- Chapter 1: Equations and Inequalities
- Chapter 10: Polar Coordinates; Vectors
- Chapter 11: Analytic Geometry
- Chapter 12: Systems of Equations and Inequalities
- Chapter 13: Sequences, Induction; and Binomial Theorem
- Chapter 14: Counting and Probability
- Chapter 2: Graphs
- Chapter 3: Functions and their Graphs
- Chapter 4: Linear and Quadratic Functions
- Chapter 5: Polynomial and Rational Functions
- Chapter 6: Exponential and Logarithmic Functions
- Chapter 7: Trigonometric Functions
- Chapter 8: Analytic Trigonometry
- Chapter 9: Applications of Trigonometric Functions
- Chapter R: Review
Algebra and Trigonometry 8th Edition - Solutions by Chapter
Full solutions for Algebra and Trigonometry | 8th Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Column space C (A) =
space of all combinations of the columns of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Every v in V is orthogonal to every w in W.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Constant down each diagonal = time-invariant (shift-invariant) filter.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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