- 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; One-to-One Functions
- Chapter 4.4: One-to-One 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
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).
Tv = Av + Vo = linear transformation plus shift.
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.
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].
Incidence matrix of a directed graph.
The m by n edge-node 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 A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)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.
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 •
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Pseudoinverse A+ (Moore-Penrose 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 ].
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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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