×
×

# Solutions for Chapter 8.2: Arithmetic Sequences

## Full solutions for College Algebra | 7th Edition

ISBN: 9780134469164

Solutions for Chapter 8.2: Arithmetic Sequences

Solutions for Chapter 8.2
4 5 0 356 Reviews
27
4
##### ISBN: 9780134469164

Since 90 problems in chapter 8.2: Arithmetic Sequences have been answered, more than 32547 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780134469164. Chapter 8.2: Arithmetic Sequences includes 90 full step-by-step solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 7.

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).

• Augmented matrix [A b].

Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

• Big formula for n by n determinants.

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• 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).

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.

• Indefinite matrix.

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

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• 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).

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• 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.

• Pivot.

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

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Rank r (A)

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

• Solvable system Ax = b.

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

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

• Trace of A

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.