- Chapter 1: Speaking Mathematically
- Chapter 10: Graphs and Trees
- Chapter 2: THE LOGIC OF COMPOUND STATEMENTS
- Chapter 3: The Logic of Quantied Statements
- Chapter 4: Elementary Number Theory and Methods of Proof
- Chapter 5: Sequences, Mathematical Induction, and Recursion
- Chapter 6: Set Theory
- Chapter 7: Functions
- Chapter 8: Relations
- Chapter 9: Counting and Probability
Discrete Mathematics: Introduction to Mathematical Reasoning 1st Edition - Solutions by Chapter
Full solutions for Discrete Mathematics: Introduction to Mathematical Reasoning | 1st Edition
Discrete Mathematics: Introduction to Mathematical Reasoning | 1st Edition - Solutions by ChapterGet Full Solutions
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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!).
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
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).
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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.