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# Solutions for Chapter 9.1: SEQUENCES AND THEIR NOTATIONS

## Full solutions for College Algebra | 1st Edition

ISBN: 9781938168383

Solutions for Chapter 9.1: SEQUENCES AND THEIR NOTATIONS

Solutions for Chapter 9.1
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##### ISBN: 9781938168383

This textbook survival guide was created for the textbook: College Algebra, edition: 1. Since 71 problems in chapter 9.1: SEQUENCES AND THEIR NOTATIONS have been answered, more than 32126 students have viewed full step-by-step solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9781938168383. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.1: SEQUENCES AND THEIR NOTATIONS includes 71 full step-by-step solutions.

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

Tv = Av + Vo = linear transformation plus shift.

• 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).

• 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].

• 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.

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

Use AT for complex A.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Lucas numbers

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

• 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.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• 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.

• Plane (or hyperplane) in Rn.

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

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

P = aaT laTa has rank l.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• 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!

• Singular Value Decomposition

(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

• Skew-symmetric matrix K.

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

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.

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