 A1.A1.1: Prove Theorem 3, which states that the multiplicative identity elem...
 A1.A1.2: Prove Theorem 4, which states that for every nonzero real number x,...
 A1.A1.3: Prove that for all real numbers x and y, (x) y = x . (y) = (x y).
 A1.A1.4: Prove that for all real numbers x and y, (x + y) = (x) + (y).
 A1.A1.5: Prove that for all real numbers x and y, (x) . ( y) = x ' y.
 A1.A1.6: Prove that for all real numbers x, y, and z, if x + Z = y+z, then x...
 A1.A1.7: Prove that for every real number x, (x) = x.
 A1.A1.8: Prove that for all real numbers x and y, x = y if and only ifx  y ...
 A1.A1.9: Prove that for all real numbers x and y, x  y = (x + y).
 A1.A1.10: Prove that for all nonzero real numbers x andy, l /(x/y) = y/x, whe...
 A1.A1.11: Prove that for all real numbers w, x, y, and z, if x I 0 and z I ...
 A1.A1.12: Prove that for every positive real number x, l /x is also a positiv...
 A1.A1.13: Prove that for all positive real numbers x and y, x . Y is also a p...
 A1.A1.14: Prove that for all real numbers x and y, if x > 0 and y < 0, then x...
 A1.A1.15: Prove that for all real numbers x, y, and z, if x > y and z < 0, th...
 A1.A1.16: Prove that for every real number x, x I 0 if and only if x2 > O.
 A1.A1.17: Prove that for all real numbers w, x, y, and z, if w < x and y < z,...
 A1.A1.18: Prove that for all positive real numbers x and y, if x < y, then l/...
 A1.A1.19: Prove that for every positive real number x, there exists a positiv...
 A1.A1.20: Prove that between every two distinct real numbers there is a ratio...
 A1.A1.21: Define a relation rv on the set of ordered pairs of positive intege...
 A1.A1.22: Define a relation on ordered pairs of integers with second entry no...
Solutions for Chapter A1: Axioms for the Real Numbers and the Positive Integers
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter A1: Axioms for the Real Numbers and the Positive Integers
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter A1: Axioms for the Real Numbers and the Positive Integers includes 22 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Since 22 problems in chapter A1: Axioms for the Real Numbers and the Positive Integers have been answered, more than 34436 students have viewed full stepbystep solutions from this chapter.

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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!

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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