- Chapter 1: Graphs, Functions, and Models
- Chapter 1.1: Introduction to Graphing
- Chapter 1.2: Functions and Graphs
- Chapter 1.3: Linear Functions, Slope, and Applications
- Chapter 1.4: Equations of Lines and Modeling
- Chapter 1.5: Linear Equations, Functions, Zeros, and Applications
- Chapter 1.6: Solving Linear Inequalities
- Chapter 2: More on Functions
- Chapter 2.1: Increasing, Decreasing, and Piecewise Functions; Applications
- Chapter 2.2: The Algebra of Functions
- Chapter 2.3: The Composition of Functions
- Chapter 2.4: Symmetry
- Chapter 2.5: Transformations
- Chapter 2.6: Variation and Applications
- Chapter 3: Quadratic Functions and Equations; Inequalities
- Chapter 3.1: The Complex Numbers
- Chapter 3.2: Quadratic Equations, Functions, Zeros, and Models
- Chapter 3.3: Analyzing Graphs of Quadratic Functions
- Chapter 3.4: Solving Rational Equations and Radical Equations
- Chapter 3.5: Solving Equations and Inequalities with Absolute Value
- Chapter 4: Polynomial Functions and Rational Functions
- Chapter 4.1: Polynomial Functions and Modeling
- Chapter 4.2: Graphing Polynomial Functions
- Chapter 4.3: Polynomial Division; The Remainder Theorem and the Factor Theorem
- Chapter 4.4: Theorems about Zeros of Polynomial Functions
- Chapter 4.5: Rational Functions
- Chapter 4.6: Polynomial Inequalities and Rational Inequalities
- Chapter 5: Exponential Functions and Logarithmic Functions
- Chapter 5.1: Inverse Functions
- Chapter 5.2: Exponential Functions and Graphs
- Chapter 5.3: Logarithmic Functions and Graphs
- Chapter 5.4: Properties of Logarithmic Functions
- Chapter 5.5: Solving Exponential Equations and Logarithmic Equations
- Chapter 5.6: Applications and Models: Growth and Decay; Compound Interest
- Chapter 6: Systems of Equations and Matrices
- Chapter 6.1: Systems of Equations in Two Variables
- Chapter 6.2: Systems of Equations in Three Variables
- Chapter 6.3: Matrices and Systems of Equations
- Chapter 6.4: Matrix Operations
- Chapter 6.5: Inverses of Matrices
- Chapter 6.6: Determinants and Cramers Rule
- Chapter 6.7: Systems of Inequalities and Linear Programming
- Chapter 6.8: Partial Fractions
- Chapter 7: Conic Sections
- Chapter 7.1: The Parabola
- Chapter 7.2: The Circle and the Ellipse
- Chapter 7.3: The Hyperbola
- Chapter 7.4: Nonlinear Systems of Equations and Inequalities
- Chapter 8: Sequences, Series, and Combinatorics
- Chapter 8.1: Sequences and Series
- Chapter 8.2: Arithmetic Sequences and Series
- Chapter 8.3: Geometric Sequences and Series
- Chapter 8.4: Mathematical Induction
- Chapter 8.5: Combinatorics: Permutations
- Chapter 8.6: Combinatorics: Combinations
- Chapter 8.7: The Binomial Theorem
- Chapter 8.8: Probability
- Chapter R: Basic Concepts of Algebra
- Chapter R.1: The Real-Number System
- Chapter R.2: Integer Exponents, Scientific Notation, and Order of Operations
- Chapter R.3: Addition, Subtraction, and Multiplication of Polynomials
- Chapter R.4: Factoring
- Chapter R.5: The Basics of Equation Solving
- Chapter R.6: Rational Expressions
- Chapter R.7: Radical Notation and Rational Exponents
College Algebra: Graphs and Models 5th Edition - Solutions by Chapter
Full solutions for College Algebra: Graphs and Models | 5th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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).
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(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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].
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Solvable system Ax = b.
The right side b is in the column space of A.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).