 2.4.1: Which of these sets have the inductive property?(a) (b)(c) (d) {17}...
 2.4.2: Suppose S is inductive. Which of the following must be true?(a) If ...
 2.4.3: (a) Prove that is inductive.(b) Prove that is inductive.
 2.4.4: . Evaluate or simplify each.(a) 4! (b
 2.4.5: . Give an inductive definition for each:
 2.4.6: Use the PMI to prove the following for all natural numbers n.
 2.4.7: Use the PMI to prove the following for all natural numbers:(a) is d...
 2.4.8: Use the Generalized PMI to prove the following. (a) for all(b) for ...
 2.4.9: Use the PMI to prove DeMorgans Laws for an indexed familyYou may us...
 2.4.10: Let be n points in a plane with no three points collinear. Show tha...
 2.4.11: A puzzle called the Towers of Hanoi consists of a board with 3 pegs...
 2.4.12: With a little practice, perhaps using coins of various sizes, you s...
 2.4.13: In a certain kind of tournament, every player plays every other pla...
 2.4.14: Assign a grade of A (correct), C (partially correct), or F (failure...
Solutions for Chapter 2.4: Mathematical Induction
Full solutions for A Transition to Advanced Mathematics  7th Edition
ISBN: 9780495562023
Solutions for Chapter 2.4: Mathematical Induction
Get Full SolutionsSince 14 problems in chapter 2.4: Mathematical Induction have been answered, more than 6148 students have viewed full stepbystep 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 stepbystep solutions.

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, offdiagonal 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 = Q1 = 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.