- 13.1: When the sum of 8 and 5 is subtracted from the product of 8 and 5,w...
- 13.2: The Moon is about two hundred fifty thousand miles from the Earth. ...
- 13.3: Use words to write 521,000,000,000.
- 13.4: Use digits to write five million, two hundred thousand.
- 13.5: Robin entered a tennis tournament when she was three-scoreyears old...
- 13.6: The auditorium at the Community Cultural Center has seats for1000 p...
- 13.7: It is 405 miles from Minneapolis, Minnesota to Chicago,Illinois. It...
- 13.8: Use mental math to solve exercises 8 and 9. Describe the mentalmath...
- 13.9: Use mental math to solve exercises 8 and 9. Describe the mentalmath...
- 13.10: Which digit is in the thousands place in 54,321?
- 13.11: What is the place value of the 1 in 1,234,567,890?
- 13.12: The three sides of an equilateral triangle areequal in length. What...
- 13.13: 5432 100
- 13.14: 60,00030
- 13.15: 1000 7
- 13.16: $4.56 3
- 13.17: Compare: 3 + 2 + 1 + 0 3 2 1 0
- 13.18: The rule for the sequence below is different from the rulesfor addi...
- 13.19: What is 12 of 5280?
- 13.20: 365 w = 365
- 13.21: (5 + 6 + 7) 3
- 13.22: Use a ruler to find the length in inches of the rectangle below.
- 13.23: Write two ways to find the perimeter of a square: one way byadding ...
- 13.24: Multiply to find the answer to this addition problem:125 + 125 + 12...
- 13.25: At what temperature on the Fahrenheit scale does water boil?
- 13.26: Show three ways to write 21 divided by 7.
- 13.27: Find each unknown number. Check your work.8a = 816
- 13.28: Find each unknown number. Check your work.b4 12
- 13.29: Find each unknown number. Check your work.12c 4
- 13.30: Find each unknown number. Check your work.d - 16 = 61
Solutions for Chapter 13: Problems About Comparing Elapsed-Time Problems
Full solutions for Saxon Math, Course 1 | 1st Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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.
peA) = det(A - AI) has peA) = zero matrix.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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.
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.
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.
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
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.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.