- 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).
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
A sequence of steps intended to approach the desired solution.
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).
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Every v in V is orthogonal to every w in W.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
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.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
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.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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