 Chapter 1.1: Statements, Symbolic Representation, and Tautologies
 Chapter 1.2: Propositional Logic
 Chapter 1.3: Quantifiers, Predicates, and Validity
 Chapter 1.4: Predicate Logic
 Chapter 1.5: Logic Programming
 Chapter 1.6: Logic Programming
 Chapter 2.1: Proof Techniques
 Chapter 2.2: Induction
 Chapter 2.3: More on Proof of Correctness
 Chapter 2.4: Number Theory
 Chapter 3.1: Recursive Definitions
 Chapter 3.2: Recurrence Relations
 Chapter 3.3: Analysis of Algorithms
 Chapter 4.1: Sets
 Chapter 4.2: Counting
 Chapter 4.3: Principle of Inclusion and Exclusion; Pigeonhole Principle
 Chapter 4.4: Permutations and Combinations
 Chapter 5.1: Relations
 Chapter 5.2: Topological Sorting
 Chapter 5.3: Relations and Databases
 Chapter 5.4: Functions
 Chapter 5.5: Order of Magnitude
 Chapter 5.6: The Mighty Mod Function
 Chapter 5.7: Matrices
 Chapter 6.1: Graphs and Their Representations
 Chapter 6.2: Trees and Their Representations
 Chapter 6.3: Decision Trees
 Chapter 6.4: Huffman Codes
 Chapter 7.1: Directed Graphs and Binary Relations; Warshalls Algorithm
 Chapter 7.2: Euler Path and Hamiltonian Circuit
 Chapter 7.3: Shortest Path and Minimal Spanning Tree
 Chapter 7.4: Traversal Algorithms
 Chapter 7.5: Articulation Points and Computer Networks
 Chapter 8.1: Boolean Algebra Structure
 Chapter 8.2: Logic Networks
 Chapter 8.3: Minimization
 Chapter 9.1: Algebraic Structures
 Chapter 9.2: Coding Theory
 Chapter 9.3: FiniteState Machines
 Chapter 9.4: Turing Machines
 Chapter 9.5: Formal Languages
Mathematical Structures for Computer Science 7th Edition  Solutions by Chapter
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Mathematical Structures for Computer Science  7th Edition  Solutions by Chapter
Get Full SolutionsThe full stepbystep solution to problem in Mathematical Structures for Computer Science were answered by , our top Math solution expert on 01/18/18, 05:04PM. Since problems from 41 chapters in Mathematical Structures for Computer Science have been answered, more than 37248 students have viewed full stepbystep answer. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107. This expansive textbook survival guide covers the following chapters: 41. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fundamental Theorem.
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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.