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# Solutions for Chapter 10.5: Discrete Mathematics and Its Applications 7th Edition

## Full solutions for Discrete Mathematics and Its Applications | 7th Edition

ISBN: 9780073383095

Solutions for Chapter 10.5

Solutions for Chapter 10.5
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##### ISBN: 9780073383095

Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 66 problems in chapter 10.5 have been answered, more than 199000 students have viewed full step-by-step solutions from this chapter. Chapter 10.5 includes 66 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

Key Math Terms and definitions covered in this textbook
• Adjacency matrix of a graph.

Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Big formula for n by n determinants.

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• Free variable Xi.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

• 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.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

• Orthogonal matrix Q.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• 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.

• Singular matrix A.

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

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Transpose matrix AT.

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

• 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.

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