- Chapter 1: Equations and Inequalities
- Chapter 1.1: Linear Equations
- Chapter 1.2: Quadratic Equations
- Chapter 1.3: Complex Numbers; Quadratic Equations in the Complex Number System
- Chapter 1.4: Radical Equations; Equations Quadratic in Form; Factorable Equations
- Chapter 1.5: Solving Inequalities
- Chapter 1.6: Equations and Inequalities Involving Absolute Value
- Chapter 1.7: Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications
- Chapter 10.1: Counting
- Chapter 10.2: Permutations and Combinations
- Chapter 10.3: Probability
- Chapter 2: Graphs
- Chapter 2.1: The Distance and Midpoint Formulas
- Chapter 2.2: Graphs of Equations in Two Variables; Intercepts; Symmetry
- Chapter 2.3: Lines
- Chapter 2.4: Circles
- Chapter 2.5: Variation
- Chapter 3: Functions and Their Graphs
- Chapter 3.1: Functions
- Chapter 3.2: The Graph of a Function
- Chapter 3.3: Properties of Functions
- Chapter 3.4: Library of Functions; Piecewise-defined Functions
- Chapter 3.5: Graphing Techniques: Transformations
- Chapter 3.6: Mathematical Models: Building Functions
- Chapter 4: Linear and Quadratic Functions
- Chapter 4.1: Linear Functions and Their Properties
- Chapter 4.2: Linear Models: Building Linear Functions from Data
- Chapter 4.3: Quadratic Functions and Their Properties
- Chapter 4.4: Build Quadratic Models from Verbal Descriptions and from Data
- Chapter 4.5: Inequalities Involving Quadratic Functions
- Chapter 5.1: Polynomial Functions and Models
- Chapter 6: Exponential and Logarithmic Functions
- Chapter 6.1: Composite Functions
- Chapter 6.2: One-to-One Functions; Inverse Functions
- Chapter 6.3: Exponential Functions
- Chapter 6.4: Logarithmic Functions
- Chapter 6.5: Properties of Logarithms
- Chapter 6.6: Logarithmic and Exponential Equations
- Chapter 6.7: Financial Models
- Chapter 6.8: Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay Models
- Chapter 6.9: Building Exponential, Logarithmic, and Logistic Models from Data
- Chapter 7: Analytic Geometry
- Chapter 7.2: The Parabola
- Chapter 7.3: The Ellipse
- Chapter 7.4: The Hyperbola
- Chapter 8.1: Systems of Linear Equations: Substitution and Elimination
- Chapter 8.2: Systems of Linear Equations: Matrices
- Chapter 8.3: Systems of Linear Equations: Determinants
- Chapter 8.4: Matrix Algebra
- Chapter 8.5: Partial Fraction Decomposition
- Chapter 8.6: Systems of Nonlinear Equations
- Chapter 8.7: Systems of Inequalities
- Chapter 9.1: Sequences
- Chapter 9.2: Arithmetic Sequences
- Chapter 9.3: Geometric Sequences; Geometric Series
- Chapter 9.4: Mathematical Induction
- Chapter 9.5: The Binomial Theorem
- Chapter Chapter 10: Counting and Probability
- Chapter Chapter 8: Systems of Equations and Inequalities
- Chapter Chapter 9: Sequences; Induction; the Binomial Theorem
- Chapter R.1: Real Numbers
- Chapter R.2: Algebra Essentials
- Chapter R.3: Geometry Essentials
- Chapter R.4 : Polynomials
- Chapter R.5 : Factoring Polynomials
- Chapter R.6 : Synthetic Division
- Chapter R.7 : Rational Expressions
- Chapter R.8 : nth Roots; Rational Exponents
College Algebra 9th Edition - Solutions by Chapter
Full solutions for College Algebra | 9th 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).
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
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).
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Outer product uv T
= column times row = rank one matrix.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Solvable system Ax = b.
The right side b is in the column space of A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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.