- 10.1: State the rule of the following sequence. Then find the nextthree t...
- 10.2: Find how many years there were from the year the Pilgrims landedin ...
- 10.3: Is the number 1492 even or odd? How can you tell?
- 10.4: What weight is indicated on this scale?
- 10.5: If the perimeter of a square is 40 mm, how long is each sideof the ...
- 10.6: How much money is12 of $6.50?
- 10.7: Compare: 4 3 + 2 4 (3 + 2)
- 10.8: Use words and digits towrite the fraction of this circle that is no...
- 10.9: What is the a. product of 100 and 100? b. sum of 100 and 100?
- 10.10: 365 100
- 10.11: 146 240
- 10.12: 78 48
- 10.13: 907 36
- 10.14: 426010
- 10.15: 426020
- 10.16: 426015
- 10.17: 28,347 9,637
- 10.18: $8 + w = $11.49
- 10.19: $10 $0.75
- 10.20: $0.56 60
- 10.21: $6.20 4
- 10.22: Find each unknown number. Check your work.56 + 28 + 37 + n = 200
- 10.23: Find each unknown number. Check your work.a 67 = 49
- 10.24: Find each unknown number. Check your work.67 b = 49
- 10.25: Find each unknown number. Check your work.8c = 120
- 10.26: Find each unknown number. Check your work.d8 = 24
- 10.27: Here are three ways to write 12 divided by 4.4 12d8 2 4 124Show thr...
- 10.28: What number is one third of 36?
- 10.29: Arrange the numbers 346, 463, and 809 to form two additionequations...
- 10.30: At what temperature on the Fahrenheit scale does water freeze?
Solutions for Chapter 10: Sequences Scales
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).
peA) = det(A - AI) has peA) = zero matrix.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
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.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to 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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
A sequence of steps intended to approach the desired solution.
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.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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!
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.