- 51.1: When the third multiple of 8 is subtracted from the fourthmultiple ...
- 51.2: From Monas home to school is 3.5 miles. How far does Mona travelrid...
- 51.3: Napoleon I was born in 1769. How old was he when he wascrowned empe...
- 51.4: Shelly purchased a music CD for $12.89. The sales-tax rate was 8%. ...
- 51.5: Malcom used a compass to draw a circle with a radius of 3 inches. a...
- 51.6: How can you round 12.75 to the nearest whole number?
- 51.7: 0.125 + 0.25 + 0.375
- 51.8: 0.399 + w = 0.4
- 51.9: 40.25
- 51.10: 4 0.5
- 51.11: 3.25 1100
- 51.12: 3 512 1 712
- 51.13: 58 ?24
- 51.14: 52 1734
- 51.15: (0.19)(0.21)
- 51.16: Write 0.01 as a fraction.
- 51.17: Write (6 10) (7 1100) as a decimal number.
- 51.18: The area of a square is 64 cm2. What is the perimeter of thesquare?
- 51.19: What is the least common multiple of 2, 3, and 4?
- 51.20: 5 310 6 910
- 51.21: 103 12
- 51.22: A collection of paperback books was stacked 12 inches high.Each boo...
- 51.23: Estimate the quotient when 4876 is divided by 98.
- 51.24: What factors do 16 and 24 have in common?
- 51.25: Find the product of 11.8 and 3.89 by rounding the factors tothe nea...
- 51.26: Find the average of the decimal numbers that correspond topoints x ...
- 51.27: 2 2 3 3 52 2 3 5
- 51.28: Mentally calculate the total price of ten pounds of bananas at$0.79...
- 51.29: Rename 3423 and 34 as fractions with 12 as the denominator. Then ad...
- 51.30: a. Jasons first nine test scores are shown below. Find themedian an...
Solutions for Chapter 51: Rounding Decimal Numbers
Full solutions for Saxon Math, Course 1 | 1st Edition
Tv = Av + Vo = linear transformation plus shift.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
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.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
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.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(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.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.