- 63.1: What is the difference between the sum of 0.6 and 0.4 andthe produc...
- 63.2: Mt. Whitney, the highest point in California, has an elevationof 14...
- 63.3: It was 39 outside at 1 p.m. By 7 p.m. the temperature haddropped 11...
- 63.4: Write the mixed number 4 23 as an improper fraction.
- 63.5: Round $678.25 to the nearest ten dollars. Describe how youdecided u...
- 63.6: What time is 2 12 hours after 10:15 a.m.? How did you find youranswer?
- 63.7: (30 15) (30 15)
- 63.8: Compare: 5823
- 63.9: w 323 112
- 63.10: 68 34
- 63.11: 614 558
- 63.12: 34 25
- 63.13: 34 25
- 63.14: (1 0.4)(1 + 0.4)
- 63.15: How much money is 60% of $45?
- 63.16: 0.4 8
- 63.17: 8 0.4
- 63.18: What is the next number in this sequence?0.2, 0.4, 0.6, 0.8,
- 63.19: What is the tenth prime number?
- 63.20: What is the perimeter of this rectangle?
- 63.21: A triangular prism has how many a. faces? b. edges? c. vertices?
- 63.22: Write 212 and 115 as improper fractions. Then multiply the improper...
- 63.23: This rectangle is divided into two congruentregions. What is the ar...
- 63.24: A ton is 2000 pounds. How many pounds is 2 12 tons?
- 63.25: Which arrow could be pointing to 0.2 on this number line?
- 63.26: The paper cup would not rollstraight. One end was 7 cm in diameter,...
- 63.27: Jefferson got a hit 30% of the 240 times he went to bat during thes...
- 63.28: Jena has run 11.5 miles of a 26.2-mile race. Find the remaining dis...
- 63.29: The sales-tax rate was 7%. The two CDs cost $15.49 each. What wasth...
- 63.30: Rosa is mixing paint in ceramics class. She mixes 12 teaspoon of ye...
Solutions for Chapter 63: Subtracting Mixed Numbers with Regrouping, Part 2
Full solutions for Saxon Math, Course 1 | 1st Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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 .
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
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).
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
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}.