- 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
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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 .
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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.
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.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
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 ().
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as 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.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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