×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide

Solutions for Chapter 7.6: Double-angle and Half-angle Formulas

Precalculus Enhanced with Graphing Utilities | 6th Edition | ISBN: 9780132854351 | Authors: Michael Sullivan

Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition

ISBN: 9780132854351

Precalculus Enhanced with Graphing Utilities | 6th Edition | ISBN: 9780132854351 | Authors: Michael Sullivan

Solutions for Chapter 7.6: Double-angle and Half-angle Formulas

Solutions for Chapter 7.6
4 5 0 300 Reviews
17
4
Textbook: Precalculus Enhanced with Graphing Utilities
Edition: 6
Author: Michael Sullivan
ISBN: 9780132854351

This expansive textbook survival guide covers the following chapters and their solutions. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Since 113 problems in chapter 7.6: Double-angle and Half-angle Formulas have been answered, more than 54371 students have viewed full step-by-step solutions from this chapter. Chapter 7.6: Double-angle and Half-angle Formulas includes 113 full step-by-step solutions.

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

    Tv = Av + Vo = linear transformation plus shift.

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

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

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • Diagonalization

    A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

  • Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

    Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Full row rank r = m.

    Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

  • Hankel matrix H.

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

  • Hermitian matrix A H = AT = A.

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

  • Orthogonal subspaces.

    Every v in V is orthogonal to every w in W.

  • Outer product uv T

    = column times row = rank one matrix.

  • Pivot columns of A.

    Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Rank r (A)

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

  • Singular matrix A.

    A square matrix that has no inverse: det(A) = o.

  • 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}.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Transpose matrix AT.

    Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide
×
Reset your password