

* indicates the problem will be collected 


827  Review
SO: 1.1 1.4 SO: 2.1 2.9 SO: 3.1 3.9 LA: Preface to the Student 
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 

829  Complex Numbers
LA: pp 23. SO: 1.7 
SO 1. 66, *68
LA 1. *1, *2 

93  Fields (Definition and examples) 
Discuss briefly how you determined your response. Verify that Q[sqr(2)] is a field. Show that a^{2} + a + 1 is not 0. 
Find all invertible matrices 3 by 3 matrices with entries only 0 or 1.  
95  LA: pp 410
SO: 4.1,4.2, 4.3 
SO: 4. 72, *74, *76  
98  LA:1114
SO:4.1,4.5 
SO: 4. 77, *78, *80, *81a, 82
LA:: *4, *6 

912  LA:1418
SO:4.10 
LA: *8,10,*13,*15
SO: 4. *118, 124 

915  LA:2127
SO: 4.7 
LA: 1: *11, *14
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 #Y (the direct sum). First partnership summary of work through 912 is due by 5 pm. 

919 
LA: 2129 SO: 4.4, 4.7, 4.8 
LA: 2: *1,*2, *5, SO: 4. 83,84, 86, 89, *90a, *91, 92 

922 
LA: 2129 
LA: 2: *8,*9, *12, 13,14, 17 SO: 4: 97 a, 99a, *101, 103 

926 
LA: 2734; 3841 SO: 4.8, 4.9, 4.11 
LA : 2: *17 SO: 4.: 104a, 107, 110a, *128, *132 

929 
LA: 3841 SO: 5.2, 5.3 
LA: 3: 1,*2 SO: 5. 45, 47, *49a, *51, *60 *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. 

103 
LA:4147 SO: 5.4, 5.6, 
LA: 3: 3, *4, *5 SO: 5. 56, 60, *70,*71, *72a 

1010 
LA: 4447, 5357 SO: 5.4, 5.5, 5.6 
LA: 3: *7, 8, *9, 14, 22, 23 SO: *64, *65, 69, 75a,b, 76 a,b, *83a *1. 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). *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, the z+W = z'+W if and only if z'  z is a vector in W. 

1013 
LA:48 58 SOS: 5.2, 5.7, 6.2,6.5 
LA:3: *10, 20, 21,*23,24 SO: 5. *88, 89; 6. *37b, *62 

113 
LA: 6471, 76 83,8790 179182 SOS: 9.1,9.2, 9.7(in part) 
LA: 4: 1, *2, *4; LA: 5: *5, *8, SO: 9. *41a, 46, 49a 

1112 
Continue with previous assignment readings plus SOS:10.3, 10.4 
LA : 5: 9, *10, *12, 15, 16 LA: 8. 21,*22 SO: 9. 52, 53 SO:10. *36, 38 * Complete the two parts of the proof of the lemma from 115 about the polynomial g. 

1119 
LA: 186187, 97116 SOS:: 7.2, 7.3, 7.5, 7.6, 7.7 
LA: 8.*30, *31 LA: 6. *4, *6 , 9 SOS: 7. 57, 59, 64, *72, *75 

123 
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 

1210 
TBA 
SOS: 8. 39, 41, *60, *69 LA: 7. *21 LA: 10 . *20, *21 