- 3-1.1: Write the word name for the decimal. 0.582
- 3-1.2: Write the word name for the decimal. 0.21
- 3-1.3: Write the word name for the decimal. 1.0009
- 3-1.4: Write the word name for the decimal. 2.83
- 3-1.5: Write the word name for the decimal. 782.07
- 3-1.6: Write the number that represents the decimal. Thirty-five hundredths
- 3-1.7: Write the number that represents the decimal. Three hundred twelve ...
- 3-1.8: Write the number that represents the decimal. Sixty and twenty-eigh...
- 3-1.9: Write the number that represents the decimal. Five and three hundre...
- 3-1.10: Round to the nearest dollar $493.91
- 3-1.11: Round to the nearest dollar $785.03
- 3-1.12: Round to the nearest dollar $19.80
- 3-1.13: Round to the nearest cent $0.5239
- 3-1.14: Round to the nearest cent $21.09734
- 3-1.15: Round to the nearest cent $32,048.87219
- 3-1.16: Round to the nearest tenth. 42.3784
- 3-1.17: Round to the nearest tenth. 17.03752
- 3-1.18: Round to the nearest tenth. 4.293
- 3-1.19: Tel-Sales, Inc., a prepaid phone card company in Oklahoma City, sel...
- 3-1.20: Destiny Telecom of Oakland, California, introduced a Braille prepai...
- 3-1.21: GameStop reported a quarterly gross margin of 839.18 dollars in mil...
- 3-1.22: Gannett Company reported a quarterly income before tax of negative ...
Solutions for Chapter 3-1: DECIMALS AND THE PLACE-VALUE SYSTEM
Full solutions for Business Math, | 9th Edition
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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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, off-diagonal entries = 0.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
= 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.
Every v in V is orthogonal to every w in W.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
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).
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 ].
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)j-I and det V = product of (Xk - Xi) for k > i.