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Solutions for Chapter 4.5: Linear Algebra and Its Applications 4th Edition

Linear Algebra and Its Applications | 4th Edition | ISBN: 9780321385178 | Authors: David C. Lay

Full solutions for Linear Algebra and Its Applications | 4th Edition

ISBN: 9780321385178

Linear Algebra and Its Applications | 4th Edition | ISBN: 9780321385178 | Authors: David C. Lay

Solutions for Chapter 4.5

Solutions for Chapter 4.5
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Textbook: Linear Algebra and Its Applications
Edition: 4
Author: David C. Lay
ISBN: 9780321385178

Since 34 problems in chapter 4.5 have been answered, more than 28124 students have viewed full step-by-step solutions from this chapter. Chapter 4.5 includes 34 full step-by-step solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Cayley-Hamilton Theorem.

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

  • 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.)

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

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

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

  • Diagonalizable matrix A.

    Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Hilbert matrix hilb(n).

    Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

  • lA-II = l/lAI and IATI = IAI.

    The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

  • 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.

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

    Vector addition and scalar multiplication.

  • 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.

  • Outer product uv T

    = column times row = rank one matrix.

  • Partial pivoting.

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

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • 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.

  • Solvable system Ax = b.

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

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

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