- 31.1: When the third multiple of 4 is divided by the fourth multipleof 3,...
- 31.2: The distance the Earth travels around the Sun each year isabout fiv...
- 31.3: Convert 103 to a mixed number.
- 31.4: How many square stickers with sides 1 centimeter longwould be neede...
- 31.5: How many floor tiles with sides 1 foot long would be needed to cove...
- 31.6: What is the area of a rectangle 12 inches long and 8 inches wide?
- 31.7: Describe the rule for this sequence. What is the next term? 1, 4, 9...
- 31.8: What number is 23 of 24? Draw a diagram to illustrate the problem.
- 31.9: Find the unknown number. Remember to check your work. 24 + f = 42
- 31.10: Write each answer in simplest form:18 18
- 31.11: Write each answer in simplest form:56 16
- 31.12: Write each answer in simplest form:23 12
- 31.13: Write each answer in simplest form:23 5
- 31.14: Estimate the product of 387 and 514
- 31.15: $20.00 10
- 31.16: (63)47
- 31.17: 4623 22
- 31.18: What is the reciprocal of the smallest odd prime number?
- 31.19: Two thirds of a circle is what percent of a circle?
- 31.20: Which of these numbers is closest to 100? A 90 B 89 C 111 D 109
- 31.21: For most of its orbit, Pluto is the farthest planet from the Sun in...
- 31.22: The diameter of the pizza was 14 inches. What was the ratio of ther...
- 31.23: Three of the nine softball players play outfield. What fraction of ...
- 31.24: Use an inch ruler to find the length of the line segment below.
- 31.25: 310 310
- 31.26: How many 34 s are in 1?
- 31.27: Write a fraction equal to 1 with a denominator of 8.
- 31.28: Five sixths of the 24 students in the class scored 80% or higheron ...
- 31.29: a. Name an angle in the figure at right thatmeasures less than 90. ...
- 31.30: Using a ruler, how could you calculate the floor area of your class...
Solutions for Chapter 31: Areas of Rectangles
Full solutions for Saxon Math, Course 1 | 1st Edition
Tv = Av + Vo = linear transformation plus shift.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row 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
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.