- 27.1: What is the product of the sum of 55 and 45 and thedifference of 55...
- 27.2: Potatoes are three-fourths water. If a sack of potatoes weighs20 po...
- 27.3: There were 306 students in the cafeteria. After somewent outside, t...
- 27.4: a. If the diameter of a circle is 5 in., what is the radius of the ...
- 27.5: Which of these numbers is divisible by both 2 and 3? A 122 B 123 C 132
- 27.6: Round 1,234,567 to the nearest ten thousand.
- 27.7: If ten pounds of apples costs $12.90, what is the price perpound? W...
- 27.8: What is the denominator of 2324?
- 27.9: What number is 35 of 65? Draw a diagram to illustrate theproblem.
- 27.10: How much money is 23 of $15? Draw a diagram to illustrate theproblem.
- 27.11: Use your fraction manipulatives to help answer problems 1118.6 26 36
- 27.12: Use your fraction manipulatives to help answer problems 1118.78 38
- 27.13: Use your fraction manipulatives to help answer problems 1118.66 56
- 27.14: Use your fraction manipulatives to help answer problems 1118.28 58
- 27.15: Use your fraction manipulatives to help answer problems 1118.a. How...
- 27.16: Use your fraction manipulatives to help answer problems 1118.Reduce...
- 27.17: Use your fraction manipulatives to help answer problems 1118.What f...
- 27.18: Use your fraction manipulatives to help answer problems 1118.What f...
- 27.19: Divide 2100 by 52 and write the answer with a remainder.
- 27.20: If a 36-inch-long string is made into the shape of a square, how lo...
- 27.21: Convert 76 to a mixed number.
- 27.22: 43218
- 27.23: (55 + 45) (55 45)
- 27.24: Which of these numbers is divisible by both 2 and 5? A 502 B 205 C 250
- 27.25: Describe a method for determining whether a number isdivisible by 9.
- 27.26: Which prime number is not an odd number?
- 27.27: What is the name for the perimeter of a circle?
- 27.28: What is the ratio of even numbers to odd numbers in thesquare below...
- 27.29: 37 % 12% 1212%
- 27.30: 3313% 1623 37 %
Solutions for Chapter 27: Measures of a Circle
Full solutions for Saxon Math, Course 1 | 1st Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Column space C (A) =
space of all combinations of the columns of A.
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.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Invert A by row operations on [A I] to reach [I A-I].
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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.
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.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.