Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. This expansive textbook survival guide covers the following chapters and their solutions. Since 76 problems in chapter 5: INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS have been answered, more than 176846 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. Chapter 5: INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS includes 76 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook

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!
Cayley-Hamilton Theorem.
peA) = det(A - AI) has peA) = zero matrix.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.
Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Solvable system Ax = b.
The right side b is in the column space of A.

Solutions for Chapter 5: INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS

Textbook Survival Guides