- Chapter 1: The Real Number System
- Chapter 1-3: Linear Equations and Inequalities in Two Variables; Functions
- Chapter 1.1: Fractions
- Chapter 1.2: Exponents, Order of Operations, and Inequality
- Chapter 1.3: Variables, Expressions, and Equations
- Chapter 1.4: Real Numbers and the Number Line
- Chapter 1.5: Adding and Subtracting Real Numbers
- Chapter 1.6: Multiplying and Dividing Real Numbers
- Chapter 1.7: Properties of Real Numbers
- Chapter 1.8: Simplifying Expressions
- Chapter 2: Linear Equations and Inequalities in One Variable
- Chapter 2.1: The Addition Property of Equality
- Chapter 2.2: The Multiplication Property of Equality
- Chapter 2.3: More on Solving Linear Equations
- Chapter 2.4: An Introduction to Applications of Linear Equations
- Chapter 2.5: Formulas and Additional Applications from Geometry
- Chapter 2.6: Ratio, Proportion, and Percent
- Chapter 2.7: Further Applications of Linear Equations
- Chapter 2.8: Solving Linear Inequalities
- Chapter 3: Linear Equations and Inequalities in Two Variables; Functions
- Chapter 3.1: Linear Equations in Two Variables; The Rectangular Coordinate System
- Chapter 3.2: Graphing Linear Equations in Two Variables
- Chapter 3.3: The Slope of a Line
- Chapter 3.4: Writing and Graphing Equations of Lines
- Chapter 3.5: Graphing Linear Inequalities in Two Variables
- Chapter 3.6: Introduction to Functions
- Chapter 4: Systems of Linear Equations and Inequalities
- Chapter 4.1: Solving Systems of Linear Equations by Graphing
- Chapter 4.2: Solving Systems of Linear Equations by Substitution
- Chapter 4.3: Solving Systems of Linear Equations by Elimination
- Chapter 4.4: Applications of Linear Systems
- Chapter 4.5: Solving Systems of Linear Inequalities
- Chapter 5: Exponents and Polynomials
- Chapter 5.1: The Product Rule and Power Rules for Exponents
- Chapter 5.2: Integer Exponents and the Quotient Rule
- Chapter 5.3: An Application of Exponents: Scientific Notation
- Chapter 5.4: Adding and Subtracting Polynomials; Graphing Simple Polynomials
- Chapter 5.5: Multiplying Polynomials
- Chapter 5.6: Special Products
- Chapter 5.7: Dividing Polynomials
- Chapter 6: Factoring and Applications
- Chapter 6.1: The Greatest Common Factor; Factoring by Grouping
- Chapter 6.2: Factoring Trinomials
- Chapter 6.3: More on Factoring Trinomials
- Chapter 6.4: Special Factoring Techniques
- Chapter 6.5: Solving Quadratic Equations by Factoring
- Chapter 6.6: Applications of Quadratic Equations
- Chapter 7: Rational Expressions and Applications
- Chapter 7.1: The Fundamental Property of Rational Expressions
- Chapter 7.2: Multiplying and Dividing Rational Expressions
- Chapter 7.3: Least Common Denominators
- Chapter 7.4: Adding and Subtracting Rational Expressions
- Chapter 7.5: Complex Fractions
- Chapter 7.6: Solving Equations with Rational Expressions
- Chapter 7.7: Applications of Rational Expressions
- Chapter 7.8: Variation
- Chapter 8: Roots and Radicals
- Chapter 8.1: Evaluating Roots
- Chapter 8.2: Multiplying, Dividing, and Simplifying Radicals
- Chapter 8.3: Adding and Subtracting Radicals
- Chapter 8.4: Rationalizing the Denominator
- Chapter 8.5: More Simplifying and Operations with Radicals
- Chapter 8.6: Solving Equations with Radicals
- Chapter 8.7: Using Rational Numbers as Exponents
- Chapter 9: Quadratic Equations
- Chapter 9.1: Solving Quadratic Equations by the Square Root Property
- Chapter 9.2: Solving Quadratic Equations by Completing the Square
- Chapter 9.3: Solving Quadratic Equations by the Quadratic Formula
- Chapter 9.4: Complex Numbers
- Chapter 9.5: More on Graphing Quadratic Equations; Quadratic Functions
Beginning Algebra 11th Edition - Solutions by Chapter
Full solutions for Beginning Algebra | 11th Edition
Tv = Av + Vo = linear transformation plus shift.
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.
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).
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
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
= Xl (column 1) + ... + xn(column n) = combination of columns.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Outer product uv T
= column times row = rank one matrix.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Constant down each diagonal = time-invariant (shift-invariant) filter.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.