- 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).
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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).
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].
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
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.
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].
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Gram-Schmidt 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.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
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