- 59.1: What is the product of the decimal numbers four tenths andfour hund...
- 59.2: Larry looked at the clock. It was 9:45 p.m. The bus for his class t...
- 59.3: Pluto orbits the sun at an average distance of about five billion,n...
- 59.4: 112 223
- 59.5: 112 223
- 59.6: Compare:1235
- 59.7: Compare:2369
- 59.8: 815 345
- 59.9: 34 52
- 59.10: How many 12 s are in 25? (25 12)
- 59.11: (0.875)(40)
- 59.12: 0.07 4
- 59.13: 30 d = 0.6
- 59.14: What number is halfway between 0.1 and 0.24?
- 59.15: Round 36,428,591 to the nearest million.
- 59.16: What temperature is 23 less than 8F?
- 59.17: Miguela wound a garden hose around a circular reel. If thediameter ...
- 59.18: How many square inches are needed to cover a square foot?
- 59.19: One centimeter is what fraction of one meter?
- 59.20: Mentally calculate each answer. Describe how you performedeach ment...
- 59.21: a. With one toss of a single number cube, what is the probability o...
- 59.22: Compare: (0.8)2 0.8
- 59.23: Refer to the line graph below to answer problems 2325.What was the ...
- 59.24: Refer to the line graph below to answer problems 2325.What was the ...
- 59.25: Refer to the line graph below to answer problems 2325.Write a quest...
- 59.26: Nana can pack a bag of groceries in six tenths of a minute.At that ...
- 59.27: One eighth is equivalent to 1212%. To what percent is three eighths...
- 59.28: Mentally calculate the total cost of 10 gallons of gas pricedat $2....
- 59.29: Arrange these three numbers in order from least to greatest:34, the...
- 59.30: If P = 2l + 2w, and if l = 4 and w = 3, what does Pequal?
Solutions for Chapter 59: Adding Mixed Numbers
Full solutions for Saxon Math, Course 1 | 1st Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Column space C (A) =
space of all combinations of the columns of A.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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.
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].
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 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.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
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.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.