- 110.1: When the greatest four-digit number is divided by the greatest two-...
- 110.2: The ratio of the length to the width of the Alamo is about 5 to 3. ...
- 110.3: A box of crackers in the shape of a squareprism had a length, width...
- 110.4: A full turn is 360. How many degrees is 16 of a turn?
- 110.5: Refer to these triangles to answer problems 57:Does the equilateral...
- 110.6: Refer to these triangles to answer problems 57:Sketch a triangle si...
- 110.7: Refer to these triangles to answer problems 57:What is the area of ...
- 110.8: Draw a ratio box for this problem. Then solve the problem usinga pr...
- 110.9: Complete the table to answer problems 911.234 a. b.
- 110.10: Complete the table to answer problems 911.a. 1.1 b.
- 110.11: Complete the table to answer problems 911.a. b. 64%
- 110.12: 2416 2313 2212
- 110.13: a115 2b 123
- 110.14: 9 (6.2 + 2.79)
- 110.15: 0.36m = $63.00
- 110.16: Find 6.5% of $24.89 by multiplying 0.065 by $24.89. Round the produ...
- 110.17: Round the quotient to the nearest thousandth:0.065 4
- 110.18: Write the prime factorization of 1000 using exponents.
- 110.19: All squares are similar. True or false?
- 110.20: 33 32 3 3 3
- 110.21: What is the perimeter of this polygon?
- 110.22: What is the area of this polygon?
- 110.23: Triangles I and II are congruent. Refer to these triangles to answe...
- 110.24: Triangles I and II are congruent. Refer to these triangles to answe...
- 110.25: The first Ferris wheel was built in 1893 for the worlds fairin Chic...
- 110.26: Use a ruler to draw AB 1 34 inches long. Then draw a dot at the mid...
- 110.27: Use a compass to draw a circle on a coordinate plane. Makethe cente...
- 110.28: What is the area of the circle in problem 27? (Use 3.14 for .)
- 110.29: 3 + 4 5 +7
- 110.30: If Freddy tosses a coin four times, what is the probability that th...
Solutions for Chapter 110: Symmetry
Full solutions for Saxon Math, Course 1 | 1st Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.