- 20.1: What is the difference between the product of 12 and 8 and the sum ...
- 20.2: Saturns average distance from the Sun is one billion, fourhundred t...
- 20.3: Which digit in 497,325,186 is in the ten-millions place?
- 20.4: Jill has exactly 427 beads, but when Dwayne asked her howmany beads...
- 20.5: The morning temperature was 3C. By afternoon it had warmed to8C. Ho...
- 20.6: In three basketball games Allen scored 31, 52, and 40 points. What ...
- 20.7: Find the greatest common factor of 12 and 20.
- 20.8: Find the GCF of 9, 15, and 21.
- 20.9: How much money is 14 of $3.24?
- 20.10: 5432 10
- 20.11: 28 4214
- 20.12: 56,042 + 49,985
- 20.13: 37,080 12
- 20.14: $6.47 10
- 20.15: 5 4 3 2 1
- 20.16: Find each unknown number. Check your work.w 76 = 528
- 20.17: Find each unknown number. Check your work.14,009 w = 9670
- 20.18: Find each unknown number. Check your work.6w = 90
- 20.19: Find each unknown number. Check your work.q 365 = 365
- 20.20: Find each unknown number. Check your work.365 p = 365
- 20.21: Find the missing number in the following sequence:, 10, 16, 22, 28,...
- 20.22: Compare: 50 1 49 + 1
- 20.23: The first positive odd number is 1. What is the tenth positiveodd n...
- 20.24: The perimeter of a square is 100 cm. Describe how to find thelength...
- 20.25: Estimate the length of this key to the nearest inch. Thenuse a rule...
- 20.26: A bit is 18 of a dollar. a. How many bits are in a dollar? b. How m...
- 20.27: In four boxes there are 12, 24, 36, and 48 golf balls respectively....
- 20.28: Which of the numbers below is a prime number? A 5 B 15 C 25
- 20.29: List the whole-number factors of 24. How did you find youranswer?
- 20.30: Ten billion is how much less than one trillion?
Solutions for Chapter 20: Greatest Common Factor (GCF
Full solutions for Saxon Math, Course 1 | 1st Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
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).
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.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
= Xl (column 1) + ... + xn(column n) = combination of columns.
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.
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.
Constant down each diagonal = time-invariant (shift-invariant) filter.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).