- 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 10: Matrices and Determinants
- Chapter 10.1: Matrices and Systems of Equations
- Chapter 10.2: Operations with Matrices
- Chapter 10.3: The Inverse of a Square Matrix
- Chapter 10.4: The Determinant of a Square Matrix
- Chapter 10.5: Applications of Matrices and Determinants
- Chapter 11: Sequences, Series and Probability
- Chapter 11.1: Sequences and Series
- Chapter 11.2: Arithmetic Sequences and Partial Sums
- Chapter 11.3: Geometric Sequences and Series
- Chapter 11.4: Mathematical Induction
- Chapter 11.5: The Binomial Theorem
- Chapter 11.6: Counting Principles
- Chapter 11.7: Probability
- 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: Trigonometry
- Chapter 6.1: Angles and Their Measure
- Chapter 6.2: Right Triangle Trigonometry
- Chapter 6.3: Trigonometric Functions of Any Angle
- Chapter 6.4: Graphs of Sine and Cosine Functions
- Chapter 6.5: Graphs of Trigonometric Functions
- Chapter 6.6: Inverse Trigonometric Functions
- Chapter 6.7: Applications and Models
- Chapter 7: Analytic Trigonometry
- Chapter 7.1: Using Fundamental Identities
- Chapter 7.2: Verifying Trigonometric Identities
- Chapter 7.3: Solving Trigonometric Equations
- Chapter 7.4: Sum and Difference Formulas
- Chapter 7.5: Multiple-Angle and Product-to-Sum Formulas
- Chapter 8: Additional Topics in Trigonometry
- Chapter 8.1: Law of Sines
- Chapter 8.2: Law of Cosines
- Chapter 8.3: Vectors in The Plane
- Chapter 8.4: Vectors and Dot Products
- Chapter 8.5: Trigonometric Form of A Complex Number
- Chapter 9: Systems of Equations and Inequlities
- Chapter 9.1: Linear and Nonlinear Systems of Equations
- Chapter 9.2: Two-Variable Linear Systems
- Chapter 9.3: Multivariable Linear Systems
- Chapter 9.4: Partial Fractions
- Chapter 9.5: Systems of Inequalities
- Chapter 9.6: Linear Programming
- Chapter A: Appendix A Errors and the Algebra of Calculus
- 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
Algebra and Trigonometry 8th Edition - Solutions by Chapter
Full solutions for Algebra and Trigonometry | 8th Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
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.
peA) = det(A - AI) has peA) = zero matrix.
Remove row i and column j; multiply the determinant by (-I)i + j •
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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].
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or firstname.lastname@example.org
Forgot password? Reset it here