Assignment Number 


* indicates the problem will be collected 


825/27 
Preface to course: LA: Preface to the Instructor! LA: A.1; B.3 ; B.4 

#1 
93 
Review
SO: 1.1 1.4 SO: 2.1 2.9 SO: 3.1 3.9 LA: 1.1.11.2.5 LA: 2.12.4 
SO 1. 41(ad), *45, *48a
SO 2. 37  40; *41(a,b), 50, 53 (A , B), *61a SO 3. 51(a,b), *53a, 61b, 63 LA: 1.1:5, 6 ; 1.2: 23 (ae) LA: 2.1: 8; 2.4: 5, 16, 20 (a,b), 21b, 22a 

#2 
912 
Complex Numbers
LA: 2.1.3; SO: 1.7 SO: p 116 notations. 
SO 1. 66, *68


Fields
(Definition and examples) LA: pp 175178 LA: Appendix B.2B.4 
Discuss briefly how you determined your response. Verify that Q[sqr(2)] is a field. 
Find all invertible matrices 3 by 3 matrices with entries only 0 or 1.  
#3 
919/22 
SO: 4.1,4.2, 4.3, 4.5 , A.12(added 917) LA: 4.1, 4.2 
SO: 4. 72, *74, *76 LA: 4.1: 6,9 ; 4.2: 13, *6, *11, *12 *From Notes on Properties of Vector spaces

Suppose V is a vector space over the field F and U is a family of subspaces of V. [The family may have an infinite number of distinct subspaces for members.] Let W = {v in V : v is an element of every subspace that is a member of U,} Prove: W is a subspace of V 

Summary 
924 
*Partnership Summary #1 of work till 919 1 page 2 sides or 2 pages 1 side One submission per partnership. 

#4 
924/26? 
SO:4.1,4.5 SO:4.10 
SO: 4. 77, *78, *80, *81a, 82
SO: 4. *118, 124 

#4 
926 
SO: 4.7 
SO:4. * 119
*Show that C, the complex numbers, is a vector space over R, the real numbers, with subspaces X={a+0i: a in R} and Y= {0 + bi: b in R}. Show C = X `oplus` Y (the direct sum). 

#5 
106 
SO: 4.4, 4.7, 4.8 LA: 4.3, 4.4 
SO: 4. 83 ,84, 89, *90a, *91, 92 LA: 4.3: 10, *19, *33 SO: 4: 97 a, 99a, *101, 103 

#6 
1013 
SO: 4.8, 4.9, 4.11 LA: 4.4 
SO: 4.: 104a, 107, 110a, *128, *132 *1. Show that the dimension of C, the complex numbers, as a vector space over R is 2. *2. Suppose that V is a 3 dimensional vector space over Z_{2}. Prove that V has exactly 8 elements. 
Generalize the statement of problem 2 and prove your generalization is correct.  
#7 
1020 
SO: 5.2, 5.3 LA: 5.1, 5.2 
SO: 5. 45, 47, *49a, *51, *60 LA: 5.1: 7; 5.2: *7, 9, *10, *11, *16 

#8 
1031 
SO: 5.4,5.5, 5.6,6.2,6.5 LA: 5:2, 5.3, 5.5 
SO: 5. 56, *70,*71, *72a LA: 5.3: *7; 5.5: 13, 14 *1. Suppose V is a vector space and P: V > V is a linear operator where PP = P. Prove: (i) If w is in the range of P then P(w) = w. (ii) for any v in V, vP(v) is in the Null Space of P. *2. Suppose V is a vector space and W<V. For z a vector in V, let z+W = { v in V where v = z +w for some w in W}. Prove if z' is a vector in V, then z+W = z'+W if and only if z'  z is a vector in W. 

#9 
1119 
SOS: 5.2, 5.7, 9.1,9.2, 9.7(in part) LA: 6.1.1, 6.1.3; 6.2.2; theorem 6.2; 6.3 
SO: 5. *64, *65, 69, 75a,b, 76 a,b, *83a,*88 6. *37b, *62 9. *41a, 46, 49a LA: *6.1.8, *6.1.14, 6.4.3, 6.4.6 

#10 
125 
Continue with previous assignment readings plus SOS:9.4, 9.7,9.8 10.3, 10.4,10.5(?),10.6, 10.7 
SO: 9. 53,54 SO:10. *36, 38, *47 * Complete the two parts of the proof of the lemma from 1119 about the polynomial g. 

SOS:: 7.2, 7.3, 7.5, 7.6, 7.7 
SOS: 7. 57, 59, 64, *72, *75 

SOS:7.8 
*1.Suppose T is the matrix for a Markov Chain. Prove:Powers of T have real number entries that are never larger than 1. *2. Discuss in detail the long run behavior of the Markov chain with matrix:
SOS:7. *83, *86  
TBA 
SOS: 8. 39, 41, *60, *69 