- 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: Five-Number Summaries and Box Plots
- Chapter 1.4: Histograms and Stem-and-Leaf Plots
- Chapter 1.6: Two-Variable 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: Multiple-Stage 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: Time-Distance 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: Point-Slope Form of a Linear Equation
- Chapter 4.4: Equivalent Algebraic Equations
- Chapter 4.5: Writing Point-Slope 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 Real-World 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
Discovering Algebra: An Investigative Approach | 2nd Edition - Solutions by ChapterGet Full Solutions
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.
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.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
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.
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 = Q-1 = Q.
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 ().
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 = U-I.
Orthonormal columns (complex analog of Q).