- 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
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 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.
Column space C (A) =
space of all combinations of the columns of A.
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].
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
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.
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.
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.
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).
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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.
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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