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Solutions for Chapter 11: Introduction to Trigonometry

Elementary Geometry for College Students | 6th Edition | ISBN: 9781285195698 | Authors: Daniel C. Alexander, Geralyn M. Koeberlein

Full solutions for Elementary Geometry for College Students | 6th Edition

ISBN: 9781285195698

Elementary Geometry for College Students | 6th Edition | ISBN: 9781285195698 | Authors: Daniel C. Alexander, Geralyn M. Koeberlein

Solutions for Chapter 11: Introduction to Trigonometry

Solutions for Chapter 11
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Textbook: Elementary Geometry for College Students
Edition: 6
Author: Daniel C. Alexander, Geralyn M. Koeberlein
ISBN: 9781285195698

Elementary Geometry for College Students was written by and is associated to the ISBN: 9781285195698. This textbook survival guide was created for the textbook: Elementary Geometry for College Students, edition: 6. Chapter 11: Introduction to Trigonometry includes 47 full step-by-step solutions. Since 47 problems in chapter 11: Introduction to Trigonometry have been answered, more than 3153 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • Back substitution.

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

  • Cayley-Hamilton Theorem.

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

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

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

  • Conjugate Gradient Method.

    A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • Fourier matrix F.

    Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

  • 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.

  • Hankel matrix H.

    Constant along each antidiagonal; hij depends on i + j.

  • Krylov subspace Kj(A, b).

    The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

  • Multiplication Ax

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

  • 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.

  • 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.

  • 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.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • 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!

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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