- 72.1: What is the average of 4.2, 2.61, and 3.6?
- 72.2: Four tablespoons equals 14 cup. How many tablespoonswould equal one...
- 72.3: The temperature on the moon ranges from a high of about 130Cto a lo...
- 72.4: Four of the 12 marbles in the bag are blue. If one marble is taken ...
- 72.5: The diameter of a circle is 1 meter. The circumference is how manyc...
- 72.6: What fraction of a dollar is a nickel?
- 72.7: Find each unknown number:n 12 35
- 72.8: Find each unknown number:1 - w 712
- 72.9: Find each unknown number:w 212 313
- 72.10: Find each unknown number:1 w = 0.23
- 72.11: Write the standard decimal number for the following:(6 10) a4 110b ...
- 72.12: Which of these numbers is closest to 1? A 1 B 0.1 C 10
- 72.13: What is the largest prime number that is less than 100?
- 72.14: Which of these figures is not a parallelogram? A C B D
- 72.15: A loop of string two feet around is formed to make a square. a. How...
- 72.16: Figure ABCD is a rectangle. a. Name an angle complementary to DCM. ...
- 72.17: Refer to this menu and the information that follows to answerproble...
- 72.18: Refer to this menu and the information that follows to answerproble...
- 72.19: Refer to this menu and the information that follows to answerproble...
- 72.20: If A = lw, and if l equals 2.5 and w equals 0.4, what does A equal?
- 72.21: Write the prime factorization of both the numerator and the denomin...
- 72.22: Refer to the coordinate plane below to answer problems 22 and 23.Id...
- 72.23: Refer to the coordinate plane below to answer problems 22 and 23.Na...
- 72.24: Draw a pair of parallel lines. Then draw a second pair of parallell...
- 72.25: 12 56 35
- 72.26: 3 x 1 12 223
- 72.27: 34 2
- 72.28: 112 123
- 72.29: (0.12)(0.24)
- 72.30: 0.6 0.25
Solutions for Chapter 72: Fractions Chart Multiplying Three Fractions
Full solutions for Saxon Math, Course 1 | 1st Edition
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
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].
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
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 ().
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.