 5.3.1: What is the 28th term in the sequence of positive rationals produce...
 5.3.2: Use a counting process similar to that described in the discussion ...
 5.3.3: Prove Theorem 5.3.5 by induction on the number of elements in the f...
 5.3.4: Complete the proof of Theorem 5.3.6 by showing that the function h ...
 5.3.5: The Infinite Hotel is undergoing some remodeling, and consequently ...
 5.3.6: . Without referring to Theorem 5.3.8, prove part (a) of Corollary 5...
 5.3.7: Without referring to Theorem 5.3.8, prove part (b) of Corollary 5.3...
 5.3.8: Without referring to Theorem 5.3.8, prove part (c) of Corollary 5.3...
 5.3.9: Use the theorems of this section to prove that(a) an infinite subse...
 5.3.10: Prove or disprove:(a) If and B is denumerable, then A is denumerabl...
 5.3.11: Prove that if is a denumerable family of pairwise disjoint distinct...
 5.3.12: Give an example, if possible, of a family of sets such that(a) each...
 5.3.13: (a) Let S be the set of all sequences of 0s and 1s. For example,and...
 5.3.14: Let A be a denumerable set. Prove that(a) the set of all 1element ...
 5.3.15: . Assign a grade of A (correct), C (partially correct), or F (failu...
Solutions for Chapter 5.3: Countable Sets
Full solutions for A Transition to Advanced Mathematics  7th Edition
ISBN: 9780495562023
Solutions for Chapter 5.3: Countable Sets
Get Full SolutionsSince 15 problems in chapter 5.3: Countable Sets have been answered, more than 5829 students have viewed full stepbystep solutions from this chapter. A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780495562023. Chapter 5.3: Countable Sets includes 15 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: A Transition to Advanced Mathematics, edition: 7.

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

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

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

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

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).

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

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