- 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
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.
Column space C (A) =
space of all combinations of the columns of A.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
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.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Every v in V is orthogonal to every w in W.
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 •
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.