- 90.1: What is the mean of 4.2, 4.8, and 5.1?
- 90.2: The movie is 120 minutes long. If it begins at 7:15 p.m., when will...
- 90.3: Fifteen of the 25 students in Room 20 are boys. What percent of the...
- 90.4: This triangular prism has how many moreedges than vertices?
- 90.5: The teacher cut a 12-inch diameter circle from a sheet of construct...
- 90.6: Write a description of a trapezoid
- 90.7: Arrange these numbers in order from least to greatest:1, 2, 0, 4, 12
- 90.8: Express the unknown factor as a mixed number:25n = 70
- 90.9: Refer to the triangle to answer questions 911.What is the area of t...
- 90.10: Refer to the triangle to answer questions 911.What is the perimeter...
- 90.11: Refer to the triangle to answer questions 911.What is the ratio of ...
- 90.12: Write 6.25 as a mixed number. Then subtract 58 from the mixed numbe...
- 90.13: Ali was facing north. Then he turned to his left 180. Whatdirection...
- 90.14: Write 28% as a reduced fraction.
- 90.15: n12 2030
- 90.16: 0.625 10
- 90.17: 250.8
- 90.18: 338 334
- 90.19: 5 18 178
- 90.20: 6 23 310 5 4
- 90.21: One third of the two dozen knights were on horseback. How manyknigh...
- 90.22: Weights totaling 38 ounces were placed on the left side ofthis scal...
- 90.23: The cube at right is made up of smallercubes that each have a volum...
- 90.24: Round forty-eight hundredths to the nearest tenth.
- 90.25: 1144 1121
- 90.26: The ratio of dogs to cats in the neighborhood is 6 to 5. What is th...
- 90.27: 10 + 10 10 10 10
- 90.28: The Thompsons drink a gallon of milk every two days. Thereare four ...
- 90.29: Simplify: a. 10 cm + 100 mm (Write the answer in millimeters.) b. 3...
- 90.30: On a coordinate plane draw a segment from point A (3, 1)to point B ...
Solutions for Chapter 90: Measuring Turns
Full solutions for Saxon Math, Course 1 | 1st Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Remove row i and column j; multiply the determinant by (-I)i + j •
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
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.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Invert A by row operations on [A I] to reach [I A-I].
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
A sequence of steps intended to approach the desired solution.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
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.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
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.
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 picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.