- 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 12.2: 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 13.1: 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: Systems of Equations and Inequalities
- 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: Graphing Calculator Lab: The Family of Parabolas
- 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: nth Roots
- Chapter 7.5: nth Roots
- Chapter 7.6: Fractional 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: Graphing Calculator Lab
- 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: Sampling and Error
- Chapter Chapter 13: Trigonometric Functions
- Chapter Chapter 14: Trigonometric Graphs and Identities
- Chapter Chapter 2: Linear Relations and Functions
- Chapter Chapter 3: Solving 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: Graphing Calculator Lab: Cooling
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
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).
peA) = det(A - AI) has peA) = zero matrix.
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.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (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.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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