- 4.1.1: Prove that 1 23 + 1 34 + + 1 (n+1)(n+2) = n 2n+4 for every positive...
- 4.1.2: Prove that 1 13 + 1 35 + 1 57 + + 1 (2n1)(2n+1) = n 2n+1 for every ...
- 4.1.3: Prove that 1 14 + 1 47 + 1 710 + + 1 (3n2)(3n+1) = n 3n+1 for every...
- 4.1.4: Find a formula for 2+4+6+ +2n for every positive integer n and then...
- 4.1.5: Prove that 1 + 5 + 9 + + (4n 3) = n(2n 1) for every positive intege...
- 4.1.6: Prove that 2 + 5 + 8 + + (3n 1) = n(3n + 1)/2 for every positive in...
- 4.1.7: Prove the result in Result 4.5: For every positive integer n, 12 + ...
- 4.1.8: Prove the result in Result 4.6: For every positive integer n, 13 +2...
- 4.1.9: Prove that 1 + 3 + 32 + + 3n = (3n+1 1)/2 for every positive intege...
- 4.1.10: Let r 2 be an integer. Prove that 1+r +r2 + +rn = rn+1 1 r1 for eve...
- 4.1.11: Prove that 1 2 + 2 3 + 3 4 + + n(n + 1) = n(n+1)(n+2) 3 for every p...
- 4.1.12: (a) Let k N. Prove that if k2 + k + 5 is even, then (k + 1)2 + (k +...
- 4.1.13: Consider the expression (n + 2)(n + 3)/2, where n N. (a) Is (n + 2)...
- 4.1.14: Square tiles, one foot on each side, are to be placed on a square s...
Solutions for Chapter 4.1: The Principle of Mathematical Induction
Full solutions for Discrete Mathematics | 1st Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Remove row i and column j; multiply the determinant by (-I)i + j •
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).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
A sequence of steps intended to approach the desired solution.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
A directed graph that has constants Cl, ... , Cm associated with the edges.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Constant down each diagonal = time-invariant (shift-invariant) filter.