 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
ISBN: 9781591417835
Solutions for Chapter 78: Capacity
Get Full SolutionsSaxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. Since 30 problems in chapter 78: Capacity have been answered, more than 33435 students have viewed full stepbystep solutions from this chapter. Chapter 78: Capacity includes 30 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1.

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cyclic shift
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 SI 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.

Factorization
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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Norm
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.

Pascal matrix
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 = Q1 = 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 = UI.
Orthonormal columns (complex analog of Q).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.