- 67.1: Allison is making a large collage of a beach scene. She needs 2 yar...
- 67.2: A mile is 5280 feet. A nautical mile is about 6080 feet. Anautical ...
- 67.3: Instead of dividing $1.50 by $0.05, Marcus formed anequivalent divi...
- 67.4: Find each unknown number:6 cm + k = 11 cm
- 67.5: Find each unknown number:8g = 9.6
- 67.6: Find each unknown number:710 w 12
- 67.7: Find each unknown number:35 n100
- 67.8: The perimeter of a quadrilateral is 172 inches. What is the average...
- 67.9: $100.00 ($46.75 + $9.68)
- 67.10: (2 0.3) (0.2 0.3)
- 67.11: 4 29 14 278
- 67.12: 223 4 29
- 67.13: 313 234
- 67.14: 113 214
- 67.15: 1.44 60
- 67.16: $6.00 $0.15
- 67.17: Five dollars was divided evenly among 4 people. How much money dide...
- 67.18: The area of a regular quadrilateral is 100 square inches. Whatis it...
- 67.19: Write the prime factorizations of 625 and of 1000. Thenreduce 6251000.
- 67.20: What is the area of the rectangle shown below?
- 67.21: Thirty-six of the 88 piano keys are black. What fraction of the pia...
- 67.22: Draw a rectangular prism. Begin by drawing two congruentrectangles.
- 67.23: 112 = 1
- 67.24: There are 1000 meters in a kilometer. How many meters are in2.5 kil...
- 67.25: Which arrow could be pointing to 0.1 on the number line?
- 67.26: If the tip of the minute hand is 6 inches from the center of theclo...
- 67.27: A basketball is an example of what geometric solid?
- 67.28: Write 51% as a fraction. Then write the fraction as a decimalnumber.
- 67.29: What is the probability of rollinga prime number with one toss of a...
- 67.30: This quadrilateral has one pairof parallel sides. What kind of quad...
Solutions for Chapter 67: Using Prime Factorization to Reduce Fractions
Full solutions for Saxon Math, Course 1 | 1st Edition
Column space C (A) =
space of all combinations of the columns of A.
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
A symmetric matrix with eigenvalues of both signs (+ and - ).
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.
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.
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.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
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.
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.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).