- 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
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.)
Remove row i and column j; multiply the determinant by (-I)i + j •
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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!).
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Solvable system Ax = b.
The right side b is in the column space of A.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).