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# Solutions for Chapter 13.1: Sequences; Induction; the Binomial Theorem

## Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition

ISBN: 9780132854351

Solutions for Chapter 13.1: Sequences; Induction; the Binomial Theorem

Solutions for Chapter 13.1
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##### ISBN: 9780132854351

Chapter 13.1: Sequences; Induction; the Binomial Theorem includes 66 full step-by-step solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Since 66 problems in chapter 13.1: Sequences; Induction; the Binomial Theorem have been answered, more than 59555 students have viewed full step-by-step solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• 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].

• 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.

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• 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).

• Hermitian matrix A H = AT = A.

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

• 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.

• 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.

• Nullspace matrix N.

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

• Pivot columns of A.

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

• Pseudoinverse A+ (Moore-Penrose inverse).

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

• 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.

• 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.

• Skew-symmetric matrix K.

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

• 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.

• 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).

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