- 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Remove row i and column j; multiply the determinant by (-I)i + j •
Column space C (A) =
space of all combinations of the columns of A.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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.
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.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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.