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# Solutions for Chapter 11.3: Discrete Mathematics with Applications 4th Edition

## Full solutions for Discrete Mathematics with Applications | 4th Edition

ISBN: 9780495391326

Solutions for Chapter 11.3

Solutions for Chapter 11.3
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##### ISBN: 9780495391326

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Since 28 problems in chapter 11.3 have been answered, more than 28060 students have viewed full step-by-step solutions from this chapter. Chapter 11.3 includes 28 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Cholesky factorization

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

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• Cramer's Rule for Ax = b.

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

• Cyclic shift

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

• Dimension of vector space

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

• Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

• 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.

• 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.

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• 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.

• 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.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Projection matrix P onto subspace S.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• 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.

• Trace of A

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

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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