- 86.1: What is the quotient when the decimal number ten and six tenths isd...
- 86.2: The time in Los Angeles is 3 hours earlier than the time in New Yor...
- 86.3: Geraldine paid with a $10 bill for 1 dozen keychains that cost 75 e...
- 86.4: 32 + 1.8(50)
- 86.5: If each block has a volume ofone cubic inch, what is the volume of ...
- 86.6: The ratio of hardbacks to paperbacks in the school library was5 to ...
- 86.7: Nate missed three of the 20 questions on the test. What percent of ...
- 86.8: The credit card company charges 1.5% (0.015) interest onthe unpaid ...
- 86.9: a. Write 4 5 as a decimal number. b. Write 45 as a percent.
- 86.10: Serena is stuck on a multiple-choice question that has four choices...
- 86.11: 512 378
- 86.12: 3 b 14 58
- 86.13: a412ba23 3 b
- 86.14: 1212 100
- 86.15: 5 1 12
- 86.16: 56 of $30
- 86.17: Find each unknown number:4.72 + 12 + n = 50.4
- 86.18: Find each unknown number:$10 m = $9.87
- 86.19: Find each unknown number:3n = 0.48
- 86.20: Find each unknown number:w8 2520
- 86.21: What are the next three terms in this sequence of perfectsquares?1,...
- 86.22: This parallelogram is divided intotwo congruent triangles. a. What ...
- 86.23: Sydney drew a circle with a radius of 10 cm.What was the approximat...
- 86.24: Choose the appropriate unit for the area of a garage. A square inch...
- 86.25: Which two ratios form a proportion? How do you know?91281412212036
- 86.26: The wheel of the covered wagon turned around once in about 12 feet....
- 86.27: Anabel drove her car 348 miles in 6 hours. Divide the distance by t...
- 86.28: The opposite angles of aparallelogram are congruent. The adjacentan...
- 86.29: The diameter of each wheel on the lawn mower is 10 inches.How far m...
- 86.30: The coordinates of three vertices of a parallelogram are(3, 3), (2,...
Solutions for Chapter 86: Area of a Circle
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.
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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.
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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Constant down each diagonal = time-invariant (shift-invariant) filter.