- 43.1: The bike cost $120. The sales-tax rate was 8%. What was thetotal co...
- 43.2: If one hundred fifty knights could sit at the Round Table andonly o...
- 43.3: During the 1996 Summer Olympics in Atlanta, Georgia, the Americanat...
- 43.4: In problems 4 and 5, multiply by a fraction equal to 1 to complete ...
- 43.5: In problems 4 and 5, multiply by a fraction equal to 1 to complete ...
- 43.6: Find each unknown number. Remember to check your work.23n 1
- 43.7: Find each unknown number. Remember to check your work.6 w 145
- 43.8: Find each unknown number. Remember to check your work.m 414 634
- 43.9: Find each unknown number. Remember to check your work.c 2.45 = 3
- 43.10: Find each unknown number. Remember to check your work.12 d = 1.43
- 43.11: Find each unknown number. Remember to check your work.58 15
- 43.12: Find each unknown number. Remember to check your work.34 5
- 43.13: Find each unknown number. Remember to check your work.378 138
- 43.14: Which of these numbers is not a prime number? A 23 B 33 C 43
- 43.15: Compare: 2222 22
- 43.16: In football a loss of yardage is often expressed as a negative numb...
- 43.17: Write the decimal number for nine and twelve hundredths.
- 43.18: Round 67,492,384 to the nearest million.
- 43.19: 0.37 102
- 43.20: 0.6 0.4 0.2
- 43.21: The perimeter of a square room is 80 feet. The area of the room is ...
- 43.22: Divide 100 by 16 and write the answer as a mixed number. Reduce the...
- 43.23: a. Instead of dividing 100 by 16, Sandy divided the dividendand div...
- 43.24: What is the least common multiple (LCM) of 4, 6, and 8?
- 43.25: What are the next three numbers in this sequence?22 22 22116,18, 31...
- 43.26: Find the length of the segment below to the nearest eighth of an inch.
- 43.27: What mixed number is indicated on the number line below?
- 43.28: Write 12 and 15 as fractions with denominators of 10. Then add ther...
- 43.29: Forty percent of the 20 seats on the bus were occupied. Write40% as...
- 43.30: Describe each angle in the figure as acute,right, or obtuse. a. ang...
Solutions for Chapter 43: Equivalent Division Problems Finding Unknowns in Fraction and Decimal Problems
Full solutions for Saxon Math, Course 1 | 1st Edition
Solutions for Chapter 43: Equivalent Division Problems Finding Unknowns in Fraction and Decimal ProblemsGet Full Solutions
peA) = det(A - AI) has peA) = zero matrix.
A = CTC = (L.J]))(L.J]))T for positive definite 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).
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].
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.