- 104.1: A pyramid with a square base has how many more edgesthan vertices?
- 104.2: Becki weighed 7 lb 8 oz when she was born and 12 lb 6 oz at 3 month...
- 104.3: There are 6 fish and 10 snails in the aquarium. What is the ratio o...
- 104.4: A teams win-loss ratio was 3 to 2. If the team had played 20 gamesw...
- 104.5: If Molly tosses a coin and rolls a number cube, what is theprobabil...
- 104.6: a. What is the perimeter of thisparallelogram? b. What is the area ...
- 104.7: If each acute angle of a parallelogram measures 59, thenwhat is the...
- 104.8: The center of this circle is theorigin. The circle passes through (...
- 104.9: Which ratio forms a proportion with 23?A 24 B 34 C 46 D 32
- 104.10: Complete this proportion: 68 a12
- 104.11: What is the perimeter of the hexagon atright? Dimensions are in cen...
- 104.12: Complete the table to answer problems 1214.320 a. b.
- 104.13: Complete the table to answer problems 1214.a. 1.2 b.
- 104.14: Complete the table to answer problems 1214.a. b. 10%
- 104.15: Sharon bought a notebook for 40% off the regular price of $6.95. Wh...
- 104.16: Between which two consecutive whole numbers is 200?
- 104.17: Compare:a12b3 the probability of 3 consecutive heads coin tosses
- 104.18: Divide 0.624 by 0.05 and round the quotient to the nearestwhole num...
- 104.19: The average of three numbers is 20. What is the sum of the threenum...
- 104.20: Write the prime factorization of 450 using exponents.
- 104.21: 3 + 5 4 +2
- 104.22: 34 52 4 2100 23
- 104.23: How many blocks 1 inch on each edge would it take to fill a shoe bo...
- 104.24: Three fourths of the 60 athletes played in the game. How many athle...
- 104.25: The distance a car travels can be found by multiplying the speed of...
- 104.26: Use the figure on the right toanswer ac.a. What is the area of the ...
- 104.27: Colby measured the circumference and diameter of fourcircles. Then ...
- 104.28: Hector was thinking of a two-digit counting number, and heasked Sim...
- 104.29: The coordinates of three vertices of a triangle are (3, 5), (1, 5),...
- 104.30: 2 gal1 4 qt1 gal 2 pt1 qt
Solutions for Chapter 104: Algebraic Addition Activity
Full solutions for Saxon Math, Course 1 | 1st Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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].
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
A sequence of steps intended to approach the desired solution.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
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.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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.