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Solutions for Chapter 4.4: Properties of Bases

Full solutions for Linear Algebra with Applications | 8th Edition

ISBN: 9781449679545

Solutions for Chapter 4.4: Properties of Bases

Solutions for Chapter 4.4
4 5 0 270 Reviews
Textbook: Linear Algebra with Applications
Edition: 8
Author: Gareth Williams
ISBN: 9781449679545

Chapter 4.4: Properties of Bases includes 33 full step-by-step solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Since 33 problems in chapter 4.4: Properties of Bases have been answered, more than 8756 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545.

Key Math Terms and definitions covered in this textbook
  • Distributive Law

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

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

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

  • Gram-Schmidt orthogonalization A = QR.

    Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

  • Hankel matrix H.

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

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • 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.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • Linearly dependent VI, ... , Vn.

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

  • Multiplier eij.

    The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

  • 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.

  • Partial pivoting.

    In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

  • 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.

  • Reflection matrix (Householder) Q = I -2uuT.

    Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Toeplitz matrix.

    Constant down each diagonal = time-invariant (shift-invariant) filter.

  • 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.

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B II·

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