 Chapter 1.1: Propositions and Connectives
 Chapter 1.2: Conditionals and Biconditionals
 Chapter 1.3: Quantifiers
 Chapter 1.4: Basic Proof Methods I
 Chapter 1.5: Basic Proof Methods II
 Chapter 1.6: Proofs Involving Quantifiers
 Chapter 1.7: Additional Examples of Proofs
 Chapter 2.1: Basic Concepts of Set Theory
 Chapter 2.2: Set Theory
 Chapter 2.3: Set Theory
 Chapter 2.4: Mathematical Induction
 Chapter 2.5: Equivalent Forms of Induction
 Chapter 2.6: Principles of Counting
 Chapter 3.1: Relations and Partitions
 Chapter 3.2: Equivalence Relations
 Chapter 3.3: Partitions
 Chapter 3.4: Relations and Partitions
 Chapter 3.5: Graphs
 Chapter 4.1: Functions as Relations
 Chapter 4.2: Constructions of Functions
 Chapter 4.3: Functions That Are Onto; OnetoOne Functions
 Chapter 4.4: OnetoOne Correspondences and Inverse Functions
 Chapter 4.5: Images of Sets
 Chapter 4.6: Sequences
 Chapter 5.1: Equivalent Sets; Finite Sets
 Chapter 5.2: Infinite Sets
 Chapter 5.3: Countable Sets
 Chapter 5.4: The Ordering of Cardinal Numbers
 Chapter 5.5: Comparability of Cardinal Numbers and the Axiom of Choice
 Chapter 6.1: Algebraic Structures
 Chapter 6.2: Groups
 Chapter 6.3: Subgroups
 Chapter 6.4: Operation Preserving Maps
 Chapter 6.5: Rings and Fields
 Chapter 7.1: Completeness of the Real Numbers
 Chapter 7.2: The HeineBorel Theorem
 Chapter 7.3: The BolzanoWeierstrass Theorem
 Chapter 7.4: The Bounded Monotone Sequence Theorem
 Chapter 7.5: Equivalents of Completeness
A Transition to Advanced Mathematics 7th Edition  Solutions by Chapter
Full solutions for A Transition to Advanced Mathematics  7th Edition
ISBN: 9780495562023
A Transition to Advanced Mathematics  7th Edition  Solutions by Chapter
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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 ].

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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