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Solutions for Chapter 2: Fundamentals of Differential Equations and Boundary Value Problems 6th Edition

Fundamentals of Differential Equations and Boundary Value Problems | 6th Edition | ISBN: 9780321747747 | Authors: Kent Nagle

Full solutions for Fundamentals of Differential Equations and Boundary Value Problems | 6th Edition

ISBN: 9780321747747

Fundamentals of Differential Equations and Boundary Value Problems | 6th Edition | ISBN: 9780321747747 | Authors: Kent Nagle

Solutions for Chapter 2

Solutions for Chapter 2
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Textbook: Fundamentals of Differential Equations and Boundary Value Problems
Edition: 6
Author: Kent Nagle
ISBN: 9780321747747

Chapter 2 includes 41 full step-by-step solutions. This textbook survival guide was created for the textbook: Fundamentals of Differential Equations and Boundary Value Problems, edition: 6. Fundamentals of Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780321747747. This expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 2 have been answered, more than 1827 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

  • Cholesky factorization

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

  • Cofactor Cij.

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

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • Hankel matrix H.

    Constant along each antidiagonal; hij depends on i + j.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Independent vectors VI, .. " vk.

    No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

  • Inverse matrix A-I.

    Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

  • Kronecker product (tensor product) A ® B.

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

  • 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.

  • Orthogonal subspaces.

    Every v in V is orthogonal to every w in W.

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Positive definite matrix A.

    Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

  • Reflection matrix (Householder) Q = I -2uuT.

    Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Row space C (AT) = all combinations of rows of A.

    Column vectors by convention.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • 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·

  • Wavelets Wjk(t).

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

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