- 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
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.