- 26.26.1: We list several pairs of functions f and g. For each pair, please d...
- 26.26.2: Consider functions f and g. Prove that f D g (as sets) if and only ...
- 26.26.3: Let A and B be sets. Prove that A D B if and only if idA D idB.
- 26.26.4: What is the difference between the identity function defined on a s...
- 26.26.5: Complete the proof of Proposition 26.8.
- 26.26.6: Prove Proposition 26.9.
- 26.26.7: Suppose A and B are sets, and f and g are functions with f W A ! B ...
- 26.26.8: Suppose f W A ! B is a bijection. Explain why the following are inc...
- 26.26.9: Suppose A, B, and C are sets and f W A ! B and g W B ! C. Prove the...
- 26.26.10: Find a pair of functions f and g, from set A to itself, such that f...
- 26.26.11: Let A be a set and f a function with f W A ! A. a. Suppose f is one...
- 26.26.12: Suppose f W A ! A and g W A ! A are both bijections. a. Prove or di...
- 26.26.13: Let A be a set and let f W A ! A. Then f f is also a function from ...
- 26.26.14: For each of the following sequences, find a formula for the n-th it...
Solutions for Chapter 26: Composition
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
Upper triangular systems are solved in reverse order Xn to Xl.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
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.
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.