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# Solutions for Chapter 11.5: Infinite Geometric Series

## Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving | 1st Edition

ISBN: 9780078778568

Solutions for Chapter 11.5: Infinite Geometric Series

Solutions for Chapter 11.5
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##### ISBN: 9780078778568

This expansive textbook survival guide covers the following chapters and their solutions. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. Since 76 problems in chapter 11.5: Infinite Geometric Series have been answered, more than 44931 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. Chapter 11.5: Infinite Geometric Series includes 76 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Adjacency matrix of a graph.

Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• Column picture of Ax = b.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

• Diagonal matrix D.

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

• 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.

• Distributive Law

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

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Use AT for complex A.

• Fourier matrix F.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

• Hermitian matrix A H = AT = A.

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

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Multiplication Ax

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

• Normal equation AT Ax = ATb.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

• Orthogonal subspaces.

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

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

• Vector v in Rn.

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

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