- 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
peA) = det(A - AI) has peA) = zero matrix.
Column space C (A) =
space of all combinations of the columns of A.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
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.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.