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Solutions for Chapter 8: College Algebra and Trigonometry 7th Edition

College Algebra and Trigonometry | 7th Edition | ISBN: 9781439048603 | Authors: Richard N. Aufmann

Full solutions for College Algebra and Trigonometry | 7th Edition

ISBN: 9781439048603

College Algebra and Trigonometry | 7th Edition | ISBN: 9781439048603 | Authors: Richard N. Aufmann

Solutions for Chapter 8

Solutions for Chapter 8
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Textbook: College Algebra and Trigonometry
Edition: 7
Author: Richard N. Aufmann
ISBN: 9781439048603

College Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. Since 71 problems in chapter 8 have been answered, more than 21193 students have viewed full step-by-step solutions from this chapter. Chapter 8 includes 71 full step-by-step solutions. This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • 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.

  • Hilbert matrix hilb(n).

    Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • 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 .

  • Least squares solution X.

    The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • 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.

  • Nullspace matrix N.

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

  • Outer product uv T

    = column times row = rank one matrix.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Pseudoinverse A+ (Moore-Penrose inverse).

    The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

  • Row picture of Ax = b.

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

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • Singular matrix A.

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

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).