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Solutions for Chapter 54: ISOMORPHISM THEOREMS AND SOLVABLE GROUPS

Modern Algebra: An Introduction | 6th Edition | ISBN: 9780470384435 | Authors: John R. Durbin

Full solutions for Modern Algebra: An Introduction | 6th Edition

ISBN: 9780470384435

Modern Algebra: An Introduction | 6th Edition | ISBN: 9780470384435 | Authors: John R. Durbin

Solutions for Chapter 54: ISOMORPHISM THEOREMS AND SOLVABLE GROUPS

Chapter 54: ISOMORPHISM THEOREMS AND SOLVABLE GROUPS includes 11 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 11 problems in chapter 54: ISOMORPHISM THEOREMS AND SOLVABLE GROUPS have been answered, more than 8059 students have viewed full step-by-step solutions from this chapter. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6.

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

    Tv = Av + Vo = linear transformation plus shift.

  • Cayley-Hamilton Theorem.

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

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Complete solution x = x p + Xn to Ax = b.

    (Particular x p) + (x n in nullspace).

  • 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].

  • 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.

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • Hankel matrix H.

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

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • 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.

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

    Vector addition and scalar multiplication.

  • Multiplier eij.

    The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

  • Network.

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

  • 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.

  • Orthogonal subspaces.

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

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

    Column vectors by convention.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as 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.

  • Toeplitz matrix.

    Constant down each diagonal = time-invariant (shift-invariant) filter.

  • Vector addition.

    v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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