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Solutions for Chapter 3.2: Differential Equations and Linear Algebra 3rd Edition

Differential Equations and Linear Algebra | 3rd Edition | ISBN: 9780136054252 | Authors: C. Henry Edwards, David E. Penney

Full solutions for Differential Equations and Linear Algebra | 3rd Edition

ISBN: 9780136054252

Differential Equations and Linear Algebra | 3rd Edition | ISBN: 9780136054252 | Authors: C. Henry Edwards, David E. Penney

Solutions for Chapter 3.2

Solutions for Chapter 3.2
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Textbook: Differential Equations and Linear Algebra
Edition: 3
Author: C. Henry Edwards, David E. Penney
ISBN: 9780136054252

Since 30 problems in chapter 3.2 have been answered, more than 11471 students have viewed full step-by-step solutions from this chapter. Differential Equations and Linear Algebra was written by and is associated to the ISBN: 9780136054252. This textbook survival guide was created for the textbook: Differential Equations and Linear Algebra, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.2 includes 30 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • Block matrix.

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

    The n roots are the eigenvalues of A.

  • Cross product u xv in R3:

    Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

  • Eigenvalue A and eigenvector x.

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

  • 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 column rank r = n.

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

  • 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.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • 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)ยท(b - Ax) = o.

  • Normal matrix.

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

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Projection matrix P onto subspace S.

    Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

  • 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.

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • Spectral Theorem A = QAQT.

    Real symmetric A has real A'S and orthonormal q's.

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

  • 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.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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