 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 SlopeForm 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
ISBN: 9780321758934
Intermediate Algebra for College Students  6th Edition  Solutions by Chapter
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GramSchmidt 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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.