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# Solutions for Chapter 5.7: Number Theory and the Real Number System

## Full solutions for A Survey of Mathematics with Applications | 9th Edition

ISBN: 9780321759665

Solutions for Chapter 5.7: Number Theory and the Real Number System

Solutions for Chapter 5.7
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##### ISBN: 9780321759665

Since 91 problems in chapter 5.7: Number Theory and the Real Number System have been answered, more than 79893 students have viewed full step-by-step solutions from this chapter. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.7: Number Theory and the Real Number System includes 91 full step-by-step solutions. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9.

Key Math Terms and definitions covered in this textbook
• 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.

• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• 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.

• 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!).

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• 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.

• Hilbert matrix hilb(n).

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

• Hypercube matrix pl.

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

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Left nullspace N (AT).

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

• 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.

• Random matrix rand(n) or randn(n).

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

• Simplex method for linear programming.

The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

• 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.

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