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Solutions for Chapter 2.4: Mathematical Induction

A Transition to Advanced Mathematics | 7th Edition | ISBN: 9780495562023 | Authors: Douglas Smith, Maurice Eggen, Richard St. Andre

Full solutions for A Transition to Advanced Mathematics | 7th Edition

ISBN: 9780495562023

A Transition to Advanced Mathematics | 7th Edition | ISBN: 9780495562023 | Authors: Douglas Smith, Maurice Eggen, Richard St. Andre

Solutions for Chapter 2.4: Mathematical Induction

Since 14 problems in chapter 2.4: Mathematical Induction have been answered, more than 6148 students have viewed full step-by-step solutions from this chapter. A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780495562023. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: A Transition to Advanced Mathematics, edition: 7. Chapter 2.4: Mathematical Induction includes 14 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • 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).

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

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

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

  • 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.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • Multiplication Ax

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

  • 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.

  • Outer product uv T

    = column times row = rank one matrix.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • 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).

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • 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.

  • Saddle point of I(x}, ... ,xn ).

    A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

  • 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.

  • 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.

  • Vector v in Rn.

    Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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