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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 11425 students have viewed full step-by-step solutions from this chapter. Linear Algebra and Its Applications was written by Sieva Kozinsky 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
  • Determinant IAI = det(A).

    Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • 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.

  • lA-II = l/lAI and IATI = IAI.

    The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

  • 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)·(b - Ax) = o.

  • Nullspace matrix N.

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

  • Random matrix rand(n) or randn(n).

    MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

  • Rank r (A)

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

  • 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.

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • 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.

  • Spanning set.

    Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

  • 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.

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • Volume of box.

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

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