 31.1: Write the word name for the decimal. 0.582
 31.2: Write the word name for the decimal. 0.21
 31.3: Write the word name for the decimal. 1.0009
 31.4: Write the word name for the decimal. 2.83
 31.5: Write the word name for the decimal. 782.07
 31.6: Write the number that represents the decimal. Thirtyfive hundredths
 31.7: Write the number that represents the decimal. Three hundred twelve ...
 31.8: Write the number that represents the decimal. Sixty and twentyeigh...
 31.9: Write the number that represents the decimal. Five and three hundre...
 31.10: Round to the nearest dollar $493.91
 31.11: Round to the nearest dollar $785.03
 31.12: Round to the nearest dollar $19.80
 31.13: Round to the nearest cent $0.5239
 31.14: Round to the nearest cent $21.09734
 31.15: Round to the nearest cent $32,048.87219
 31.16: Round to the nearest tenth. 42.3784
 31.17: Round to the nearest tenth. 17.03752
 31.18: Round to the nearest tenth. 4.293
 31.19: TelSales, Inc., a prepaid phone card company in Oklahoma City, sel...
 31.20: Destiny Telecom of Oakland, California, introduced a Braille prepai...
 31.21: GameStop reported a quarterly gross margin of 839.18 dollars in mil...
 31.22: Gannett Company reported a quarterly income before tax of negative ...
Solutions for Chapter 31: DECIMALS AND THE PLACEVALUE SYSTEM
Full solutions for Business Math,  9th Edition
ISBN: 9780135108178
Solutions for Chapter 31: DECIMALS AND THE PLACEVALUE SYSTEM
Get Full SolutionsChapter 31: DECIMALS AND THE PLACEVALUE SYSTEM includes 22 full stepbystep solutions. This textbook survival guide was created for the textbook: Business Math, , edition: 9. Business Math, was written by and is associated to the ISBN: 9780135108178. This expansive textbook survival guide covers the following chapters and their solutions. Since 22 problems in chapter 31: DECIMALS AND THE PLACEVALUE SYSTEM have been answered, more than 13102 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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!

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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

Pseudoinverse A+ (MoorePenrose 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).

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).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Solvable system Ax = b.
The right side b is in the column space of A.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.