- 62.1: In music there are whole notes, half notes, quarter notes, and eigh...
- 62.2: Don is 5 feet 212 inches tall. How many inches tall is that?
- 62.3: Which of these numbers is not a prime number? A 11 B 21 C 31 D 41
- 62.4: Write 1 13 and 1 12 as improper fractions, and multiply theimproper...
- 62.5: If the chance of rain is 20%, what is the chance that it will not r...
- 62.6: The prices for three pairs of skates were $36.25, $41.50, and $43.7...
- 62.7: Instead of dividing 15 by 2 1 2, Solomon doubled both numbersand th...
- 62.8: Find each unknown number:m 438 314
- 62.9: Find each unknown number:n 310 35
- 62.10: Find each unknown number:6d = 0.456
- 62.11: Find each unknown number:0.04w = 1.5
- 62.12: Find each unknown number:12 34 58
- 62.13: Find each unknown number:56 12
- 62.14: Find each unknown number:12 45
- 62.15: Find each unknown number:23 12
- 62.16: Find each unknown number:1 (0.2 0.03)
- 62.17: Find each unknown number:(0.14)(0.16)
- 62.18: One centimeter equals 10 millimeters. How many millimeters does2.5 ...
- 62.19: List all of the common factors of 18 and 24. Then circle the greate...
- 62.20: Ten marbles are in a bag. Four of the marbles are red.a. If one mar...
- 62.21: If the perimeter of a square is 40 mm, what is the area of thesquare?
- 62.22: At 6 a.m. the temperature was 6F. At noon the temperature was 14F.F...
- 62.23: Lisa used a compass to draw a circle with a radius of 1 12 inches. ...
- 62.24: The circle graph below shows the favorite sport of 100 people. Refe...
- 62.25: The circle graph below shows the favorite sport of 100 people. Refe...
- 62.26: The circle graph below shows the favorite sport of 100 people. Refe...
- 62.27: The circle graph below shows the favorite sport of 100 people. Refe...
- 62.28: What number is 40% of 200?
- 62.29: Here we show 18 written as a product of prime numbers:2 3 3 Write 2...
- 62.30: Judges awarded Sandra these scores for her performance onthe vault:...
Solutions for Chapter 62: Writing Mixed Numbers as Improper Fractions
Full solutions for Saxon Math, Course 1 | 1st Edition
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!
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of 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).
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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 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.
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.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.