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Solutions for Chapter 9: Translate Figures and Use Vectors

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 9: Translate Figures and Use Vectors

Solutions for Chapter 9
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Textbook: Geometry (Holt McDougal Larson Geometry)
Edition: 1
Author: Ron Larson Laurie Boswell Timothy D. Kanold, Lee Stiff
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 354 problems in chapter 9: Translate Figures and Use Vectors have been answered, more than 26331 students have viewed full step-by-step solutions from this chapter. Geometry (Holt McDougal Larson Geometry) was written by and is associated to the ISBN: 9780618595402. Chapter 9: Translate Figures and Use Vectors includes 354 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Column space C (A) =

    space of all combinations of the columns of 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).

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • 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 .

  • Lucas numbers

    Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • Multiplier eij.

    The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

  • Outer product uv T

    = column times row = rank one matrix.

  • 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.

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Singular matrix A.

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

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • Spanning set.

    Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

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

    T- 1 has rank 1 above and below diagonal.

  • 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.

  • Wavelets Wjk(t).

    Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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