- Chapter 1.1: Inductive and Deductive Reasoning
- Chapter 1.2: Estimation, Graphs, and Mathematical Models
- Chapter 1.3: Problem Solving
- Chapter 10.1: Points, Lines, Planes, and Angles
- Chapter 10.2: Triangles
- Chapter 10.3: Polygons, Perimeter, and Tessellations
- Chapter 10.4: Area and Circumference
- Chapter 10.5: Volume and Surface Area
- Chapter 10.6: Right Triangle Trigonometry
- Chapter 10.7: Beyond Euclidean Geometry
- Chapter 11.1: The Fundamental Counting Principle
- Chapter 11.2: Permutations
- Chapter 11.3: Combinations
- Chapter 11.4: Fundamentals of Probability
- Chapter 11.5: Probability with the Fundamental Counting Principle, Permutations, and Combinations
- Chapter 11.6: Events Involving Not and Or; Odds
- Chapter 11.7: Events Involving And; Conditional Probability
- Chapter 11.8: Expected Value
- Chapter 12.1: Sampling, Frequency Distributions, and Graphs
- Chapter 12.2: Measures of Central Tendency
- Chapter 12.3: Measures of Dispersion
- Chapter 12.4: The Normal Distribution
- Chapter 12.5: Problem Solving with the Normal Distribution
- Chapter 12.6: Scatter Plots, Correlation, and Regression Lines
- Chapter 13.1: Voting Methods
- Chapter 13.2: Flaws of Voting Methods
- Chapter 13.3: Apportionment Methods
- Chapter 13.4: Flaws of Apportionment Methods
- Chapter 14.1: Graphs, Paths, and Circuits
- Chapter 14.2: Euler Paths and Euler Circuits
- Chapter 14.3: Hamilton Paths and Hamilton Circuits
- Chapter 14.4: Trees
- Chapter 2.1: Basic Set Concepts
- Chapter 2.2: Subsets
- Chapter 2.3: Venn Diagrams and Set Operations
- Chapter 2.4: Set Operations and Venn Diagrams with Three Sets
- Chapter 2.5: Survey Problems
- Chapter 3.1: Statements, Negations, and Quantified Statements
- Chapter 3.2: Compound Statements and Connectives
- Chapter 3.3: Truth Tables for Negation, Conjunction, and Disjunction
- Chapter 3.4: Truth Tables for the Conditional and the Biconditional
- Chapter 3.5: Equivalent Statements and Variations of Conditional Statements
- Chapter 3.6: Negations of Conditional Statements and De Morgans Laws
- Chapter 3.7: Arguments and Truth Tables
- Chapter 3.8: Arguments and Euler Diagrams
- Chapter 4.1: Our Hindu-Arabic System and Early Positional Systems
- Chapter 4.2: Number Bases in Positional Systems
- Chapter 4.3: Computation in Positional Systems
- Chapter 4.4: Looking Back at Early Numeration Systems
- Chapter 5.1: Number Theory: Prime and Composite Numbers
- Chapter 5.2: The Integers; Order of Operations
- Chapter 5.3: The Rational Numbers
- Chapter 5.4: The Irrational Numbers
- Chapter 5.5: Real Numbers and Their Properties; Clock Addition
- Chapter 5.6: Exponents and Scientific Notation
- Chapter 5.7: Arithmetic and Geometric Sequences
- Chapter 6.1: Algebraic Expressions and Formulas
- Chapter 6.2: Linear Equations in One Variable and Proportions
- Chapter 6.3: Applications of Linear Equations
- Chapter 6.4: Linear Inequalities in One Variable
- Chapter 6.5: Quadratic Equations
- Chapter 7.1: Graphing and Functions
- Chapter 7.2: Linear Functions and Their Graphs
- Chapter 7.3: Systems of Linear Equations in Two Variables
- Chapter 7.4: Linear Inequalities in Two Variables
- Chapter 7.5: Linear Programming
- Chapter 7.6: Modeling Data: Exponential, Logarithmic, and Quadratic Functions
- Chapter 8.1: Percent, Sales Tax, and Discounts
- Chapter 8.2: Income Tax
- Chapter 8.3: Simple Interest
- Chapter 8.4: Compound Interest
- Chapter 8.5: Annuities, Methods of Saving, and Investments
- Chapter 8.6: Cars
- Chapter 8.7: The Cost of Home Ownership
- Chapter 8.8: Credit Cards
- Chapter 9.1: Measuring Length; The Metric System
- Chapter 9.2: Measuring Area and Volume
- Chapter 9.3: Measuring Weight and Temperature
- Chapter Chaprter 5 : Number Theory and the Real Number System
- Chapter Chapter 1: Problem Solving and Critical Thinking
- Chapter Chapter 10: Geometry
- Chapter Chapter 11: Counting Methods and Probability Theory
- Chapter Chapter 12: Statistics
- Chapter Chapter 13: Voting and Apportionment
- Chapter Chapter 14: Graph Theory
- Chapter Chapter 2: Set Theory
- Chapter Chapter 3: Logic
- Chapter Chapter 4: Number ,Representation and Calculation 211
- Chapter Chapter 5: Number Theory and the Real Number System
- Chapter Chapter 6: Algebra: Equations and Inequalities
- Chapter Chapter 7: Algebra:Graphs, Functions, and Linear Systems
- Chapter Chapter 8: Personal Finance
- Chapter Chapter 9: Measurements
Thinking Mathematically 6th Edition - Solutions by Chapter
Full solutions for Thinking Mathematically | 6th Edition
Tv = Av + Vo = linear transformation plus shift.
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!
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + 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.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.