- Chapter 1.1: Algebraic Expressions, real Numbers, and Interval Novation
- Chapter 1.2: Operations with Real Numbers and Simplifying Algebraic Expressions
- Chapter 1.3: Graphing Equations
- Chapter 1.4: Solving Linear Equations
- Chapter 1.5: Problem Solving and Using Formulas
- Chapter 1.6: Properties of Integral Exponents
- Chapter 1.7: Scientific Notation
- Chapter 10.1: Distance and Midpoint Formulas; Circles
- Chapter 10.2: The Ellipse
- Chapter 10.3: The Hyperbola
- Chapter 10.4: The Parabola; Identifying Conic Sections
- Chapter 10.5: Systems of Nonlinear Equations in Two Variables
- Chapter 11.1: Sequences and Summation Notation
- Chapter 11.2: Arithmetic Sequences
- Chapter 11.3: Geometric Sequences and Series
- Chapter 11.4: The Binomial Theorem
- Chapter 2.1: Introduction to Functions
- Chapter 2.2: Graphs of Functions
- Chapter 2.3: The Algebra of Functions
- Chapter 2.4: Linear Functions and Slope
- Chapter 2.5: The Point Slope-Form of the Equation of a Line
- Chapter 3.1: Systems of Linear Equations in Two Variables
- Chapter 3.2: Problem Solving and Business Applications Using Systems of Equations
- Chapter 3.3: Systems of Linear Equations in Three Variables
- Chapter 3.4: Matrix Solutions of Linear Systems
- Chapter 3.5: Determinants and Cramers Rule
- Chapter 4.1: Solving Linear Inequalities
- Chapter 4.2: Compound Inequalities
- Chapter 4.3: Equations and Inequalities Involving Absolute Value
- Chapter 4.4: Linear Inequalities in Two Variables
- Chapter 4.5: Linear Programming
- Chapter 5.1: Introduction to Polynomials and Polynomial Functions
- Chapter 5.2: Multiplication of Polynomials
- Chapter 5.3: Greatest Common Factors and Factoring by Grouping
- Chapter 5.4: Factoring Trinomials
- Chapter 5.5: Factoring Special Forms
- Chapter 5.6: A General Factoring Strategy
- Chapter 5.7: Polynomial Equations and Their Applications
- Chapter 6.1: Rational Expressions and Functions: Multiplying and Dividing
- Chapter 6.2: Adding and Subtracting Rational Expressions
- Chapter 6.3: Complex Rational Expressions
- Chapter 6.4: Division of Polynomials
- Chapter 6.5: Synthetic Division and the Remainder Theorem
- Chapter 6.6: Rational Equations
- Chapter 6.7: Formulas and Applications of Rational Equations
- Chapter 6.8: Modeling Using Variation
- Chapter 7.1: Radical Expressions and Functions
- Chapter 7.2: Rational Exponents
- Chapter 7.3: Multiplying and Simplifying Radical Expressions
- Chapter 7.4: Adding, Subtracting, and Dividing Radical Expressions
- Chapter 7.5: Multiplying with More Than One Term and Rationalizing Denominators
- Chapter 7.6: Radical Equations
- Chapter 8.1: The Square Root Property and Completing the Square
- Chapter 8.2: The Quadratic Formula
- Chapter 8.3: Quadratic Functions and Their Graphs
- Chapter 8.4: Equations Quadratic in Form
- Chapter 8.5: Polynomial and Rational Inequalities
- Chapter 9.1: Exponential Functions
- Chapter 9.2: Composite and Inverse Functions
- Chapter 9.3: Logarithmic Functions
- Chapter 9.4: Properties of Logarithms
- Chapter 9.5: Exponential and Logarithmic Equations
- Chapter 9.6: Exponential Growth and Decay; Modeling Data
- Chapter Chapter 1: Algebra, Mathematical Models, and Problem Solving
- Chapter Chapter 10: Conic Sections and Systems of Nonlinear Equations
- Chapter Chapter 11: Sequences, Series, and the Binomial Theorem
- Chapter Chapter 2: Functions and Linear Equations
- Chapter Chapter 3: Systems of Linear Equations
- Chapter Chapter 4: Inequalities and Problem Solving
- Chapter Chapter 5: Polynomials, Polynomial Functions, and Factoring
- Chapter Chapter 6: Rational Expressions, Functions, and Equations
- Chapter Chapter 7: Radicals, Radical Functions, and Rational Exponents
- Chapter Chapter 8: Quadratic Equations and Functions
- Chapter Chapter 9: Exponential and Logarithmic Functions
Intermediate Algebra for College Students 6th Edition - Solutions by Chapter
Full solutions for Intermediate Algebra for College Students | 6th Edition
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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.
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).
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Solvable system Ax = b.
The right side b is in the column space of A.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Constant down each diagonal = time-invariant (shift-invariant) filter.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.