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# Solutions for Chapter Chapter 5: Number Theory and the Real Number System

## Full solutions for Thinking Mathematically | 6th Edition

ISBN: 9780321867322

Solutions for Chapter Chapter 5: Number Theory and the Real Number System

Solutions for Chapter Chapter 5
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##### ISBN: 9780321867322

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. Since 30 problems in chapter Chapter 5: Number Theory and the Real Number System have been answered, more than 67341 students have viewed full step-by-step solutions from this chapter. Chapter Chapter 5: Number Theory and the Real Number System includes 30 full step-by-step solutions.

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).

• Back substitution.

Upper triangular systems are solved in reverse order Xn to Xl.

A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

• Dimension of vector space

dim(V) = number of vectors in any basis for V.

• Elimination matrix = Elementary matrix Eij.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• 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.

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• 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.

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• 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.

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Partial pivoting.

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

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• 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.

• Stiffness matrix

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

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