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Solutions for Chapter 1.7: Existence and Uniqueness of Solutions

Advanced Engineering Mathematics | 9th Edition | ISBN: 9780471488859 | Authors: Erwin Kreyszig

Full solutions for Advanced Engineering Mathematics | 9th Edition

ISBN: 9780471488859

Advanced Engineering Mathematics | 9th Edition | ISBN: 9780471488859 | Authors: Erwin Kreyszig

Solutions for Chapter 1.7: Existence and Uniqueness of Solutions

Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 10 problems in chapter 1.7: Existence and Uniqueness of Solutions have been answered, more than 46329 students have viewed full step-by-step solutions from this chapter. Chapter 1.7: Existence and Uniqueness of Solutions includes 10 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • 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).

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • 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.

  • 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.

  • Normal matrix.

    If N NT = NT N, then N has orthonormal (complex) eigenvectors.

  • Nullspace matrix N.

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

  • Outer product uv T

    = column times row = rank one matrix.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • 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).

  • Row picture of Ax = b.

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

  • Standard basis for Rn.

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

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • 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.

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