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# Solutions for Chapter 11.2: Parabolas

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

ISBN: 9780495559719

Solutions for Chapter 11.2: Parabolas

Solutions for Chapter 11.2
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##### ISBN: 9780495559719

Chapter 11.2: Parabolas includes 54 full step-by-step solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. Since 54 problems in chapter 11.2: Parabolas have been answered, more than 33611 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• 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.

• Column picture of Ax = b.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

• 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.

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• 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.

• 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.

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = O.

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

• Nilpotent matrix N.

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

• Normal matrix.

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

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Orthogonal matrix Q.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• 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.

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

Spectral radius = max of IAi I.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B IIĀ·

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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