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Solutions for Chapter 2.2: Linear Algebra and Its Applications 4th Edition

Linear Algebra and Its Applications | 4th Edition | ISBN: 9780321385178 | Authors: David C. Lay

Full solutions for Linear Algebra and Its Applications | 4th Edition

ISBN: 9780321385178

Linear Algebra and Its Applications | 4th Edition | ISBN: 9780321385178 | Authors: David C. Lay

Solutions for Chapter 2.2

Solutions for Chapter 2.2
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Textbook: Linear Algebra and Its Applications
Edition: 4
Author: David C. Lay
ISBN: 9780321385178

This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. Chapter 2.2 includes 42 full step-by-step solutions. Since 42 problems in chapter 2.2 have been answered, more than 34725 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Complex conjugate

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

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • 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.

  • 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!).

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Left nullspace N (AT).

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

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • 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.

  • 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.

  • 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.

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

  • Vector space V.

    Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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