 4.7.1E: A calculator display shows that = 1.414213562, and This suggests th...
 4.7.2E: Example (h) illustrates a technique for showing that any repeating ...
 4.7.3E: Determine which statement are true and which are false. Prove those...
 4.7.4E: Determine which statement are true and which are false. Prove those...
 4.7.5E: Determine which statement are true and which are false. Prove those...
 4.7.6E: Determine which statement are true and which are false. Prove those...
 4.7.7E: Determine which statement are true and which are false. Prove those...
 4.7.8E: Determine which statement are true and which are false. Prove those...
 4.7.9E: Determine which statement are true and which are false. Prove those...
 4.7.10E: Determine which statement are true and which are false. Prove those...
 4.7.11E: Determine which statement are true and which are false. Prove those...
 4.7.12E: Determine which statement are true and which are false. Prove those...
 4.7.13E: Determine which statement are true and which are false. Prove those...
 4.7.14E: Consider the following sentence: If x is rational then is irrationa...
 4.7.15E: a. Prove that for all integers a, if a3 is even then a is even.b. P...
 4.7.16E: a. Use proof by contradiction to show that for any integer n, it is...
 4.7.17E: Give an example to show that if d is not prime and n2 is divisible ...
 4.7.18E: The quotientremainder theorem says not only that there exist quoti...
 4.7.19E: Prove that is irrational.
 4.7.20E: Prove that for any integer a, 9 (a2 ? 3).
 4.7.21E: An alternative proof of the irrationality of counts the number of 2...
 4.7.22E: Use the proof technique illustrated in exercise to prove that if n ...
 4.7.23E: Prove that is irrational.
 4.7.24E: Prove that log5(2) is irrational. (Hint: Use the unique factorisati...
 4.7.25E: Let N = 2 • 3 • 5 • 7 + 1. What remainder is obtained when N is div...
 4.7.26E: Suppose a is an integer and p is a prime number such that p  a and...
 4.7.27E: Let p1, p2, p3, … be a list of all prime numbers in ascending order...
 4.7.28E: For exercise, use the fact that for all integersn, n! =n(n ? 1)… 3 ...
 4.7.29E: For exercise, use the fact that for all integersn, n! =n(n ? 1)… 3 ...
 4.7.30E: Prove that if p1, p2, …, and pn are distinct prime numbers with p1 ...
 4.7.31E: a. Fermat’s last theorem says that for all integers n > 2, the equa...
 4.7.32E: For exercise note that to show there is a unique object with a cert...
 4.7.33E: For exercise note that to show there is a unique object with a cert...
 4.7.34E: For exercise note that to show there is a unique object with a cert...
 4.7.35E: For exercise note that to show there is a unique object with a cert...
Solutions for Chapter 4.7: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.7
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions. Since 35 problems in chapter 4.7 have been answered, more than 24872 students have viewed full stepbystep solutions from this chapter. Chapter 4.7 includes 35 full stepbystep solutions.

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.

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

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).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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 .

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.