- 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
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).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
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.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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).
Outer product uv T
= column times row = rank one matrix.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
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 ().
Solvable system Ax = b.
The right side b is in the column space of A.