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Solutions for Chapter 8: Find Angle Measures in Polygons

Geometry (Holt McDougal Larson Geometry) | 1st Edition | ISBN: 9780618595402 | Authors: Ron Larson Laurie Boswell Timothy D. Kanold, Lee Stiff

Full solutions for Geometry (Holt McDougal Larson Geometry) | 1st Edition

ISBN: 9780618595402

Geometry (Holt McDougal Larson Geometry) | 1st Edition | ISBN: 9780618595402 | Authors: Ron Larson Laurie Boswell Timothy D. Kanold, Lee Stiff

Solutions for Chapter 8: Find Angle Measures in Polygons

Solutions for Chapter 8
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Textbook: Geometry (Holt McDougal Larson Geometry)
Edition: 1
Author: Ron Larson Laurie Boswell Timothy D. Kanold, Lee Stiff
ISBN: 9780618595402

Geometry (Holt McDougal Larson Geometry) was written by and is associated to the ISBN: 9780618595402. This textbook survival guide was created for the textbook: Geometry (Holt McDougal Larson Geometry), edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 347 problems in chapter 8: Find Angle Measures in Polygons have been answered, more than 26700 students have viewed full step-by-step solutions from this chapter. Chapter 8: Find Angle Measures in Polygons includes 347 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • Cholesky factorization

    A = CTC = (L.J]))(L.J]))T for positive definite A.

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

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

  • Cramer's Rule for Ax = b.

    B j has b replacing column j of A; x j = det B j I det A

  • 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).

  • Elimination matrix = Elementary matrix Eij.

    The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

  • 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.

  • Incidence matrix of a directed graph.

    The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

  • Kronecker product (tensor product) A ® B.

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

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • Normal equation AT Ax = ATb.

    Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • 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 •

  • Partial pivoting.

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

  • Positive definite matrix A.

    Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

  • Projection matrix P onto subspace S.

    Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

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

    Spectral radius = max of IAi I.

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • 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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