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Solutions for Chapter 6: Equations

Trigonometry | ISBN: 9780495108351 | Authors: Charles P McKeague

Full solutions for Trigonometry

ISBN: 9780495108351

Trigonometry | ISBN: 9780495108351 | Authors: Charles P McKeague

Solutions for Chapter 6: Equations

Solutions for Chapter 6
4 5 0 331 Reviews
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Textbook: Trigonometry
Edition:
Author: Charles P McKeague
ISBN: 9780495108351

This expansive textbook survival guide covers the following chapters and their solutions. Since 37 problems in chapter 6: Equations have been answered, more than 31979 students have viewed full step-by-step solutions from this chapter. Chapter 6: Equations includes 37 full step-by-step solutions. This textbook survival guide was created for the textbook: Trigonometry, edition: . Trigonometry was written by and is associated to the ISBN: 9780495108351.

Key Math Terms and definitions covered in this textbook
  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Identity matrix I (or In).

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

  • Kronecker product (tensor product) A ® B.

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

  • 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.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • Multiplication Ax

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

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • 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.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

  • Spanning set.

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

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • 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.

  • Vector v in Rn.

    Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

  • Wavelets Wjk(t).

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

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