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# Solutions for Chapter 7.5: Multiple-Angle and Product-to-Sum Formulas

## Full solutions for Algebra and Trigonometry | 8th Edition

ISBN: 9781439048474

Solutions for Chapter 7.5: Multiple-Angle and Product-to-Sum Formulas

Solutions for Chapter 7.5
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##### ISBN: 9781439048474

Since 140 problems in chapter 7.5: Multiple-Angle and Product-to-Sum Formulas have been answered, more than 144087 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. Chapter 7.5: Multiple-Angle and Product-to-Sum Formulas includes 140 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their 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).

• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Circulant matrix C.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

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.

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Jordan form 1 = M- 1 AM.

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

• Left nullspace N (AT).

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

• Length II x II.

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

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• 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.

• Normal matrix.

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

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!