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Solutions for Chapter 4.2: Linear Algebra and Its Applications 5th Edition

Linear Algebra and Its Applications | 5th Edition | ISBN: 9780321982384 | Authors: David C. Lay; Steven R. Lay; Judi J. McDonald

Full solutions for Linear Algebra and Its Applications | 5th Edition

ISBN: 9780321982384

Linear Algebra and Its Applications | 5th Edition | ISBN: 9780321982384 | Authors: David C. Lay; Steven R. Lay; Judi J. McDonald

Solutions for Chapter 4.2

Solutions for Chapter 4.2
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Textbook: Linear Algebra and Its Applications
Edition: 5
Author: David C. Lay; Steven R. Lay; Judi J. McDonald
ISBN: 9780321982384

Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since 40 problems in chapter 4.2 have been answered, more than 41243 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Chapter 4.2 includes 40 full step-by-step solutions.

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

    Tv = Av + Vo = linear transformation plus shift.

  • Block matrix.

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

  • Cholesky factorization

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

  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • 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.

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • 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.

  • Gauss-Jordan method.

    Invert A by row operations on [A I] to reach [I A-I].

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • 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.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Kronecker product (tensor product) A ® B.

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

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • Multiplication Ax

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

  • 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).

  • 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.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Singular matrix A.

    A square matrix that has no inverse: det(A) = o.

  • 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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