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Solutions for Chapter 11.3: Algebra and Trigonometry 9th Edition

Algebra and Trigonometry | 9th Edition | ISBN: 9780321716569 | Authors: Michael Sullivan

Full solutions for Algebra and Trigonometry | 9th Edition

ISBN: 9780321716569

Algebra and Trigonometry | 9th Edition | ISBN: 9780321716569 | Authors: Michael Sullivan

Solutions for Chapter 11.3

Solutions for Chapter 11.3
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Textbook: Algebra and Trigonometry
Edition: 9
Author: Michael Sullivan
ISBN: 9780321716569

Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Since 85 problems in chapter 11.3 have been answered, more than 58162 students have viewed full step-by-step solutions from this chapter. Chapter 11.3 includes 85 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • 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.

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)ยท(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 .

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • Normal matrix.

    If N NT = NT N, then N has orthonormal (complex) eigenvectors.

  • Nullspace matrix N.

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

  • 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.

  • Permutation matrix P.

    There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

  • Pivot.

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

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Rank r (A)

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

  • Row space C (AT) = all combinations of rows of A.

    Column vectors by convention.

  • Saddle point of I(x}, ... ,xn ).

    A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

  • Solvable system Ax = b.

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

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

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