×
Get Full Access to Math - Textbook Survival Guide
Get Full Access to Math - Textbook Survival Guide
×

# Solutions for Chapter 5.6: Exponential and Logarithmic Equations

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

ISBN: 9780495559719

Solutions for Chapter 5.6: Exponential and Logarithmic Equations

Solutions for Chapter 5.6
4 5 0 325 Reviews
13
1
##### ISBN: 9780495559719

This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. Since 70 problems in chapter 5.6: Exponential and Logarithmic Equations have been answered, more than 34901 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.6: Exponential and Logarithmic Equations includes 70 full step-by-step solutions. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719.

Key Math Terms and definitions covered in this textbook
• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Hankel matrix H.

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

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Multiplication Ax

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

• 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.

• Rotation matrix

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Subspace S of V.

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

• Transpose matrix AT.

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

×