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# Solutions for Chapter 4-6: The Algebra Of Functions

## Full solutions for Amsco's Algebra 2 and Trigonometry | 1st Edition

ISBN: 9781567657029

Solutions for Chapter 4-6: The Algebra Of Functions

Solutions for Chapter 4-6
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##### ISBN: 9781567657029

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4-6: The Algebra Of Functions includes 16 full step-by-step solutions. Since 16 problems in chapter 4-6: The Algebra Of Functions have been answered, more than 29275 students have viewed full step-by-step solutions from this chapter. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Cholesky factorization

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

• Cofactor Cij.

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

• Column space C (A) =

space of all combinations of the columns of A.

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• 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.

• 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].

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• 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).

• Orthogonal subspaces.

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

• Outer product uv T

= column times row = rank one matrix.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Plane (or hyperplane) in Rn.

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

• 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).

• 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 space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Standard basis for Rn.

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

• Trace of A

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

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