- 84.1: What is the ratio of prime numbers to composite numbers in this lis...
- 84.2: Bianca poured four cups of milk from a full half-gallon container. ...
- 84.3: 6 + 6 6 6 6
- 84.4: Write 30% as a reduced fraction. Then write the fraction as a decim...
- 84.5: Find the area of each triangle:5
- 84.6: Find the area of each triangle:6
- 84.7: a. Write 120 as a decimal number. b. Write 120 as a percent.
- 84.8: Some parallelograms are rectangles. True or false?Why?
- 84.9: What is the area of this parallelogram?
- 84.10: What is the perimeter of thisparallelogram?
- 84.11: a318 214b 112
- 84.12: 56 223 3
- 84.13: 813 100
- 84.14: (4 3.2) 10
- 84.15: 0.5 0.5 + 0.5 0.5
- 84.16: 8 0.04
- 84.17: Which digit is in the hundredths place in 12.345678?
- 84.18: How do you round 5 1 8 to the nearest whole number?
- 84.19: Write the prime factorization of 700 using exponents.
- 84.20: Two ratios form a proportion if the ratios reduce to the same fract...
- 84.21: The perimeter of a square is 1 meter. How many centimeterslong is e...
- 84.22: Fong scored 9 of the teams 45 points. a. What fraction of the teams...
- 84.23: What time is 5 hours 30 minutes after 9:30 p.m.?
- 84.24: Write and complete this proportion: Six is to four as whatnumber is...
- 84.25: Figure ABCD is a parallelogram.Its opposite angles (A and C, B andD...
- 84.26: If each small cube has a volume of 1 cm3,what is the volume of this...
- 84.27: Simplify:2 ft + 24 in. (Write the sum in inches.)
- 84.28: Simplify:a. 100 cm210 cmb. 180 pages4 days
- 84.29: A triangle has vertices at the coordinates (4, 4) and (4, 0) andat ...
- 84.30: This year Moises has read 24 books. Sixteen of the bookswere non-fi...
Solutions for Chapter 84: Order of Operations, Part 2
Full solutions for Saxon Math, Course 1 | 1st Edition
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.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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].
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.
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].
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
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.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Solvable system Ax = b.
The right side b is in the column space of A.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).