- 78.1: What is the difference when the product of 12 and 12 is subtracted ...
- 78.2: The claws of a Siberian tiger are 10 centimeters long. How manymill...
- 78.3: Sue was thinking of a number between 40 and 50 that is amultiple of...
- 78.4: Diagram this statement. Then answer the questions that follow. Four...
- 78.5: Which counting number is neither a prime number nor acomposite number?
- 78.6: Find each unknown number:45m 1
- 78.7: Find each unknown number:45 w 1
- 78.8: Find each unknown number:45 x 1
- 78.9: Find each unknown number:34 n100
- 78.10: a. What fraction of the rectangle below is shaded? b. Write the ans...
- 78.11: Convert the decimal number 1.15 to a mixed number.
- 78.12: Compare:a. 3 5 0.35 b. 100 1 4 + 23
- 78.13: 56 12
- 78.14: 414 313
- 78.15: 12 23 56
- 78.16: 112 223
- 78.17: 112 223
- 78.18: 223 112
- 78.19: a. What is the perimeter of this square? b. What is the area of thi...
- 78.20: The opposite sides of a rectangle are parallel. True orfalse?
- 78.21: What is the average of 33 and 52 ?
- 78.22: The diameter of the small wheel was 7 inches. The circumference was...
- 78.23: How many inches is 2 12 feet?
- 78.24: Which arrow below could be pointing to 0.1?
- 78.25: Draw a quadrilateral that is not a rectangle.
- 78.26: Find the prime factorization of 900 by using a factor tree.Then wri...
- 78.27: Three vertices of a rectangle have the coordinates (5, 3), (5, 1), ...
- 78.28: Refer to this table to answer a and b.a. A teaspoon of soup is what...
- 78.29: A liter is closest in size to which of the following? A pint B quar...
- 78.30: In 1881 Clara Barton founded the American Red Cross, an organizatio...
Solutions for Chapter 78: Capacity
Full solutions for Saxon Math, Course 1 | 1st Edition
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.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
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.
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!).
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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).
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.