- Chapter 1: Equations and Inequalities
- 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: Conic Sections
- 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: Sequences and Series
- 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: Sampling and Error
- Chapter 12.1: The Counting Principle
- 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 Distribution
- Chapter 12.9: Binomial Experiments
- Chapter 13: Trigonometric Functions
- 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: Trigonometric Graphs and Identities
- 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: Linear Relations and Functions
- 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: Solving Equations and Inequalities
- Chapter 3.1: Solving Systems of Equations by Graphing
- Chapter 3.2: Solving Systems of Equations Algebraically
- Chapter 3.3: Systems of Equations and Inequalities
- Chapter 3.4: Linear Programming
- Chapter 3.5: Solving Systems of Equations in Three Variables
- Chapter 4: Matrices
- 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: Quadratic Functions and Inequalities
- 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: Graphing Calculator Lab: The Family of Parabolas
- Chapter 5.8: Graphing and Solving Quadratic Inequalities
- Chapter 6: Polynomial Functions
- 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: Radical Equations and Inequalities
- Chapter 7.1: Operations on Functions
- Chapter 7.2: Inverse Functions and Relations
- Chapter 7.3: Square Root Functions and Inequalities
- Chapter 7.4: nth Roots
- Chapter 7.5: nth Roots
- Chapter 7.6: Fractional Exponents
- Chapter 7.7: Solving Radical Equations and Inequalities
- Chapter 8: Rational Expressions and Equations
- 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: Graphing Calculator Lab
- Chapter 9: Graphing Calculator Lab: Cooling
- 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
California Algebra 2: Concepts, Skills, and Problem Solving 1st Edition - Solutions by Chapter
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving | 1st Edition
California Algebra 2: Concepts, Skills, and Problem Solving | 1st Edition - Solutions by ChapterGet Full Solutions
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.)
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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.
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.
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.
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.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.