: Write a single summation.Exercise
 5.1.43E: Write using summation or product notation.Exercise12 –22 + 32 –42 +...
 5.1.44E: Write using summation or product notation.Exercise (13 –1) –(23 –1)...
 5.1.45E: Write using summation or product notation.Exercise (22 –1)• (32 –1)...
 5.1.46E: Write using summation or product notation.Exercise
 5.1.47E: Write using summation or product notation.Exercise1 –r + r2 –r3 + r...
 5.1.48E: Write using summation or product notation.Exercise (1 –t) • (1 –t2)...
 5.1.49E: Write using summation or product notation.Exercise13 + 23 + 33 + … ...
 5.1.50E: Write using summation or product notation.Exercise
 5.1.51E: Write using summation or product notation.Exercisen + (n –1) + (n –...
 5.1.52E: Write using summation or product notation.Exercise
 5.1.53E: Transform making the change of variable i = k + 1.Exercise
 5.1.54E: Transform making the change of variable i = k + 1.Exercise
 5.1.55E: Transform making the change of variable j = i –1.Exercise
 5.1.56E: Transform making the change of variable j = i –1.Exercise
 5.1.57E: Transform making the change of variable j = i –1.Exercise
 5.1.58E: Transform making the change of variable j = i –1.Exercise
 5.1.59E: Write a single summation or product.Exercise
 5.1.60E: Write a single summation or product.Exercise
 5.1.61E: Write a single summation or product.Exercise
 5.1.62E: Assume the values of the variables are restricted so that the expre...
 5.1.63E: Assume the values of the variables are restricted so that the expre...
 5.1.64E: Assume the values of the variables are restricted so that the expre...
 5.1.65E: Assume the values of the variables are restricted so that the expre...
 5.1.66E: Assume the values of the variables are restricted so that the expre...
 5.1.67E: Assume the values of the variables are restricted so that the expre...
 5.1.68E: Assume the values of the variables are restricted so that the expre...
 5.1.69E: Assume the values of the variables are restricted so that the expre...
 5.1.70E: Assume the values of the variables are restricted so that the expre...
 5.1.71E: Assume the values of the variables are restricted so that the expre...
 5.1.72E: Assume the values of the variables are restricted so that the expre...
 5.1.73E: Assume the values of the variables are restricted so that the expre...
 5.1.74E: Assume the values of the variables are restricted so that the expre...
 5.1.75E: Assume the values of the variables are restricted so that the expre...
 5.1.76E: Assume the values of the variables are restricted so that the expre...
 5.1.77E: Assume the values of the variables are restricted so that the expre...
 5.1.78E: Prove that for all nonnegative integers n and r with
 5.1.79E: Prove that if p is a prime number and r is an integer with 0<r then...
 5.1.80E: Suppose is a onedimensional array and consider the following algor...
 5.1.81E: Use repeated division by 2 to convert (by hand) the integers in 81–...
 5.1.82E: Use repeated division by 2 to convert (by hand) the integers in 81–...
 5.1.83E: Use repeated division by 2 to convert (by hand) the integers in 81–...
 5.1.84E: Make a trace table to trace the action of Algorithm 5.1.1 on the in...
 5.1.85E: Make a trace table to trace the action of Algorithm 5.1.1 on the in...
 5.1.86E: Make a trace table to trace the action of Algorithm 5.1.1 on the in...
 5.1.87E: Write an informal description of an algorithm (using repeated divis...
 5.1.88E: Use the algorithm you developed for exercise 87 to convert the inte...
 5.1.89E: Use the algorithm you developed for exercise 87 to convert the inte...
 5.1.90E: Use the algorithm you developed for exercise 87 to convert the inte...
 5.1.91E: Write a formal version of the algorithm you developed for exercise ...
Solutions for Chapter 5.1: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.1
Get Full SolutionsChapter 5.1 includes 91 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. Since 91 problems in chapter 5.1 have been answered, more than 51631 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

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.

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).

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

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].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

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

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

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Solvable system Ax = b.
The right side b is in the column space of A.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.