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Solutions for Chapter 11: Sequences, Series, and the Binomial Theorem

Full solutions for Intermediate Algebra | 6th Edition

ISBN: 9780321785046

Solutions for Chapter 11: Sequences, Series, and the Binomial Theorem

Solutions for Chapter 11
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ISBN: 9780321785046

This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 11: Sequences, Series, and the Binomial Theorem includes 148 full step-by-step solutions. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. Since 148 problems in chapter 11: Sequences, Series, and the Binomial Theorem have been answered, more than 59595 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• Covariance matrix:E.

When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

• Dimension of vector space

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

• Ellipse (or ellipsoid) x T Ax = 1.

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• 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.

• 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.

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• 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.

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• Multiplier eij.

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

• Nilpotent matrix N.

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

• Orthogonal matrix Q.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Solvable system Ax = b.

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

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

Spectral radius = max of IAi I.

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B II·

• Vector v in Rn.

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

• Wavelets Wjk(t).

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

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