 13.5.26E: Construct a Turing machine that computes the function f(n1, n2) = n...
 13.5.28E: By finding the composite of the Turing machines you constructed in ...
 13.5.1E: Let T be the Turing machine defined by the five tuples: (s0, 0, s1,...
 13.5.3E: What does the Turing machine described by the five tuples (s0, 0, s...
 13.5.2E: Let T be the Turing machine defined by the five tuples: (s0, 0, s2,...
 13.5.4E: What does the Turing machine described by the five tuples (s0, 0, s...
 13.5.5E: What does the Turing machine described by the five tuples (s0 , 1,s...
 13.5.6E: Construct a Turing machine with tape symbols 0, 1, and B that, when...
 13.5.7E: Construct a Turing machine with tape symbols 0, 1, and B that, when...
 13.5.9E: Construct a Turing machine with tape symbols 0, 1, and B that, give...
 13.5.8E: Construct a Turing machine with tape symbols 0, 1, and B that, give...
 13.5.10E: Construct a Turing machine with tape symbols 0, 1. and B that, give...
 13.5.11E: Construct a Turing machine that recognizes the set of all bit strin...
 13.5.13E: Construct a Turing machine that recognizes the set of all bit strin...
 13.5.12E: Construct a Turing machine that recognizes the set of all bit strin...
 13.5.14E: Show at each step the contents of the tape of the Turing machine in...
 13.5.17E: Construct a Turing machine that recognizes the set {02n 1n  n ? 0}.
 13.5.15E: Explain why the Turing machine in Example 3 recognizes a bit string...
 13.5.16E: Construct a Turing machine that recognizes the set {02nln  n ? 0}.
 13.5.18E: Construct a Turing machine that computes the function f(n) = n + 2 ...
 13.5.19E: Construct a Turing machine that computes the function f(n) = n  3 ...
 13.5.22E: Construct a Turing machine that computes the function f(n) = 2n for...
 13.5.20E: Construct a Turing machine that computes the function f(n) = n mod ...
 13.5.23E: Construct a Turing machine that computes the function f(n) = 3n for...
 13.5.21E: Construct a Turing machine that computes the function f(n) = 3 if ?...
 13.5.24E: Construct a Turing machine that computes the function f(n1 , n2) = ...
 13.5.25E: Construct a Turing machine that computes the function f(n1, n2) = ...
 13.5.27E: By finding the composite of the Turing machines you constructed in ...
 13.5.30E: Which of the following problems is a decision problem?a) Is the seq...
 13.5.31E: Show that B(2) is at least 4 by finding a Turing machine with two s...
 13.5.29E: Which of the following problems is a decision problem?a) What is th...
 13.5.32E: Show that the function B(n) cannot be computed by any Turing machin...
Solutions for Chapter 13.5: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 13.5
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Chapter 13.5 includes 32 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 32 problems in chapter 13.5 have been answered, more than 185770 students have viewed full stepbystep solutions from this chapter.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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.

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.