 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
ISBN: 9781591417835
Solutions for Chapter 31: Areas of Rectangles
Get Full SolutionsSince 30 problems in chapter 31: Areas of Rectangles have been answered, more than 33662 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1. Chapter 31: Areas of Rectangles includes 30 full stepbystep solutions.

Affine transformation
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).

Covariance matrix:E.
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.

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.

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 IIA1 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, ... , AjIb. 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.

Lucas numbers
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.

Multiplication Ax
= 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!

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.