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# Solutions for Chapter 11.6: Polar Equations of Conics

## Full solutions for Algebra and Trigonometry with Analytic Geometry | 12th Edition

ISBN: 9780495559719

Solutions for Chapter 11.6: Polar Equations of Conics

Solutions for Chapter 11.6
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##### ISBN: 9780495559719

This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. Chapter 11.6: Polar Equations of Conics includes 39 full step-by-step solutions. Since 39 problems in chapter 11.6: Polar Equations of Conics have been answered, more than 37391 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• 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.

• Change of basis matrix M.

The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Hankel matrix H.

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

• Kirchhoff's Laws.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• 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).

• 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 •

• 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.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Spectral Theorem A = QAQT.

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

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

• 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.

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