- Chapter 1.1: Algebraic Expressions, real Numbers, and Interval Novation
- Chapter 1.2: Operations with Real Numbers and Simplifying Algebraic Expressions
- Chapter 1.3: Graphing Equations
- Chapter 1.4: Solving Linear Equations
- Chapter 1.5: Problem Solving and Using Formulas
- Chapter 1.6: Properties of Integral Exponents
- Chapter 1.7: Scientific Notation
- Chapter 10.1: Distance and Midpoint Formulas; Circles
- Chapter 10.2: The Ellipse
- Chapter 10.3: The Hyperbola
- Chapter 10.4: The Parabola; Identifying Conic Sections
- Chapter 10.5: Systems of Nonlinear Equations in Two Variables
- Chapter 11.1: Sequences and Summation Notation
- Chapter 11.2: Arithmetic Sequences
- Chapter 11.3: Geometric Sequences and Series
- Chapter 11.4: The Binomial Theorem
- Chapter 2.1: Introduction to Functions
- Chapter 2.2: Graphs of Functions
- Chapter 2.3: The Algebra of Functions
- Chapter 2.4: Linear Functions and Slope
- Chapter 2.5: The Point Slope-Form of the Equation of a Line
- Chapter 3.1: Systems of Linear Equations in Two Variables
- Chapter 3.2: Problem Solving and Business Applications Using Systems of Equations
- Chapter 3.3: Systems of Linear Equations in Three Variables
- Chapter 3.4: Matrix Solutions of Linear Systems
- Chapter 3.5: Determinants and Cramers Rule
- Chapter 4.1: Solving Linear Inequalities
- Chapter 4.2: Compound Inequalities
- Chapter 4.3: Equations and Inequalities Involving Absolute Value
- Chapter 4.4: Linear Inequalities in Two Variables
- Chapter 4.5: Linear Programming
- Chapter 5.1: Introduction to Polynomials and Polynomial Functions
- Chapter 5.2: Multiplication of Polynomials
- Chapter 5.3: Greatest Common Factors and Factoring by Grouping
- Chapter 5.4: Factoring Trinomials
- Chapter 5.5: Factoring Special Forms
- Chapter 5.6: A General Factoring Strategy
- Chapter 5.7: Polynomial Equations and Their Applications
- Chapter 6.1: Rational Expressions and Functions: Multiplying and Dividing
- Chapter 6.2: Adding and Subtracting Rational Expressions
- Chapter 6.3: Complex Rational Expressions
- Chapter 6.4: Division of Polynomials
- Chapter 6.5: Synthetic Division and the Remainder Theorem
- Chapter 6.6: Rational Equations
- Chapter 6.7: Formulas and Applications of Rational Equations
- Chapter 6.8: Modeling Using Variation
- Chapter 7.1: Radical Expressions and Functions
- Chapter 7.2: Rational Exponents
- Chapter 7.3: Multiplying and Simplifying Radical Expressions
- Chapter 7.4: Adding, Subtracting, and Dividing Radical Expressions
- Chapter 7.5: Multiplying with More Than One Term and Rationalizing Denominators
- Chapter 7.6: Radical Equations
- Chapter 8.1: The Square Root Property and Completing the Square
- Chapter 8.2: The Quadratic Formula
- Chapter 8.3: Quadratic Functions and Their Graphs
- Chapter 8.4: Equations Quadratic in Form
- Chapter 8.5: Polynomial and Rational Inequalities
- Chapter 9.1: Exponential Functions
- Chapter 9.2: Composite and Inverse Functions
- Chapter 9.3: Logarithmic Functions
- Chapter 9.4: Properties of Logarithms
- Chapter 9.5: Exponential and Logarithmic Equations
- Chapter 9.6: Exponential Growth and Decay; Modeling Data
- Chapter Chapter 1: Algebra, Mathematical Models, and Problem Solving
- Chapter Chapter 10: Conic Sections and Systems of Nonlinear Equations
- Chapter Chapter 11: Sequences, Series, and the Binomial Theorem
- Chapter Chapter 2: Functions and Linear Equations
- Chapter Chapter 3: Systems of Linear Equations
- Chapter Chapter 4: Inequalities and Problem Solving
- Chapter Chapter 5: Polynomials, Polynomial Functions, and Factoring
- Chapter Chapter 6: Rational Expressions, Functions, and Equations
- Chapter Chapter 7: Radicals, Radical Functions, and Rational Exponents
- Chapter Chapter 8: Quadratic Equations and Functions
- Chapter Chapter 9: Exponential and Logarithmic Functions
Intermediate Algebra for College Students 6th Edition - Solutions by Chapter
Full solutions for Intermediate Algebra for College Students | 6th Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
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.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Column space C (A) =
space of all combinations of the columns of A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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].
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
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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.