- 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: Counting and Probability
- 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: Systems of Equations and Inequalities
- 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: Sequences; Induction; the Binomial Theorem
- 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 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
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
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).
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].
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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.
A directed graph that has constants Cl, ... , Cm associated with the edges.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.