 22.1: Perform the indicated operations. Write the sum as a fraction, whol...
 22.2: Perform the indicated operations. Write the sum as a fraction, whol...
 22.3: Perform the indicated operations. Write the sum as a fraction, whol...
 22.4: Perform the indicated operations. Write the sum as a fraction, whol...
 22.5: Perform the indicated operations. Write the sum as a fraction, whol...
 22.6: Perform the indicated operations. Write the sum as a fraction, whol...
 22.7: Perform the indicated operations. Write the sum as a fraction, whol...
 22.8: Perform the indicated operations. Write the sum as a fraction, whol...
 22.9: Perform the indicated operations. Write the sum as a fraction, whol...
 22.10: Perform the indicated operations. Write the sum as a fraction, whol...
 22.11: Find the difference. Write the difference in lowest terms.
 22.12: Find the difference. Write the difference in lowest terms.
 22.13: Find the difference. Write the difference in lowest terms.
 22.14: Find the difference. Write the difference in lowest terms.
 22.15: Find the difference. Write the difference in lowest terms.
 22.16: Find the difference. Write the difference in lowest terms.
 22.17: Find the difference. Write the difference in lowest terms.
 22.18: Find the difference. Write the difference in lowest terms.
 22.19: Find the difference. Write the difference in lowest terms.
 22.20: Find the difference. Write the difference in lowest terms.
 22.21: Loretta McBride is determining the amount of fabric required for wi...
 22.22: Marveen McCready, a commercial space designer, has taken these meas...
 22.23: Rob Farinelli is building a gazebo and plans to use for the floor t...
 22.24: Tenisha Gist cuts brass plates for an engraving job. From a sheet o...
 22.25: The fabric Loretta McBride has selected for the window treatment in...
 22.26: Rob Farinelli purchased two boards that are 12 feet and will cut th...
 22.27: Rob Farinelli purchased four boards that are each 14 feet to make 1...
 22.28: Charlie Carr has a sheet of brass that is 36 inches wide and cuts t...
Solutions for Chapter 22: ADDING AND SUBTRACTING FRACTIONS
Full solutions for Business Math,  9th Edition
ISBN: 9780135108178
Solutions for Chapter 22: ADDING AND SUBTRACTING FRACTIONS
Get Full SolutionsBusiness Math, was written by and is associated to the ISBN: 9780135108178. Chapter 22: ADDING AND SUBTRACTING FRACTIONS includes 28 full stepbystep solutions. This textbook survival guide was created for the textbook: Business Math, , edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 28 problems in chapter 22: ADDING AND SUBTRACTING FRACTIONS have been answered, more than 19404 students have viewed full stepbystep solutions from this chapter.

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.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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 •

Outer product uv T
= column times row = rank one matrix.

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.

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).