- 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
Remove row i and column j; multiply the determinant by (-I)i + j •
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
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.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A sequence of steps intended to approach the desired solution.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
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.
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).
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).
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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.
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).
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.