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Solutions for Chapter 9.4: Mathematical Induction

Full solutions for College Algebra | 9th Edition

ISBN: 9780321716811

Solutions for Chapter 9.4: Mathematical Induction

Solutions for Chapter 9.4
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Textbook: College Algebra
Edition: 9
Author: Michael Sullivan
ISBN: 9780321716811

Since 34 problems in chapter 9.4: Mathematical Induction have been answered, more than 7872 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.4: Mathematical Induction includes 34 full step-by-step solutions. College Algebra was written by and is associated to the ISBN: 9780321716811.

Key Math Terms and definitions covered in this textbook
  • Diagonalizable matrix A.

    Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

  • 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.

  • Elimination matrix = Elementary matrix Eij.

    The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

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

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

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • 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!).

  • 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.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • Krylov subspace Kj(A, b).

    The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

  • Left inverse A+.

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

  • Network.

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

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • 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.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Semidefinite matrix A.

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

  • Spanning set.

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

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B IIĀ·

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

  • 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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