 Chapter 0: Out of Chaos
 Chapter 0.1: The Same yet Smaller
 Chapter 0.2: More and More
 Chapter 0.3: Shorter yet Longer
 Chapter 0.4: Going Somewhere?
 Chapter 0.5: Out of Chaos
 Chapter 1: Data Exploration
 Chapter 1.1: Bar Graphs and Dot Plots
 Chapter 1.2: Summarizing Data with Measures of Center
 Chapter 1.3: FiveNumber Summaries and Box Plots
 Chapter 1.4: Histograms and StemandLeaf Plots
 Chapter 1.6: TwoVariable Data
 Chapter 1.7: Estimating
 Chapter 1.8: Using Matrices to Organize and Combine Data
 Chapter 10: Probability
 Chapter 10.1: Relative Frequency Graphs
 Chapter 10.2: Probability Outcomes and Trials
 Chapter 10.3: Random Outcomes
 Chapter 10.4: Counting Techniques
 Chapter 10.5: MultipleStage Experiments
 Chapter 10.6: Expected Value
 Chapter 11: Introduction to Geometry
 Chapter 11.1: Parallel and Perpendicular
 Chapter 11.2: Finding the Midpoint
 Chapter 11.3: Squares, Right Triangles, and Areas
 Chapter 11.4: The Pythagorean Theorem
 Chapter 11.5: Operations with Roots
 Chapter 11.6: A Distance Formula
 Chapter 11.7: Similar Triangles and Trigonometric Functions
 Chapter 11.8: Trigonometry
 Chapter 2: Proportional Reasoning and Variation
 Chapter 2.1: Proportions
 Chapter 2.3: Proportions and Measurement Systems
 Chapter 2.4: Direct Variation
 Chapter 2.5: Inverse Variation
 Chapter 2.7: Evaluating Expressions
 Chapter 2.8: Undoing Operations
 Chapter 3: Linear Explorations
 Chapter 3.1: Recursive Sequences
 Chapter 3.2: Linear Plots
 Chapter 3.3: TimeDistance Relationships
 Chapter 3.4: Linear Equations and the Intercept Form
 Chapter 3.5: Linear Equations and Rate of Change
 Chapter 3.6: Solving Equations Using the Balancing Method
 Chapter 4: Fitting a Line to Data
 Chapter 4.1: A Formula for Slope
 Chapter 4.2: Writing a Linear Equation to Fit Data
 Chapter 4.3: PointSlope Form of a Linear Equation
 Chapter 4.4: Equivalent Algebraic Equations
 Chapter 4.5: Writing PointSlope Equations to Fit Data
 Chapter 4.6: More on Modeling
 Chapter 4.7: Applications of Modeling
 Chapter 5: Systems of Equations and Inequalities
 Chapter 5.1: Solving Systems of Equations
 Chapter 5.2: Solving Systems of Equations Using Substitution
 Chapter 5.3: Solving Systems of Equations Using Elimination
 Chapter 5.4: Solving Systems of Equations Using Matrices
 Chapter 5.5: Inequalities in One Variable
 Chapter 5.6: Graphing Inequalities in Two Variables
 Chapter 5.7: Systems of Inequalities
 Chapter 6: Exponents and Exponential Models
 Chapter 6.1: Recursive Routines
 Chapter 6.2: Exponential Equations
 Chapter 6.3: Multiplication and Exponents
 Chapter 6.4: Scientific Notation for Large Numbers
 Chapter 6.5: Looking Back with Exponents
 Chapter 6.6: Zero and Negative Exponents
 Chapter 6.7: Fitting Exponential Models to Data
 Chapter 7: Functions
 Chapter 7.1: Secret Codes
 Chapter 7.2: Functions and Graphs
 Chapter 7.3: Graphs of RealWorld Situations
 Chapter 7.4: Function Notation
 Chapter 7.5: Defining the AbsoluteValue Function
 Chapter 7.6: Squares, Squaring, and Parabolas
 Chapter 8: Transformations
 Chapter 8.1: Translating Points
 Chapter 8.2: Translating Graphs
 Chapter 8.3: Reflecting Points and Graphs
 Chapter 8.4: Stretching and Shrinking Graphs
 Chapter 8.6: Introduction to Rational Functions
 Chapter 8.7: Transformations with Matrices
 Chapter 9: Quadratic Models
 Chapter 9.1: Solving Quadratic Equations
 Chapter 9.2: Finding the Roots and the Vertex
 Chapter 9.3: From Vertex to General Form
 Chapter 9.4: Factored Form
 Chapter 9.6: Completing the Square
 Chapter 9.7: The Quadratic Formula
 Chapter 9.8: Cubic Functions
Discovering Algebra: An Investigative Approach 2nd Edition  Solutions by Chapter
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Discovering Algebra: An Investigative Approach  2nd Edition  Solutions by Chapter
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Companion matrix.
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 nl  An).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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 •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).