 5.2.1E: Use mathematical induction (and the proof of Proposition as a model...
 5.2.2E: Use mathematical induction to show that any postage of at least 12¢...
 5.2.3E: For each positive integer n, let P(n) be the formula a. Write P(1)....
 5.2.4E: For each integer n with n ? 2, let P (n) be the formula a. Write P ...
 5.2.5E: Fill in the missing pieces in the following proof that1 + 3 + 5 + …...
 5.2.6E: Prove each statement i using mathematical induction. Do not derive ...
 5.2.7E: Prove each statement i using mathematical induction. Do not derive ...
 5.2.8E: Prove each statement i using mathematical induction. Do not derive ...
 5.2.9E: Prove each statement i using mathematical induction. Do not derive ...
 5.2.10E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.11E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.12E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.13E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.14E: for all integers n ? 0.
 5.2.15E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.16E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.17E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.18E: If x is a real number not divisible by n, then for all integers n ? 1,
 5.2.19E: (For students who have studied calculus) Use mathematical induction...
 5.2.20E: Use the formula for the sum of the first n integers and/or the form...
 5.2.21E: Use the formula for the sum of the first n integers and/or the form...
 5.2.22E: Use the formula for the sum of the first n integers and/or the form...
 5.2.23E: Use the formula for the sum of the first n integers and/or the form...
 5.2.24E: Use the formula for the sum of the first n integers and/or the form...
 5.2.25E: Use the formula for the sum of the first n integers and/or the form...
 5.2.26E: Use the formula for the sum of the first n integers and/or the form...
 5.2.27E: Use the formula for the sum of the first n integers and/or the form...
 5.2.28E: Use the formula for the sum of the first n integers and/or the form...
 5.2.29E: Use the formula for the sum of the first n integers and/or the form...
 5.2.30E: Find a formula in n, a, m, and d for the sum (a + md) + (a + (m + 1...
 5.2.31E: Find a formula in a, r, m, and n for the sumarm + arm+1 + arm+2 + …...
 5.2.32E: You have two parents, four grandparents, eight greatgrandparents, ...
 5.2.33E: Find the mistakes in the proof fragments,ExerciseTheorem: For any i...
 5.2.34E: Find the mistakes in the proof fragments,ExerciseTheorem: For any i...
 5.2.35E: Find the mistakes in the proof fragments,ExerciseTheorem: For any i...
 5.2.36E: Use Theorem to prove that if m and n are any positive integers and ...
 5.2.37E: Use Theorem and the result of exercise 10 to prove that if p is any...
Solutions for Chapter 5.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.2
Get Full SolutionsSince 37 problems in chapter 5.2 have been answered, more than 24488 students have viewed full stepbystep solutions from this chapter. Chapter 5.2 includes 37 full stepbystep solutions. This 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.

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.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite 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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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 matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

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 •

Outer product uv T
= column times row = rank one matrix.

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.

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).