- 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
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Column space C (A) =
space of all combinations of the columns of A.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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
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.
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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 .
A directed graph that has constants Cl, ... , Cm associated with the edges.
Every v in V is orthogonal to every w in W.
Outer product uv T
= column times row = rank one matrix.
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 ().
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
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