 40.1: In the circle graph in example 5, what percent of the petsare birds...
 40.2: The U.S. Constitution was ratified in 1788. In 1920 the 19thamendme...
 40.3: White Rabbit is threeandahalfhours late for a veryimportant dat...
 40.4: Look at problems 49. Predict which of the answers to thoseproblems ...
 40.5: Look at problems 49. Predict which of the answers to thoseproblems ...
 40.6: Look at problems 49. Predict which of the answers to thoseproblems ...
 40.7: Look at problems 49. Predict which of the answers to thoseproblems ...
 40.8: Look at problems 49. Predict which of the answers to thoseproblems ...
 40.9: Look at problems 49. Predict which of the answers to thoseproblems ...
 40.10: Write one and two hundredths as a decimal number.
 40.11: Write (6 10,000) + (8 100) in standard form.
 40.12: A square room has a perimeter of 32 feet. How many squarefloor tile...
 40.13: What is the least common multiple (LCM) of 2, 4, and 8?
 40.14: 623 423
 40.15: 5 338
 40.16: 58 23
 40.17: 256 526
 40.18: Compare: 12 2212 33
 40.19: 1000 w = 567
 40.20: Nine whole numbers are factors of 100. Two of the factors are1 and ...
 40.21: 92 29
 40.22: Round $4167 to the nearest hundred dollars.
 40.23: The circle graph below displays the favorite sports of a number ofs...
 40.24: Jamal earned $5.00 walking his neighbors dog for one week. He wasgi...
 40.25: Write a ratio problem that relates to the circle graph inproblem 23...
 40.26: Arrange the numbers in this multiplication fact to formanother mult...
 40.27: To solve the division problem 240 15, Elianna divided bothnumbers b...
 40.28: Forty percent of the 25 students in the class are boys. Write40% as...
 40.29: What mixed number is represented by point A on the numberline below?
 40.30: Make a circle graph that shows the portion of a full day spent in v...
Solutions for Chapter 40: Using Zero as a Placeholder Circle Graphs
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Solutions for Chapter 40: Using Zero as a Placeholder Circle Graphs
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1. Since 30 problems in chapter 40: Using Zero as a Placeholder Circle Graphs have been answered, more than 35270 students have viewed full stepbystep solutions from this chapter. Chapter 40: Using Zero as a Placeholder Circle Graphs includes 30 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.