- Chapter 1-1: Expressions and Formulas
- Chapter 1-2: Properties of Real Numbers
- Chapter 1-3: Solving Equations
- Chapter 1-4: Solving Absolute Value Equations
- Chapter 1-5: Solving Inequalities
- Chapter 1-6: Solving Compound and Absolute Value Inequalities
- Chapter 10-1: Midpoint and Distance Formulas
- Chapter 10-2: Parabolas
- Chapter 10-3: Circles
- Chapter 10-4: Ellipses
- Chapter 10-5: Hyperbolas
- Chapter 10-6: Conic Sections
- Chapter 10-7: Solving Quadratic Systems
- Chapter 11-1: Arithmetic Sequences
- Chapter 11-2: Arithmetic Series
- Chapter 11-3: Geometric Sequences
- Chapter 11-4: Geometric Series
- Chapter 11-5: Infinite Geometric Series
- Chapter 11-6: Recursion and Special Sequences
- Chapter 11-7: The Binomial Theorem
- Chapter 11-8: Proof and Mathematical Induction
- Chapter 12-1: The Counting Principle
- Chapter 12-10: Sampling and Error
- Chapter 12-2: Permutations and Combinations
- Chapter 12-3: Probability
- Chapter 12-4: Multiplying Probabilities
- Chapter 12-5: Adding Probabilities
- Chapter 12-6: Statistical Measures
- Chapter 12-7: The Normal Distribution
- Chapter 12-8: Exponential and Binomial Distribu
- Chapter 12-9: Binomial Experiments
- Chapter 13-1: Right Triangle Trigonometry
- Chapter 13-2: Angles and Angle Measure
- Chapter 13-3: Trigonometric Functions of General Angles
- Chapter 13-4: Law of Sines
- Chapter 13-5: Law of Cosines
- Chapter 13-6: Circular Functions
- Chapter 13-7: Inverse Trigonometric Functions
- Chapter 14-1: Graphing Trigonometric Functions
- Chapter 14-2: Translations of Trigonometric Graphs
- Chapter 14-3: Trigonometric Identities
- Chapter 14-4: Verifying Trigonometric Identities
- Chapter 14-5: Sum and Differences of Angles Formulas
- Chapter 14-6: Double-Angle and Half-Angle Formulas
- Chapter 14-7: Solving Trigonometric Equations
- Chapter 2-1: Relations and Functions
- Chapter 2-2: Linear Equations
- Chapter 2-3: Slope
- Chapter 2-4: Writing Linear Equations
- Chapter 2-5: Statistics: Using Scatter Plots
- Chapter 2-6: Special Functions
- Chapter 2-7: Graphing Inequalities
- Chapter 3-1: Solving Systems of Equations by Graphing
- Chapter 3-2: Solving Systems of Equations Algebraically
- Chapter 3-3: Solving Systems of Inequalities by Graphing
- Chapter 3-4: Linear Programming
- Chapter 3-5: Solving Systems of Equations in Three Variables
- Chapter 4-1: Introduction to Matrices
- Chapter 4-2: Operations with Matrices
- Chapter 4-3: Multiplying Matrices
- Chapter 4-4: Transformations with Matrices
- Chapter 4-5: Determinants
- Chapter 4-6: Cramers Rule
- Chapter 4-7: Identity and Inverse Matrices
- Chapter 4-8: Using Matrices to Solve Systems of Equations
- Chapter 5-1: Graphing Quadratic Functions
- Chapter 5-2: Solving Quadratic Equations by Graphing
- Chapter 5-3: Solving Quadratic Equations by Factoring
- Chapter 5-4: Complex Numbers
- Chapter 5-5: Completing the Square
- Chapter 5-6: The Quadratic Formula and the Discriminant
- Chapter 5-7: Analyzing Graphs of Quadratic Functions
- Chapter 5-8: Graphing and Solving Quadratic Inequalities
- Chapter 6-1: Properties of Exponents
- Chapter 6-2: Operations with Polynomials
- Chapter 6-3: Dividing Polynomials
- Chapter 6-4: Polynomial Functions
- Chapter 6-5: Analyzing Graphs of Polynomial Functions
- Chapter 6-6: Solving Polynomial Equations
- Chapter 6-7: The Remainder and Factor Theorems
- Chapter 6-8: Roots and Zeros
- Chapter 6-9: Rational Zero Theorem
- Chapter 7-1: Operations on Functions
- Chapter 7-2: Inverse Functions and Relations
- Chapter 7-3: Square Root Functions and Inequalities
- Chapter 7-4: n th Roots
- Chapter 7-5: Operations with Radical Expressions
- Chapter 7-6: Rational Exponents
- Chapter 7-7: Solving Radical Equations and Inequalities
- Chapter 8-1: Multiplying and Dividing Rational Expressions
- Chapter 8-2: Adding and Subtracting Rational Expressions
- Chapter 8-3: Graphing Rational Functions
- Chapter 8-4: Direct, Joint, and Inverse Variation
- Chapter 8-5: Classes of Functions
- Chapter 8-6: Solving Rational Equations and Inequalities
- Chapter 9-1: Exponential Functions
- Chapter 9-2: Logarithms and Logarithmic Functions
- Chapter 9-3: Properties of Logarithms
- Chapter 9-4: Common Logarithms
- Chapter 9-5: Base e and Natural Logarithms
- Chapter 9-6: Exponential Growth and Decay
- Chapter Chapter 1: Equations and Inequalities
- Chapter Chapter 10: Conic Sections
- Chapter Chapter 11: Sequences and Series
- Chapter Chapter 12: Probability and Statistics
- Chapter Chapter 13: Trigonometric Functions
- Chapter Chapter 14: Trigonometric Graphs and Identities
- Chapter Chapter 2: Linear Relations and Functions
- Chapter Chapter 3: Systems of Equations and Inequalities
- Chapter Chapter 4: Matrices
- Chapter Chapter 5: Quadratic Functions and Inequalities
- Chapter Chapter 6 : Polynomial Functions
- Chapter Chapter 7: Radical Equations and Inequalities
- Chapter Chapter 8: Rational Expressions and Equations
- Chapter Chapter 9: Exponential and Logarithmic Relations
Algebra 2, Student Edition (MERRILL ALGEBRA 2) 1st Edition - Solutions by Chapter
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition
Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition - Solutions by ChapterGet Full Solutions
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.)
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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).
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.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach 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.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
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 ().
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
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.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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.
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