 71.1: What is the least common multiple of 6 and 10?
 71.2: The highest point on land is Mt. Everest, whose peak is29,035 feet ...
 71.3: The movie lasted 105 minutes. If the movie started at 1:15 p.m., at...
 71.4: In problems 47, reduce the fractions, if possible, beforemultiplyin...
 71.5: In problems 47, reduce the fractions, if possible, beforemultiplyin...
 71.6: In problems 47, reduce the fractions, if possible, beforemultiplyin...
 71.7: In problems 47, reduce the fractions, if possible, beforemultiplyin...
 71.8: 6 334 212
 71.9: 5 318
 71.10: 514 178
 71.11: (3.5)2
 71.12: 15 $75.00
 71.13: (1 + 0.6) (1 0.6)
 71.14: Quan ordered a $4.50 bowl of soup. The tax rate was 712 % (which eq...
 71.15: What is the name for the point on the coordinate plane that has the...
 71.16: Refer to the coordinate plane below to locate the pointsindicated.N...
 71.17: Find each unknown number:1.2f = 120
 71.18: Find each unknown number:120f 1.2
 71.19: Write the prime factorization of both the numerator and the denomin...
 71.20: The perimeter of a square is 6.4 meters. What is its area?
 71.21: What fraction of this circle is notshaded?
 71.22: If the radius of this circle is 1 cm,what is the circumference of t...
 71.23: A centimeter is about as long as this segment:About how many centim...
 71.24: Water freezes at 32Fahrenheit. The temperature shown onthe thermome...
 71.25: Ray watched TV for one hour. He determined that commercials weresho...
 71.26: Name the geometric solid shownat right.
 71.27: This square and regular triangleshare a common side. The perimeter ...
 71.28: Choose the appropriate unit for the area of your state. A square in...
 71.29: a. What is the perimeter of thisparallelogram? b. What is the area ...
 71.30: In this figure BMD is a right angle.Name two angles that are a. sup...
Solutions for Chapter 71: Parallelograms
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Solutions for Chapter 71: Parallelograms
Get Full SolutionsSaxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. Chapter 71: Parallelograms includes 30 full stepbystep solutions. Since 30 problems in chapter 71: Parallelograms have been answered, more than 38532 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Outer product uv T
= column times row = rank one matrix.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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).

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Stiffness matrix
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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.