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# Solutions for Chapter 8.5: Polar Equations and Graphs

## Full solutions for Trigonometry | 10th Edition

ISBN: 9780321671776

Solutions for Chapter 8.5: Polar Equations and Graphs

Solutions for Chapter 8.5
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##### ISBN: 9780321671776

This textbook survival guide was created for the textbook: Trigonometry, edition: 10. Chapter 8.5: Polar Equations and Graphs includes 82 full step-by-step solutions. Since 82 problems in chapter 8.5: Polar Equations and Graphs have been answered, more than 33220 students have viewed full step-by-step solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780321671776. This expansive textbook survival guide covers the following chapters and their solutions.

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

Tv = Av + Vo = linear transformation plus shift.

• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• 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.

• Block matrix.

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.

• 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.

• 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.

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

Use AT for complex A.

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• 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).

• Outer product uv T

= column times row = rank one matrix.

• Partial pivoting.

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

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Rotation matrix

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Column vectors by convention.

• Semidefinite matrix A.

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

• 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.

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