×
×

# Solutions for Chapter 5.5: Multiple-Angle and Product-to-Sum Formulas

## Full solutions for Precalculus With Limits A Graphing Approach | 5th Edition

ISBN: 9780618851522

Solutions for Chapter 5.5: Multiple-Angle and Product-to-Sum Formulas

Solutions for Chapter 5.5
4 5 0 392 Reviews
26
0
##### ISBN: 9780618851522

This expansive textbook survival guide covers the following chapters and their solutions. Since 150 problems in chapter 5.5: Multiple-Angle and Product-to-Sum Formulas have been answered, more than 46014 students have viewed full step-by-step solutions from this chapter. Chapter 5.5: Multiple-Angle and Product-to-Sum Formulas includes 150 full step-by-step solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5.

Key Math Terms and definitions covered in this textbook
• 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.

• Cholesky factorization

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

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• Eigenvalue A and eigenvector x.

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

• 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.

• Fourier matrix F.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

• Gram-Schmidt orthogonalization A = QR.

Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

• Hankel matrix H.

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

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• 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.

• Left inverse A+.

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

• 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).

• Partial pivoting.

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

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

P = aaT laTa has rank l.

• Semidefinite matrix A.

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• 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!

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

×