Solutions for Chapter 1.7: 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 1.7

Solutions for Chapter 1.7
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Textbook: Linear Algebra and Its Applications
Edition: 5th
Author: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN: 9780321982384

Chapter 1.7 includes 44 full step-by-step solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5th. Since 44 problems in chapter 1.7 have been answered, more than 22957 students have viewed full step-by-step solutions from this chapter. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Adjacency matrix of a graph.

    Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Distributive Law

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

  • 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!).

  • Full row rank r = m.

    Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Indefinite matrix.

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

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • 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.

  • 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 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.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • Orthogonal matrix Q.

    Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

  • Permutation matrix P.

    There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

  • Pivot columns of A.

    Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

  • 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).

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • Subspace S of V.

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

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

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