- 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
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
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.)
A = CTC = (L.J]))(L.J]))T for positive definite A.
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).
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.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
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).
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.
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.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.