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# Solutions for Chapter 8: Relations

## Full solutions for Discrete Mathematics and Its Applications | 6th Edition

ISBN: 9780073229720

Solutions for Chapter 8: Relations

Solutions for Chapter 8
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##### ISBN: 9780073229720

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8: Relations includes 104 full step-by-step solutions. Since 104 problems in chapter 8: Relations have been answered, more than 39999 students have viewed full step-by-step solutions from this chapter. 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.

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).

• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Gram-Schmidt orthogonalization A = QR.

Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Linear combination cv + d w or L C jV j.

• 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.

• 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.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Pivot.

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

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Skew-symmetric matrix K.

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

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• 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·

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.

• Wavelets Wjk(t).

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

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