 4.3.1: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.2: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.3: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.4: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.5: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.6: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.7: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.8: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.9: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.10: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.11: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.12: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.13: Give a reason for your answer in each of 113. Assume that all varia...
 4.3.14: Fill in the blanks in the following proof that for all integers a a...
 4.3.15: Fill in the blanks in the following proof that for all integers a a...
 4.3.16: Fill in the blanks in the following proof that for all integers a a...
 4.3.17: Consider the following statement: The negative of any multiple of 3...
 4.3.18: Show that the following statement is false: For all integers a and ...
 4.3.19: For each statement in 1931, determine whether the statement is true...
 4.3.20: For each statement in 1931, determine whether the statement is true...
 4.3.21: For each statement in 1931, determine whether the statement is true...
 4.3.22: For each statement in 1931, determine whether the statement is true...
 4.3.23: For each statement in 1931, determine whether the statement is true...
 4.3.24: For each statement in 1931, determine whether the statement is true...
 4.3.25: For each statement in 1931, determine whether the statement is true...
 4.3.26: For each statement in 1931, determine whether the statement is true...
 4.3.27: For each statement in 1931, determine whether the statement is true...
 4.3.28: For each statement in 1931, determine whether the statement is true...
 4.3.29: For each statement in 1931, determine whether the statement is true...
 4.3.30: For each statement in 1931, determine whether the statement is true...
 4.3.31: For each statement in 1931, determine whether the statement is true...
 4.3.32: A fastfood chain has a contest in which a card with numbers on it ...
 4.3.33: Is it possible to have a combination of nickels, dimes, and quarter...
 4.3.34: Is it possible to have 50 coins, made up of pennies, dimes, and qua...
 4.3.35: Two athletes run a circular track at a steady pace so that the firs...
 4.3.36: It can be shown (see exercises 4448) that an integer is divisible b...
 4.3.37: Use the unique factorization theorem to write the following integer...
 4.3.38: Suppose that in standard factored form a = pe1 1 pe2 2 pek k , wher...
 4.3.39: Suppose that in standard factored form a = pe1 1 pe2 2 pek k , wher...
 4.3.40: a. If a and b are integers and 12a = 25b, does 12  b? does 25  a?...
 4.3.41: How many zeros are at the end of 458 885? Explain how you can answe...
 4.3.42: If n is an integer and n > 1, then n! is the product of n and every...
 4.3.43: In a certain town 2/3 of the adult men are married to 3/5 of the ad...
 4.3.44: Prove that if n is any nonnegative integer whose decimal representa...
 4.3.45: Prove that if n is any nonnegative integer whose decimal representa...
 4.3.46: Prove that if the decimal representation of a nonnegative integer n...
 4.3.47: Observe that 7524 = 71000 + 5100 + 210 + 4 = 7(999 + 1) + 5(99 + 1)...
 4.3.48: Prove that for any nonnegative integer n, if the sum of the digits ...
 4.3.49: Given a positive integer n written in decimal form, the alternating...
Solutions for Chapter 4.3: Direct Proof and Counterexample III: Divisibility
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.3: Direct Proof and Counterexample III: Divisibility
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Chapter 4.3: Direct Proof and Counterexample III: Divisibility includes 49 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 49 problems in chapter 4.3: Direct Proof and Counterexample III: Divisibility have been answered, more than 45124 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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

Column space C (A) =
space of all combinations of the columns of A.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

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

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 •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.