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# Solutions for Chapter 14.5: Sum and Differences of Angles Formulas

## Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving | 1st Edition

ISBN: 9780078778568

Solutions for Chapter 14.5: Sum and Differences of Angles Formulas

Solutions for Chapter 14.5
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##### ISBN: 9780078778568

California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. This expansive textbook survival guide covers the following chapters and their solutions. Since 60 problems in chapter 14.5: Sum and Differences of Angles Formulas have been answered, more than 42733 students have viewed full step-by-step solutions from this chapter. Chapter 14.5: Sum and Differences of Angles Formulas includes 60 full step-by-step solutions. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1.

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.

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• 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.

• 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

• 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.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• 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.

• 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).

• Normal equation AT Ax = ATb.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b - Ax) = o.

• Partial pivoting.

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

• Particular solution x p.

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

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

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

Column vectors by convention.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Stiffness matrix

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.