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# Solutions for Chapter 2.6: COMBINATIONS OF FUNCTIONS: COMPOSITE FUNCTIONS

## Full solutions for College Algebra | 8th Edition

ISBN: 9781439048696

Solutions for Chapter 2.6: COMBINATIONS OF FUNCTIONS: COMPOSITE FUNCTIONS

Solutions for Chapter 2.6
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##### ISBN: 9781439048696

Since 84 problems in chapter 2.6: COMBINATIONS OF FUNCTIONS: COMPOSITE FUNCTIONS have been answered, more than 31099 students have viewed full step-by-step solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9781439048696. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.6: COMBINATIONS OF FUNCTIONS: COMPOSITE FUNCTIONS includes 84 full step-by-step solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 8.

Key Math Terms and definitions covered in this textbook
• Back substitution.

Upper triangular systems are solved in reverse order Xn to Xl.

• Cayley-Hamilton Theorem.

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

• 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).

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• Elimination matrix = Elementary matrix Eij.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

• Fast Fourier Transform (FFT).

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• Indefinite matrix.

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

• 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.

• 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.

• 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.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Pivot.

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

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

• 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!

• Solvable system Ax = b.

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

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

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