- 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: Two-Step 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 Elapsed-Time 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: Fraction-Decimal-Percent Equivalents
Saxon Math, Course 1 1st Edition - Solutions by Chapter
Full solutions for Saxon Math, Course 1 | 1st Edition
Tv = Av + Vo = linear transformation plus shift.
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.
cond(A) = c(A) = IIAIlIIA-III = 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.
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. Block-diagonal: 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, off-diagonal entries = 0.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
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.
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).
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 ().
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I 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)j-I and det V = product of (Xk - Xi) for k > i.