 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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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!

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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