Assignment Number 
Date Due 

* indicates the problem will be collected ** indicates problem to be presented by partnerships 



#0 
824 
Preface to course: LA: Preface to the Instructor! LA: A.1; A.3 ; A.4 

#1 
829 
Review
SO: 1.1 1.4 SO: 2.1 2.9 SO: 3.1 3.9 LA: 1.1.11.2.5 LA: 2.22.4 
SO 1. 41(ad),
*45, *48a SO 2. 37  40; *41(a,b), 51, 54 (A , B), *62a SO 3. 51(a,b), *53a, 61b, 63 LA: 1.1:5, 6 ; 1.2: 23 (ae) LA: 2.4: 5, 16, 20 (a,b), 21b, 22a 

#2 
831 
Complex
Numbers LA: 2.1; SO: 1.7 SO: p 112 notations. 
LA: 2.1:8 SO 1. 66, *68 

97(changed
from 92) 
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.  
Summary  912  *Partnership Summary #1 of work till 99 1 page 2 sides or 2 pages 1 side One submission per partnership. 

#3 
916 (Changed 912) 
SO: 4.1,4.2, 4.3, 4.5 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 

#4 
921 
SO:4.1,4.5 SO:4.10 
SO: 4. 77, *78, */**80, *81b,
82 

#5 
923 
SO:4.10  SO: 4. *118, 124  
#6 
9 26 
SO:
4.10 
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). 

#7 
103 (Changed 928) 
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 

Summary #2 
105 
*Partnership Summary #2
of work till 930 1 page 2 sides or 2 pages 1 side One submission per partnership. 

#8 
105 
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.  
#9 
1014 
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 

#10 
1019 
SO: 5.4, 5.6  SO: 5. 56, *70,*71, *72a 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}. Suppose z and z' are vectors in V. */** Prove: z+W = z'+W if and only if z'  z is a vector in W. 

Summary #3 
1019 
*Partnership Summary #3
of work till 1014 1 page 2 sides or 2 pages 1 side One submission per partnership. 

#11 
1021 
SO: 5.4, 5.5, 5.6  SO: 5. *64, *65,
69, 75a,b, 76 a,b, *83a */** Suppose V is a vector space and P: V > V is a linear operator where PP = P. Prove: (i) If w is in R(P) then P(w) = w. (ii) for any v in V, vP(v) is in N(P). 

MIDTERM EXAMINATION 
Selfscheduled Between Monday, 1024, 3:00pm and Tuesday, 1025 8pm . See Prof. Flashman for appointment or sign up through Moodle. 
This Exam will cover
material through Assignment 11. 

#12 
112 
SOS: 5.2, 5.7, 6.2,6.5 LA: 5.35.5 
SO: 5. */**87, 88 (should refer to
87); 6. 7, 8,11,*39b, */**68, */**69 

Summary #4 
114 
*Partnership Summary #4
of work till 1031 1 page 2 sides or 2 pages 1 side One submission per partnership. 

#13 
119 
SOS: 9.1,9.2, 9.7(in part) LA: 6.1, 6.4 
SO: 9. *41a, */**42, 46 (with
solutions), 49a (With solution), *53, *54 LA: */** 5.5.14 

#14 
122 
Continue with previous assignment
readings plus SOS:10.3, 10.4 
SO: 9. 52, SO:10. *36, *38 

Future Assignments 

* Complete the two parts of the proof of the lemma from 115 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 