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# Solutions for Chapter Chapter 14: Trigonometric Graphs and Identities

## Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition

ISBN: 9780078738302

Solutions for Chapter Chapter 14: Trigonometric Graphs and Identities

Solutions for Chapter Chapter 14
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##### ISBN: 9780078738302

Chapter Chapter 14: Trigonometric Graphs and Identities includes 50 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. Since 50 problems in chapter Chapter 14: Trigonometric Graphs and Identities have been answered, more than 52450 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• 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.

• 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.

• 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.

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• Left inverse A+.

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

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• 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).

• Normal matrix.

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

• Outer product uv T

= column times row = rank one matrix.

• Particular solution x p.

Any solution to Ax = b; often x p has free variables = o.

• Pivot columns of A.

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

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

Column vectors by convention.

• Singular Value Decomposition

(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• Trace of A

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

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

• Wavelets Wjk(t).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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