- Chapter 1: Real Numbers and Algebraic Expressions
- Chapter 1.1: Tips for Success in Mathematics
- Chapter 1.2: Algebraic Expressions and Sets of Numbers
- Chapter 1.3: Operations on Real Numbers and Order of Operations
- Chapter 1.4: Properties of Real Numbers and Algebraic Expressions
- Chapter 10: Conic Sections
- Chapter 10.1: The Parabola and the Circle
- Chapter 10.2: The Ellipse and the Hyperbola
- Chapter 10.3: Solving Nonlinear Systems of Equations
- Chapter 10.4: Nonlinear Inequalities and Systems of Inequalities
- Chapter 11: Sequences, Series, and the Binomial Theorem
- Chapter 11.1: Sequences
- Chapter 11.2: Arithmetic and Geometric Sequences
- Chapter 11.3: Series
- Chapter 11.4: Partial Sums of Arithmetic and Geometric Sequences
- Chapter 11.5: The Binomial Theorem
- Chapter 2: EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING
- Chapter 2.1: Linear Equations in One Variable
- Chapter 2.1 - 2.4: Linear Inequalities and Problem Solving
- Chapter 2.2: An Introduction to Problem Solving
- Chapter 2.3: Formulas and Problem Solving
- Chapter 2.4: Linear Inequalities and Problem Solving
- Chapter 2.5: Compound Inequalities
- Chapter 2.6: Absolute Value Equations
- Chapter 2.7: Absolute Value Inequalities
- Chapter 3: GRAPHS AND FUNCTIONS
- Chapter 3.1: Graphing Equations
- Chapter 3.1-3.5: Linear Equations in Two Variables
- Chapter 3.2: Introduction to Functions
- Chapter 3.3: Graphing Linear Functions
- Chapter 3.4: The Slope of a Line
- Chapter 3.5: Equations of Lines
- Chapter 3.6: Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions
- Chapter 3.7: Graphing Linear Inequalities
- Chapter 4: Systems of Equations
- Chapter 4.1: Solving Systems of Linear Equations in Two Variables
- Chapter 4.14.3: Systems of Linear Equations
- Chapter 4.2: Solving Systems of Linear Equations in Three Variables
- Chapter 4.3: Systems of Linear Equations and Problem Solving
- Chapter 4.4: Solving Systems of Equations by Matrices
- Chapter 4.5: Systems of Linear Inequalities
- Chapter 5: Exponents, Polynomials, and Polynomial Functions
- Chapter 5.1: Exponents and Scientific Notation
- Chapter 5.1-5-5.7: OPERATIONS ON POLYNOMIALS AND FACTORING STRATEGIES
- Chapter 5.2: More Work with Exponents and Scientific Notation
- Chapter 5.3: Polynomials and Polynomial Functions
- Chapter 5.4: Multiplying Polynomials
- Chapter 5.5: The Greatest Common Factor and Factoring by Grouping
- Chapter 5.6: Factoring Trinomials
- Chapter 5.7: Factoring by Special Products
- Chapter 5.8: Solving Equations by Factoring and Problem Solving
- Chapter 6: Rational Expressions
- Chapter 6.1: Rational Functions and Multiplying and Dividing Rational Expressions
- Chapter 6.2: Adding and Subtracting Rational Expressions
- Chapter 6.3: Simplifying Complex Fractions
- Chapter 6.4: Dividing Polynomials: Long Division and Synthetic Division
- Chapter 6.5: Solving Equations Containing Rational Expressions
- Chapter 6.6: Rational Equations and Problem Solving
- Chapter 6.7: Variation and Problem Solving
- Chapter 7: Rational Exponents, Radicals, and Complex Numbers
- Chapter 7.1: Radicals and Radical Functions
- Chapter 7.1-7.5: Radical Equations and Problem Solving
- Chapter 7.2: Rational Exponents
- Chapter 7.3: Simplifying Radical Expressions
- Chapter 7.4: Adding, Subtracting, and Multiplying Radical Expressions
- Chapter 7.5: Rationalizing Denominators and Numerators of Radical Expressions
- Chapter 7.6: Radical Equations and Problem Solving
- Chapter 7.7: Complex Numbers
- Chapter 8: Quadratic Equations and Functions
- Chapter 8.1: Solving Quadratic Equations by Completing the Square
- Chapter 8.1-8.3: SUMMARY ON SOLVING QUADRATIC EQUATIONS
- Chapter 8.2: Solving Quadratic Equations by the Quadratic Formula
- Chapter 8.3: Solving Equations by Using Quadratic Methods
- Chapter 8.4: Nonlinear Inequalities in One Variable
- Chapter 8.5: Quadratic Functions and Their Graphs
- Chapter 8.6: Further Graphing of Quadratic Functions
- Chapter 9: Exponential and Logarithmic Functions
- Chapter 9.1: The Algebra of Functions; Composite Functions
- Chapter 9.1-9.6: FUNCTIONS AND PROPERTIES OF LOGARITHMS
- Chapter 9.2: Inverse Functions
- Chapter 9.3: Exponential Functions
- Chapter 9.4: Exponential Growth and Decay Functions
- Chapter 9.5: Logarithmic Functions
- Chapter 9.6: Properties of Logarithms
- Chapter 9.7: Common Logarithms, Natural Logarithms, and Change of Base
- Chapter 9.8: Exponential and Logarithmic Equations and Problem Solving
- Chapter Appendix A:
- Chapter Appendix B:
- Chapter Appendix C:
- Chapter Appendix D:
Intermediate Algebra 6th Edition - Solutions by Chapter
Full solutions for Intermediate Algebra | 6th Edition
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!
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
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.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
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).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
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.