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# Solutions for Chapter P.6: Rational Expressions

## Full solutions for College Algebra | 6th Edition

ISBN: 9780321782281

Solutions for Chapter P.6: Rational Expressions

Solutions for Chapter P.6
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##### ISBN: 9780321782281

College Algebra was written by and is associated to the ISBN: 9780321782281. This textbook survival guide was created for the textbook: College Algebra , edition: 6. Since 112 problems in chapter P.6: Rational Expressions have been answered, more than 35976 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter P.6: Rational Expressions includes 112 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• 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.

• Dimension of vector space

dim(V) = number of vectors in any basis for V.

• 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.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

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

Use AT for complex 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.

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• 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.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• 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).

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Right inverse A+.

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

• 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).

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• Trace of A

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

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B IIĀ·