- 47.1: The first positive odd number is 1. The second is 3. What is the te...
- 47.2: A passenger jet can travel 600 miles per hour. How longwould it tak...
- 47.3: Jos bought Carmen one dozen red roses, two for each month he hadkno...
- 47.4: If A = bh, what is A when b = 8 and h = 4?
- 47.5: The Commutative Property of Multiplication allows us to rearrangefa...
- 47.6: If s = 12 , what number does 4s equal?
- 47.7: Write 6.25 in expanded notation.
- 47.8: Write 99% as a fraction. Then write the fraction as a decimalnumber.
- 47.9: 12 0.18
- 47.10: 10 12.30
- 47.11: Find each missing number:w 236 62
- 47.12: Find each missing number:5y = 1.25
- 47.13: Find each missing number:n 51112 10
- 47.14: Find each missing number: 625 335
- 47.15: Find each missing number:834 534
- 47.16: Find each missing number:53 54
- 47.17: Find each missing number:34 ?20
- 47.18: Find each missing number:35=?20
- 47.19: Bobs scores on his first five tests were 18, 20, 18, 20, and 20. Hi...
- 47.20: Roberts bicycle tires are 20 inches in diameter. What is thecircumf...
- 47.21: Which factors of 20 are also factors of 30?
- 47.22: Mentally calculate the product of 6.25 and 10. Describe howyou perf...
- 47.23: Multiply as shown. Then complete the division.1.250.5 1010
- 47.24: Shelly answered 90% of the 40 questions correctly. What number is90...
- 47.25: Refer to the chart shown below to answer problems 25 and 26.Mars ta...
- 47.26: Refer to the chart shown below to answer problems 25 and 26.In the ...
- 47.27: Use an inch ruler to find the length and width of this rectangle.
- 47.28: Calculate the perimeter of the rectangle in problem 27.
- 47.29: Rename 25 so that the denominator of the renamed fractionis 10. The...
- 47.30: When we mentally multiply 15 by 10, we can simply attach azero to 1...
Solutions for Chapter 47: Circumference Pi (
Full solutions for Saxon Math, Course 1 | 1st Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Column space C (A) =
space of all combinations of the columns of A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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).
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.