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# Solutions for Chapter 11.3: Geometric Sequences and Series

## Full solutions for Intermediate Algebra for College Students | 6th Edition

ISBN: 9780321758934

Solutions for Chapter 11.3: Geometric Sequences and Series

Solutions for Chapter 11.3
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##### ISBN: 9780321758934

Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Chapter 11.3: Geometric Sequences and Series includes 124 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 124 problems in chapter 11.3: Geometric Sequences and Series have been answered, more than 45884 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Column space C (A) =

space of all combinations of the columns of A.

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• Diagonal matrix D.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

• 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).

• 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.

• Hermitian matrix A H = AT = A.

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

• 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.

• Jordan form 1 = M- 1 AM.

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• 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.

• 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 one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Simplex method for linear programming.

The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

• Spectral Theorem A = QAQT.

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

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• 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.

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