 37.37.1: In the context of Z10, please calculate: a. 3 3. b. 6 6. c. 7 3. d....
 37.37.2: Solve the following equations for x in the Zn specified. a. 3 x D 4...
 37.37.3: Solve the following equations for x in the Zn specified. Note: Thes...
 37.37.4: Here are a few more equations for you to solve in the Zn specified....
 37.37.5: For some prime numbers p, the equation x x 1 D 0 has a solution in ...
 37.37.6: Prove: For all a; b 2 Zn, .a b/ .b a/ D 0.
 37.37.7: Prove that the operations , , and are closed. This means that if a;...
 37.37.8: Prove Proposition 37.4. Why is this proposition restricted to n 2?
 37.37.9: Use Proposition 37.7 as the definition of and then prove the assert...
 37.37.10: For ordinary integers, the following is true. If ab D 0, then a D 0...
 37.37.11: Prove Proposition 37.11.
 37.37.12: Let n be a positive integer and suppose a; b 2 Zn are both invertib...
 37.37.13: Let n be an integer with n 2. Prove that in Zn the element n 1 is i...
 37.37.14: Modular exponentiation. Let b be a positive integer. The notation a...
 37.37.15: Write a computer program to calculate a b mod c where a; b; c are p...
Solutions for Chapter 37: Modular Arithmetic
Full solutions for Mathematics: A Discrete Introduction  3rd Edition
ISBN: 9780840049421
Solutions for Chapter 37: Modular Arithmetic
Get Full SolutionsChapter 37: Modular Arithmetic includes 15 full stepbystep 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 stepbystep solutions from this chapter.

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. AI is also symmetric.

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.

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