- 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
Upper triangular systems are solved in reverse order Xn to Xl.
peA) = det(A - AI) has peA) = zero matrix.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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 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!).
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
A symmetric matrix with eigenvalues of both signs (+ and - ).
= Xl (column 1) + ... + xn(column n) = combination of columns.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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·
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.