 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: 4. 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. Since 35 problems in chapter 4.7 have been answered, more than 45250 students have viewed full stepbystep solutions from this chapter. Chapter 4.7 includes 35 full stepbystep solutions.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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 p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

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