 Chapter 1: Adding Whole Numbers and Money Subtracting Whole Numbers and Money Fact Families, Part 1
 Chapter 10: Sequences Scales
 Chapter 100: Algebraic Addition of Integers
 Chapter 101: Ratio Problems Involving Totals
 Chapter 102: Mass and Weight
 Chapter 103: Perimeter of Complex Shapes
 Chapter 104: Algebraic Addition Activity
 Chapter 105: Using Proportions to Solve Percent Problems
 Chapter 106: TwoStep Equations
 Chapter 107: Area of Complex Shapes
 Chapter 108: Transformations
 Chapter 109: Corresponding Parts Similar Figures
 Chapter 11: Problems About Comparing Problems About Separating
 Chapter 110: Symmetry
 Chapter 111: Applications Using Division
 Chapter 112: Multiplying and Dividing Integers
 Chapter 113: Adding and Subtracting Mixed Measures Multiplying by Powers of Ten
 Chapter 114: Unit Multipliers
 Chapter 115: Writing Percents as Fractions, Part 2
 Chapter 116: Compound Interest
 Chapter 117: Finding a Whole When a Fraction is Known
 Chapter 118: Estimating Area
 Chapter 119: Finding a Whole When a Percent is Known
 Chapter 12: Place Value Through Trillions Multistep Problems
 Chapter 120: Volume of a Cylinder
 Chapter 13: Problems About Comparing ElapsedTime Problems
 Chapter 14: The Number Line: Negative Numbers
 Chapter 15: Problems About Equal Groups
 Chapter 16: Rounding Whole Numbers Estimating
 Chapter 17: The Number Line: Fractions and Mixed Numbers
 Chapter 18: Average Line Graphs
 Chapter 19: Factors Prime Numbers
 Chapter 2: Multiplying Whole Numbers and Money Dividing Whole Numbers and Money Fact Families, Part 2
 Chapter 20: Greatest Common Factor (GCF
 Chapter 21: Divisibility
 Chapter 22: Equal Groups Problems with Fractions
 Chapter 23: Ratio Rate
 Chapter 24: Adding and Subtracting Fractions That Have Common Denominators
 Chapter 25: Writing Division Answers as Mixed Numbers Multiples
 Chapter 26: Using Manipulatives to Reduce Fractions Adding and Subtracting Mixed Numbers
 Chapter 27: Measures of a Circle
 Chapter 28: Angles
 Chapter 29: Multiplying Fractions Reducing Fractions by Dividing by Common Factors
 Chapter 3: Unknown Numbers in Addition Unknown Numbers in Subtraction
 Chapter 30: Least Common Multiple (LCM) Reciprocals
 Chapter 31: Areas of Rectangles
 Chapter 32: Expanded Notation More on Elapsed Time
 Chapter 33: Writing Percents as Fractions, Part 1
 Chapter 34: Decimal Place Value
 Chapter 35: Writing Decimal Numbers as Fractions, Part 1 Reading and Writing Decimal Numbers
 Chapter 36: Subtracting Fractions and Mixed Numbers from Whole Numbers
 Chapter 37: Adding and Subtracting Decimal Numbers
 Chapter 38: Adding and Subtracting Decimal Numbers and Whole Numbers Squares and Square Roots
 Chapter 39: Multiplying Decimal Numbers
 Chapter 4: Unknown Numbers in Multiplication Unknown Numbers in Division
 Chapter 40: Using Zero as a Placeholder Circle Graphs
 Chapter 41: Finding a Percent of a Number
 Chapter 42: Renaming Fractions by Multiplying by 1
 Chapter 43: Equivalent Division Problems Finding Unknowns in Fraction and Decimal Problems
 Chapter 44: Simplifying Decimal Numbers Comparing Decimal Numbers
 Chapter 45: Dividing a Decimal Number by a Whole Number
 Chapter 46: Writing Decimal Numbers in Expanded Notation Mentally Multiplying Decimal Numbers by 10 and by 100
 Chapter 47: Circumference Pi (
 Chapter 48: Subtracting Mixed Numbers with Regrouping, Part 1
 Chapter 49: Dividing by a Decimal Number
 Chapter 5: Order of Operations, Part 1
 Chapter 50: Decimal Number Line (Tenths) Dividing by a Fraction
 Chapter 51: Rounding Decimal Numbers
 Chapter 52: Mentally Dividing Decimal Numbers by 10 and by 100
 Chapter 53: Decimals Chart Simplifying Fractions
 Chapter 54: Reducing by Grouping Factors Equal to 1 Dividing Fractions
 Chapter 55: Common Denominators, Part 1
 Chapter 56: Common Denominators, Part 2
 Chapter 57: Adding and Subtracting Fractions: Three Steps
 Chapter 58: Probability and Chance
 Chapter 59: Adding Mixed Numbers
 Chapter 6: Fractional Parts
 Chapter 60: Polygons
 Chapter 61: Adding Three or More Fractions
 Chapter 62: Writing Mixed Numbers as Improper Fractions
 Chapter 63: Subtracting Mixed Numbers with Regrouping, Part 2
 Chapter 64: Classifying Quadrilaterals
 Chapter 65: Prime Factorization Division by Primes Factor Trees
 Chapter 66: Multiplying Mixed Numbers
 Chapter 67: Using Prime Factorization to Reduce Fractions
 Chapter 68: Dividing Mixed Numbers
 Chapter 69: Lengths of Segments Complementary and Supplementary Angles
 Chapter 7: Lines, Segments, and Rays Linear Measure
 Chapter 70: Reducing Fractions Before Multiplying
 Chapter 71: Parallelograms
 Chapter 72: Fractions Chart Multiplying Three Fractions
 Chapter 73: Exponents Writing Decimal Numbers as Fractions, Part 2
 Chapter 74: Writing Fractions as Decimal Numbers Writing Ratios as Decimal Number
 Chapter 75: Writing Fractions and Decimals as Percents, Part 1
 Chapter 76: Comparing Fractions by Converting to Decimal Form
 Chapter 77: Finding Unstated Information in Fraction Problems
 Chapter 78: Capacity
 Chapter 79: Area of a Triangle
 Chapter 8: Perimeter
 Chapter 80: Using a Constant Factor to Solve Ratio Problems
 Chapter 81: Arithmetic with Units of Measure
 Chapter 82: Volume of a Rectangular Prism
 Chapter 83: Proportions
 Chapter 84: Order of Operations, Part 2
 Chapter 85: Using Cross Products to Solve Proportions
 Chapter 86: Area of a Circle
 Chapter 87: Finding Unknown Factors
 Chapter 88: Using Proportions to Solve Ratio Word Problems
 Chapter 89: Estimating Square Roots
 Chapter 9: The Number Line: Ordering and Comparing
 Chapter 90: Measuring Turns
 Chapter 91: Geometric Formulas
 Chapter 92: Expanded Notation with Exponents Order of Operations with Exponents
 Chapter 93: Classifying Triangles
 Chapter 94: Writing Fractions and Decimals as Percents, Part 2
 Chapter 95: Reducing Rates Before Multiplying
 Chapter 96: Functions Graphing Functions
 Chapter 97: Transversals
 Chapter 98: Sum of the Angle Measures of Triangles and Quadrilaterals
 Chapter 99: FractionDecimalPercent Equivalents
Saxon Math, Course 1 1st Edition  Solutions by Chapter
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Saxon Math, Course 1  1st Edition  Solutions by Chapter
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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 ().

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.