- Chapter 1: The Language and Tools of Algebra
- Chapter 1-1: Variables and Expressions
- Chapter 1-2: Order of Operations
- Chapter 1-3: Open Sentences
- Chapter 1-4: Identity and Equality Properties
- Chapter 1-5: The Distributive Property
- Chapter 1-6: Commutative and Associative Properties
- Chapter 1-7: Logical Reasoning and Counterexamples
- Chapter 1-8: Number Systems
- Chapter 1-9: Functions and Graphs
- Chapter 10: Radical Expressions and Triangles
- Chapter 10-1: Simplifying Radical Expressions
- Chapter 10-2: Operations with Radical Expressions
- Chapter 10-3: Radical Equations
- Chapter 10-4: The Pythagorean Theorem
- Chapter 10-5: The Distance Formula
- Chapter 10-6: Similar Triangles
- Chapter 11: Rational Expressions and Equations
- Chapter 11-1: Inverse Variation
- Chapter 11-2: Rational Expressions
- Chapter 11-3: Multiplying Rational Expressions
- Chapter 11-4: Dividing Rational Expressions
- Chapter 11-5: Dividing Polynomials
- Chapter 11-6: Rational Expressions with Like Denominators
- Chapter 11-7: Rational Expressions with Unlike Denominators
- Chapter 11-8: Mixed Expressions and Complex Fractions
- Chapter 11-9: Rational Equations and Functions
- Chapter 12: Statistics and Probability
- Chapter 12-1: Sampling and Bias
- Chapter 12-2: Counting Outcomes
- Chapter 12-3: Permutations and Combinations
- Chapter 12-4: Probability of Compound Events
- Chapter 12-5: Probability Distributions
- Chapter 12-6: Probability Simulations
- Chapter 2: Solving Linear Equations
- Chapter 2-1: Writing Equations
- Chapter 2-2: Solving Addition and Subtraction Equations
- Chapter 2-3: Solving Equations by Using Multiplication and Division
- Chapter 2-4: Solving Multi-Step Equations
- Chapter 2-5: Solving Equations with the Variable on Each Side
- Chapter 2-6: Ratios and Proportions
- Chapter 2-7: Percent of Change
- Chapter 2-8: Solving for a Specific Variable
- Chapter 2-9: Weighted Averages
- Chapter 3: Functions and Patterns
- Chapter 3-1: Modeling Relations
- Chapter 3-2: Representing Functions
- Chapter 3-3: Linear Functions
- Chapter 3-4: Arithmetic Sequences
- Chapter 3-5: Proportional and Nonproportional Relationships
- Chapter 4: Analyzing Linear Equations
- Chapter 4-1: Steepness of a Line
- Chapter 4-2: Slope and Direct Variation
- Chapter 4-3: Investigating Slope-Intercept Form
- Chapter 4-4: Writing Equations in Slope-Intercept Form
- Chapter 4-5: Writing Equations in Point-Slope Form
- Chapter 4-6: Statistics: Scatter Plots and Lines of Fit
- Chapter 4-7: Geometry: Parallel and Perpendicular Lines
- Chapter 5: Solving Systems of Linear Equations
- Chapter 5-1: Graphing Systems of Equations
- Chapter 5-2: Substitution
- Chapter 5-3: Elimination Using Addition and Subtraction
- Chapter 5-4: Elimination Using Multiplication
- Chapter 5-5: Applying Systems of Linear Equations
- Chapter 6: Solving Linear Inequalities
- Chapter 6-1: Solving Inequalities by Addition and Subtraction
- Chapter 6-2: Solving Inequalities by Multiplication and Division
- Chapter 6-3: Solving Multi-Step Inequalities
- Chapter 6-4: Solving Compound Inequalities
- Chapter 6-5: Solving Open Sentences Involving Absolute Value
- Chapter 6-6: Solving Inequalities Involving Absolute Value
- Chapter 6-7: Graphing Inequalities in Two Variables
- Chapter 6-8: Graphing Systems of Inequalities
- Chapter 7: Polynomials
- Chapter 7-1: Multiplying Monomials
- Chapter 7-2: Dividing Monomials
- Chapter 7-3: Polynomials
- Chapter 7-4: Adding and Subtracting Polynomials
- Chapter 7-5: Multiplying a Polynomial by a Monomial
- Chapter 7-6: Multiplying Polynomials
- Chapter 7-7: Special Products
- Chapter 8: Factoring
- Chapter 8-1: Monomials and Factoring
- Chapter 8-2: Factoring Using the Distributive Property
- Chapter 8-3: Factoring Trinomials: x 2 + bx + c
- Chapter 8-4: Factoring Trinomials: ax 2 + bx + c
- Chapter 8-5: Factoring Differences of Squares
- Chapter 8-6: Perfect Squares and Factoring
- Chapter 9: Quadratic and Exponential Functions
- Chapter 9-1: Graphing Quadratic Functions
- Chapter 9-2: Solving Quadratic Equations by Graphing
- Chapter 9-3: Solving Quadratic Equations by Completing the Square
- Chapter 9-4: Solving Quadratic Equations by Using the Quadratic Formula
- Chapter 9-5: Exponential Functions
- Chapter 9-6: Growth and Decay
Algebra 1, Student Edition (MERRILL ALGEBRA 1) 1st Edition - Solutions by Chapter
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition
Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition - Solutions by ChapterGet Full Solutions
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Column space C (A) =
space of all combinations of the columns of A.
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.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Outer product uv T
= column times row = rank one matrix.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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.
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