- 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Outer product uv T
= column times row = rank one matrix.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
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.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.