 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: DoubleAngle and HalfAngle 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
ISBN: 9780078778568
California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition  Solutions by Chapter
Get Full SolutionsCalifornia Algebra 2: Concepts, Skills, and Problem Solving was written by Patricia and is associated to the ISBN: 9780078778568. This expansive textbook survival guide covers the following chapters: 114. Since problems from 114 chapters in California Algebra 2: Concepts, Skills, and Problem Solving have been answered, more than 8071 students have viewed full stepbystep answer. The full stepbystep solution to problem in California Algebra 2: Concepts, Skills, and Problem Solving were answered by Patricia, our top Math solution expert on 03/09/18, 06:45PM. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.)

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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 = Q1 = Q.

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.

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.
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