- 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
Upper triangular systems are solved in reverse order Xn to Xl.
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.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Column space C (A) =
space of all combinations of the columns of A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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).
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
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].
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
Solvable system Ax = b.
The right side b is in the column space of A.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).