- 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
Remove row i and column j; multiply the determinant by (-I)i + j •
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Invert A by row operations on [A I] to reach [I A-I].
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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.
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).
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
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.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).
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