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Solutions for Chapter 1.8: 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 1.8

Solutions for Chapter 1.8
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Textbook: Linear Algebra and Its Applications
Edition: 5
Author: David C. Lay; Steven R. Lay; Judi J. McDonald
ISBN: 9780321982384

Chapter 1.8 includes 40 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since 40 problems in chapter 1.8 have been answered, more than 39848 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5.

Key Math Terms and definitions covered in this textbook
  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Cofactor Cij.

    Remove row i and column j; multiply the determinant by (-I)i + j •

  • Dimension of vector space

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

  • 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.

  • Fast Fourier Transform (FFT).

    A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Gauss-Jordan method.

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

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Jordan form 1 = M- 1 AM.

    If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Multiplication Ax

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

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • 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!

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • Subspace S of V.

    Any vector space inside V, including V and Z = {zero vector only}.

  • Sum V + W of subs paces.

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

  • Wavelets Wjk(t).

    Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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