 9.1: Tamara arranged 144 books into 8 equal stacks. How many books werei...
 9.2: Find how many years there were from 1492 to 1603 bysubtracting 1492...
 9.3: Martin is carrying groceries in from the car. If he can carry2 bags...
 9.4: Use a centimeter ruler to measure the length and width of therectan...
 9.5: How much money is 12 of $5.80?
 9.6: How many cents is 14of a dollar?
 9.7: Use words and digits to name thefraction of this triangle that is s...
 9.8: Compare:a. 5012 5120 b. 1 mm 1 cm
 9.9: Arrange these numbers in order from least to greatest: 1, 0, 12
 9.10: Compare: 100 50 25 100 (50 25)
 9.11: 4783692+ 45
 9.12: $50.00 $31.76
 9.13: $4.20 60
 9.14: 78 36
 9.15: 9 7227
 9.16: 25 7600
 9.17: 20 8014
 9.18: 7136 100
 9.19: 736 736
 9.20: Find each unknown number. Check your work.165 + a = 300
 9.21: Find each unknown number. Check your work.b 68 = 86
 9.22: Find each unknown number. Check your work.9c = 144
 9.23: Find each unknown number. Check your work.d15= 7
 9.24: Use an inch ruler to draw a line segment two inches long.Then use a...
 9.25: Which of the figures below represents a ray?
 9.26: Use digits and symbols to write this comparison:One half is greater...
 9.27: Arrange the numbers 9, 11, and 99 to form two multiplicationfacts a...
 9.28: Compare: 25 + 0 25 0
 9.29: 100 = 20 + 30 + 40 + x
 9.30: How did you choose the positions of the small and largeends of the ...
Solutions for Chapter 9: The Number Line: Ordering and Comparing
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Solutions for Chapter 9: The Number Line: Ordering and Comparing
Get Full SolutionsChapter 9: The Number Line: Ordering and Comparing includes 30 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 30 problems in chapter 9: The Number Line: Ordering and Comparing have been answered, more than 37785 students have viewed full stepbystep solutions from this chapter. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Column space C (A) =
space of all combinations of the columns of A.

Companion matrix.
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 nl  An).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

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].

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kirchhoff's Laws.
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.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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