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# Solutions for Chapter 9.3: Exponential Functions

## Full solutions for Intermediate Algebra | 6th Edition

ISBN: 9780321785046

Solutions for Chapter 9.3: Exponential Functions

Solutions for Chapter 9.3
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##### ISBN: 9780321785046

Since 80 problems in chapter 9.3: Exponential Functions have been answered, more than 67314 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Chapter 9.3: Exponential Functions includes 80 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Cyclic shift

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• Fast Fourier Transform (FFT).

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

• 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!).

• 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.

• 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.

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

• 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.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Rank r (A)

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

• Row picture of Ax = b.

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

• Solvable system Ax = b.

The right side b is in the column space of A.

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

• Wavelets Wjk(t).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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