 Chapter 1: Functions and Their Graphs
 Chapter 1.1: Functions and Their Graphs
 Chapter 1.2: Functions and Their Graphs
 Chapter 1.3: Functions and Their Graphs
 Chapter 1.4: Functions and Their Graphs
 Chapter 1.5: Functions and Their Graphs
 Chapter 1.6: Functions and Their Graphs
 Chapter 1.7: Functions and Their Graphs
 Chapter 10: Topics in Analytic Geometry
 Chapter 10.1: Topics in Analytic Geometry
 Chapter 10.2: Topics in Analytic Geometry
 Chapter 10.3: Topics in Analytic Geometry
 Chapter 10.4: Topics in Analytic Geometry
 Chapter 10.5: Topics in Analytic Geometry
 Chapter 10.6: Topics in Analytic Geometry
 Chapter 10.7: Topics in Analytic Geometry
 Chapter 2: Solving Equations and Inequalities
 Chapter 2.1: Solving Equations and Inequalities
 Chapter 2.2: Solving Equations and Inequalities
 Chapter 2.3: Solving Equations and Inequalities
 Chapter 2.4: Solving Equations and Inequalities
 Chapter 2.5: Solving Equations and Inequalities
 Chapter 2.6: Solving Equations and Inequalities
 Chapter 2.7: Solving Equations and Inequalities
 Chapter 3: Polynomial and Rational Functions
 Chapter 34: Exponential and Logarithmic Functions
 Chapter 3.1: Polynomial and Rational Functions
 Chapter 3.2: Polynomial and Rational Functions
 Chapter 3.3: Polynomial and Rational Functions
 Chapter 3.4: Polynomial and Rational Functions
 Chapter 3.5: Polynomial and Rational Functions
 Chapter 3.6: Polynomial and Rational Functions
 Chapter 3.7: Polynomial and Rational Functions
 Chapter 4: Exponential and Logarithmic Functions
 Chapter 4.1: Exponential and Logarithmic Functions
 Chapter 4.2: Exponential and Logarithmic Functions
 Chapter 4.3: Exponential and Logarithmic Functions
 Chapter 4.4: Exponential and Logarithmic Functions
 Chapter 4.5: Exponential and Logarithmic Functions
 Chapter 4.6: Exponential and Logarithmic Functions
 Chapter 5: Trigonometric Functions
 Chapter 57: Additional Topics in Trigonometry
 Chapter 5.1: Trigonometric Functions
 Chapter 5.2: Trigonometric Functions
 Chapter 5.3: Trigonometric Functions
 Chapter 5.4: Trigonometric Functions
 Chapter 5.5: Trigonometric Functions
 Chapter 5.6: Trigonometric Functions
 Chapter 5.7: Trigonometric Functions
 Chapter 6: Analytic Trigonometry
 Chapter 6.1: Analytic Trigonometry
 Chapter 6.2: Analytic Trigonometry
 Chapter 6.3: Analytic Trigonometry
 Chapter 6.4: Analytic Trigonometry
 Chapter 6.5: Analytic Trigonometry
 Chapter 7: Additional Topics in Trigonometry
 Chapter 7.1: Additional Topics in Trigonometry
 Chapter 7.2: Additional Topics in Trigonometry
 Chapter 7.3: Additional Topics in Trigonometry
 Chapter 7.4: Additional Topics in Trigonometry
 Chapter 7.5: Additional Topics in Trigonometry
 Chapter 8: Linear Systems and Matrices
 Chapter 810: Topics in Analytic Geometry
 Chapter 8.1: Linear Systems and Matrices
 Chapter 8.2: Linear Systems and Matrices
 Chapter 8.3: Linear Systems and Matrices
 Chapter 8.4: Linear Systems and Matrices
 Chapter 8.5: Linear Systems and Matrices
 Chapter 8.6: Linear Systems and Matrices
 Chapter 8.7: Linear Systems and Matrices
 Chapter 8.8: Linear Systems and Matrices
 Chapter 9: Sequences, Series, and Probability
 Chapter 9.1: Sequences, Series, and Probability
 Chapter 9.2: Sequences, Series, and Probability
 Chapter 9.3: Sequences, Series, and Probability
 Chapter 9.4: Sequences, Series, and Probability
 Chapter 9.5: Sequences, Series, and Probability
 Chapter 9.6: Sequences, Series, and Probability
 Chapter P: Prerequisites
 Chapter P2: Solving Equations and Inequalities
 Chapter P.1: Prerequisites
 Chapter P.2: Prerequisites
 Chapter P.3: Prerequisites
 Chapter P.4: Prerequisites
 Chapter P.5: Prerequisites
 Chapter P.6: Prerequisites
Algebra and Trigonometry: Real Mathematics, Real People 7th Edition  Solutions by Chapter
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Algebra and Trigonometry: Real Mathematics, Real People  7th Edition  Solutions by Chapter
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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).

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI 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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.