- Chapter 1: Equations, Inequalities, and Mathematical Modeling
- Chapter 1.1: GRAPHS OF EQUATIONS
- Chapter 1.2: LINEAR EQUATIONS IN ONE VARIABLE
- Chapter 1.3: MODELING WITH LINEAR EQUATIONS
- Chapter 1.4: QUADRATIC EQUATIONS AND APPLICATIONS
- Chapter 1.5: COMPLEX NUMBERS
- Chapter 1.6: OTHER TYPES OF EQUATIONS
- Chapter 1.7: LINEAR INEQUALITIES IN ONE VARIABLE
- Chapter 1.8: OTHER TYPES OF INEQUALITIES
- Chapter 2: Functions and Their Graphs
- Chapter 2.1: LINEAR EQUATIONS IN TWO VARIABLES
- Chapter 2.2: FUNCTIONS
- Chapter 2.3: ANALYZING GRAPHS OF FUNCTIONS
- Chapter 2.4: A LIBRARY OF PARENT FUNCTIONS
- Chapter 2.5: TRANSFORMATIONS OF FUNCTIONS
- Chapter 2.6: COMBINATIONS OF FUNCTIONS: COMPOSITE FUNCTIONS
- Chapter 2.7: INVERSE FUNCTIONS
- Chapter 3: Polynomial Functions
- Chapter 3.1: QUADRATIC FUNCTIONS AND MODELS
- Chapter 3.2: POLYNOMIAL FUNCTIONS OF HIGHER DEGREE
- Chapter 3.3: POLYNOMIAL AND SYNTHETIC DIVISION
- Chapter 3.4: ZEROS OF POLYNOMIAL FUNCTIONS
- Chapter 3.5: MATHEMATICAL MODELING AND VARIATION
- Chapter 4: Rational Functions and Conics
- Chapter 4.1: RATIONAL FUNCTIONS AND ASYMPTOTES
- Chapter 4.2: GRAPHS OF RATIONAL FUNCTIONS
- Chapter 4.3: CONICS
- Chapter 4.4: TRANSLATIONS OF CONICS
- Chapter 5: Exponential and Logarithmic Functions
- Chapter 5.1: Exponential Functions and Their Graphs
- Chapter 5.2: Logarithmic Functions and Their Graphs
- Chapter 5.3: Properties of Logarithms
- Chapter 5.4: Exponential and Logarithmic Equations
- Chapter 5.5: Exponential and Logarithmic Models
- Chapter 6: Systems of Equations and Inequalities
- Chapter 6.1: Linear and Nonlinear Systems of Equations
- Chapter 6.2: Two-Variable Linear Systems
- Chapter 6.3: Multivariable Linear Systems
- Chapter 6.4: Partial Fractions
- Chapter 6.5: Systems of Inequalities
- Chapter 6.6: Linear Programming
- Chapter 7: Matrices and Determinants
- Chapter 7.1: Matrices and Systems of Equations
- Chapter 7.2: Operations with Matrices
- Chapter 7.3: The Inverse of a Square Matrix
- Chapter 7.4: The Determinant of a Square Matrix
- Chapter 7.5: Applications of Matrices and Determinants
- Chapter 8: Sequences, Series, and Probability
- Chapter 8.1: Sequences and Series
- Chapter 8.2: Arithmetic Sequences and Partial Sums
- Chapter 8.3: Geometric Sequences and Series
- Chapter 8.4: Mathematical Induction
- Chapter 8.5: The Binomial Theorem
- Chapter 8.6: Counting Principles
- Chapter 8.7: Probability
- Chapter P: Prerequisites
- Chapter P.1: Review of Real Numbers and Their Properties
- Chapter P.2: Exponents and Radicals
- Chapter P.3: Polynomials and Special Products
- Chapter P.4: Factoring Polynomials
- Chapter P.5: Rational Expressions
- Chapter P.6: The Rectangular Coordinate System and Graphs
College Algebra 8th Edition - Solutions by Chapter
Full solutions for College Algebra | 8th Edition
Tv = Av + Vo = linear transformation plus shift.
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
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).
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
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.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
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.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.