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Solutions for Chapter Chapter 10: Geometry

Thinking Mathematically | 6th Edition | ISBN: 9780321867322 | Authors: Robert F. Blitzer

Full solutions for Thinking Mathematically | 6th Edition

ISBN: 9780321867322

Thinking Mathematically | 6th Edition | ISBN: 9780321867322 | Authors: Robert F. Blitzer

Solutions for Chapter Chapter 10: Geometry

Solutions for Chapter Chapter 10
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Textbook: Thinking Mathematically
Edition: 6
Author: Robert F. Blitzer
ISBN: 9780321867322

Chapter Chapter 10: Geometry includes 74 full step-by-step solutions. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Since 74 problems in chapter Chapter 10: Geometry have been answered, more than 70763 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322.

Key Math Terms and definitions covered in this textbook
  • Cholesky factorization

    A = CTC = (L.J]))(L.J]))T for positive definite A.

  • Condition number

    cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

  • 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.

  • Elimination matrix = Elementary matrix Eij.

    The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

  • Free variable Xi.

    Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • 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.

  • 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 .

  • Norm

    IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

  • Orthonormal vectors q 1 , ... , q n·

    Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

  • Random matrix rand(n) or randn(n).

    MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

  • Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

    Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • Row picture of Ax = b.

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

  • Skew-symmetric matrix K.

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

  • Sum V + W of subs paces.

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

  • Symmetric matrix A.

    The transpose is AT = A, and aU = a ji. A-I is also symmetric.

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • Transpose matrix AT.

    Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

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