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

## Full solutions for Algebra and Trigonometry with Analytic Geometry | 12th Edition

ISBN: 9780495559719

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

Solutions for Chapter Chapter 5
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##### ISBN: 9780495559719

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 Chapter 5: INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS have been answered, more than 34954 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 Chapter 5: INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS includes 76 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Column picture of Ax = b.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Dimension of vector space

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

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• 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.

• Fourier matrix F.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

• 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!).

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• Outer product uv T

= column times row = rank one matrix.

• Particular solution x p.

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

• Random matrix rand(n) or randn(n).

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• 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.

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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