- 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 three-and-a-half-hours 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
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 S-I AS = A = eigenvalue matrix.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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 = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Every v in V is orthogonal to every w in W.
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. A-I is also symmetric.
Unitary matrix UH = U T = U-I.
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)j-I 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.