- 64.1: When the sum of 1.3 and 1.2 is divided by the difference of 1.3 and...
- 64.2: William Shakespeare was born in 1564 and died in 1616. How manyyear...
- 64.3: Duane kicked a 45-yard field goal. How many feet is 45 yards?
- 64.4: Why is a square a regular quadrilateral?
- 64.5: A regular hexagon has a perimeter of 36 inches. How long is eachside?
- 64.6: 14 ?100
- 64.7: 8 88 8
- 64.8: 5 23 334
- 64.9: 12 23 14
- 64.10: 910 12
- 64.11: 612 278
- 64.12: Compare: 2 0.4 2 + 0.4
- 64.13: 4.8 0.35
- 64.14: 1 0.4
- 64.15: How many $0.12 pencils can Mr. Velazquez buy for $4.80?
- 64.16: Round the product of 0.33 and 0.38 to the nearesthundredth
- 64.17: Multiply the length by the width to find thearea of this rectangle.
- 64.18: Is every rectangle a parallelogram?
- 64.19: What is the twelfth prime number?
- 64.20: The area of a square is 9 cm2. a. How long is each side of the squa...
- 64.21: Refer to the box shown below to answer problems 21 and 22.This box ...
- 64.22: Refer to the box shown below to answer problems 21 and 22.If this b...
- 64.23: There are 100 centimeters in a meter. How many centimeters equal2.5...
- 64.24: Write the mixed numbers 1 12 and 2 12 as improper fractions. Then m...
- 64.25: The numbers 2, 3, 5, 7, and 11 are prime numbers. Thenumbers 4, 6, ...
- 64.26: Write 75% as an unreduced fraction. Then write the fraction as adec...
- 64.27: Reduce: 2 2 2 3 32 2 3 5 5
- 64.28: Find the missing distance d in the equation below.16.6 mi + d = 26....
- 64.29: Refer to the double-line graph below to answer problems 29 and 30.a...
- 64.30: Refer to the double-line graph below to answer problems 29 and 30.I...
Solutions for Chapter 64: Classifying Quadrilaterals
Full solutions for Saxon Math, Course 1 | 1st Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.