- 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
Upper triangular systems are solved in reverse order Xn to Xl.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
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 n-l - An).
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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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).
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.
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.
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).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.