- Chapter 1.1: Real Numbers
- Chapter 1.2: Exponents and Radicals
- Chapter 1.3: Algebraic Expressions
- Chapter 1.4: Fractional Expressions
- Chapter 10.1: Inf inite Sequences and Summation Notation
- Chapter 10.2: Arithmetic Sequences
- Chapter 10.3: Geometric Sequences
- Chapter 10.4: Mathematical Induction
- Chapter 10.5: The Binomial Theorem
- Chapter 10.6: Permutations
- Chapter 10.7: Distinguishable Permutations and Combinations
- Chapter 10.8: Probability
- Chapter 11.1: Parabolas
- Chapter 11.2: Parabolas
- Chapter 11.3: Hyperbolas
- Chapter 11.4: Plane Curves and Parametric Equations
- Chapter 11.5: Polar Coordinates
- Chapter 11.6: Polar Equations of Conics
- Chapter 2.1: Equations
- Chapter 2.2: Applied Problems
- Chapter 2.3: Quadratic Equations
- Chapter 2.4: Complex Numbers
- Chapter 2.5: Other Types of Equations
- Chapter 2.6: Inequalities
- Chapter 2.7: More on Inequalities
- Chapter 3.1: Rectangular Coordinate Systems
- Chapter 3.2: Graphs of Equations
- Chapter 3.3: Lines
- Chapter 3.4: Definition of Function
- Chapter 3.5: Graphs of Functions
- Chapter 3.6: Quadratic Functions
- Chapter 3.7: Operations on Functions
- Chapter 4.1: Polynomial Functions of Degree Greater Than 2
- Chapter 4.2: Properties of Division
- Chapter 4.3: Zeros of Polynomials
- Chapter 4.4: Complex and Rational Zeros of Polynomials
- Chapter 4.5: Rational Functions
- Chapter 4.6: Variation
- Chapter 5.1: Inverse Functions
- Chapter 5.2: Exponential Functions
- Chapter 5.3: The Natural Exponential Function
- Chapter 5.4: Logarithmic Functions
- Chapter 5.5: Properties of Logarithms
- Chapter 5.6: Exponential and Logarithmic Equations
- Chapter 6.1: Angles
- Chapter 6.2: Trigonometric Functions of Angles
- Chapter 6.3: Trigonometric Functions of Real Numbers
- Chapter 6.4: Values of the Trigonometric Functions
- Chapter 6.5: Trigonometric Graphs
- Chapter 6.6: Additional Trigonometric Graphs
- Chapter 6.7: Applied Problems
- Chapter 7.1: Verifying Trigonometry Identities
- Chapter 7.2: Tr igonome tr ic Equations
- Chapter 7.3: The Addition and Subtr action For mulas
- Chapter 7.4: Multiple- Angle For mulas
- Chapter 7.5: Product-t o-Sum and Sum-t o-Product For mulas
- Chapter 7.6: The Inver se Tr igonome tr ic Functions
- Chapter 8.1: The Law of Sines
- Chapter 8.2: The Law of Cosines
- Chapter 8.3: Vectors
- Chapter 8.4: The Dot Product
- Chapter 8.5: The Dot Product
- Chapter 8.6: De Moivres Theorem and n th Roots of Complex Numbers
- Chapter 8.7: APPLICATIONS OF TRIGONOMETRY
- Chapter 9.1: Systems of Equations
- Chapter 9.10: Partial Fractions
- Chapter 9.2: Systems of Linear Equations in Two Variables
- Chapter 9.3: Systems of Inequalities
- Chapter 9.4: Linear Programming
- Chapter 9.5: Systems of Linear Equations in More Than Two Variables
- Chapter 9.6: The Algebra of Matrices
- Chapter 9.7: The Inverse of a Matrix
- Chapter 9.8: Determinants
- Chapter 9.9: Properties of Determinants
- Chapter 9.90: Partial Fractions
- Chapter Chapter 1: Express as a simplified rational number:
- Chapter Chapter 10: SEQUENCES, SERIES, AND PROBABILIT Y
- Chapter Chapter 2: In the design of certain small turbo-prop aircraft, the landing speed V (in ) is determined by the formula , where W is the gross weight (in pounds) of the aircraft and S is the surface area (in ) of the wings. If the gross weight of the aircraft is betwe
- Chapter Chapter 3: FUNCTIONS AND GRAPH
- Chapter Chapter 4: POLYNOMIAL AND RATIONAL FUNCTIONS
- Chapter Chapter 5: INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS
- Chapter Chapter 6: THE TRIGONOMETRIC FUNCTIONS
- Chapter Chapter 7: ANALYTIC TRIGONOMETRY
- Chapter Chapter 9: SYSTEMS OF EQUATIONS AND INEQUALITIES
- Chapter Chapter11: TOPICS FROM ANALY TIC GEOMETRY
Algebra and Trigonometry with Analytic Geometry 12th Edition - Solutions by Chapter
Full solutions for Algebra and Trigonometry with Analytic Geometry | 12th Edition
Algebra and Trigonometry with Analytic Geometry | 12th Edition - Solutions by ChapterGet Full Solutions
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).
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
A = CTC = (L.J]))(L.J]))T for positive definite A.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
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.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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.
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