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# Solutions for Chapter 37: Modular Arithmetic

## Full solutions for Mathematics: A Discrete Introduction | 3rd Edition

ISBN: 9780840049421

Solutions for Chapter 37: Modular Arithmetic

Solutions for Chapter 37
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##### ISBN: 9780840049421

Chapter 37: Modular Arithmetic includes 15 full step-by-step solutions. Mathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421. This textbook survival guide was created for the textbook: Mathematics: A Discrete Introduction, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Since 15 problems in chapter 37: Modular Arithmetic have been answered, more than 9255 students have viewed full step-by-step solutions from this chapter.

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).

• 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.

• Block matrix.

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Indefinite matrix.

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

• 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.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• 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).

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Normal equation AT Ax = ATb.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Outer product uv T

= column times row = rank one matrix.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Spectral radius = max of IAi I.

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Trace of A

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

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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