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Solutions for Chapter 11.2: Parabolas

Algebra and Trigonometry with Analytic Geometry | 12th Edition | ISBN: 9780495559719 | Authors: Earl Swokowski, Jeffery A. Cole

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

ISBN: 9780495559719

Algebra and Trigonometry with Analytic Geometry | 12th Edition | ISBN: 9780495559719 | Authors: Earl Swokowski, Jeffery A. Cole

Solutions for Chapter 11.2: Parabolas

Solutions for Chapter 11.2
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Textbook: Algebra and Trigonometry with Analytic Geometry
Edition: 12
Author: Earl Swokowski, Jeffery A. Cole
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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