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Solutions for Chapter 7.9: Nonhomogeneous Linear Systems

Elementary Differential Equations and Boundary Value Problems | 11th Edition | ISBN: 9781119256007 | Authors: Boyce, Diprima, Meade

Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition

ISBN: 9781119256007

Elementary Differential Equations and Boundary Value Problems | 11th Edition | ISBN: 9781119256007 | Authors: Boyce, Diprima, Meade

Solutions for Chapter 7.9: Nonhomogeneous Linear Systems

Solutions for Chapter 7.9
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Textbook: Elementary Differential Equations and Boundary Value Problems
Edition: 11
Author: Boyce, Diprima, Meade
ISBN: 9781119256007

Chapter 7.9: Nonhomogeneous Linear Systems includes 17 full step-by-step solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Since 17 problems in chapter 7.9: Nonhomogeneous Linear Systems have been answered, more than 12334 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • Covariance matrix:E.

    When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

  • 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.

  • 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.

  • Full column rank r = n.

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

  • Hermitian matrix A H = AT = A.

    Complex analog a j i = aU of a symmetric matrix.

  • 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.

  • 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.

  • Nilpotent matrix N.

    Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • 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.

  • 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.

  • Singular Value Decomposition

    (SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • Vandermonde matrix V.

    V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

  • Vector addition.

    v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

  • Wavelets Wjk(t).

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

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