## Martin Flashman's Courses MATH 344 Linear Algebra Fall, 2003 MWF 2:00-2:50 ROOM: SH128  Assignments  SO = Schaum's Outline Linear Algebra LA = Linear Algebra Done Right

No credit for answers without neatly organized work!
Optional Problems will be graded for bonus assignment points.
Due Date
Problems
* indicates the problem will be collected
Optional
8-27 Review
SO: 1.1- 1.4
SO: 2.1- 2.9
SO: 3.1- 3.9
LA: Preface to the Student
SO 1. 41(a-d), *45,  *48a
SO 2. 37 - 40; *41(a,b), 50, 53 (A , B), *61a
SO 3. 51(a,b), *53a, 61b, 63

8-29 Complex Numbers
LA: pp 2-3.
SO: 1.7
SO 1.  66, *68
LA 1.  *1, *2

9-3   Fields (Definition and examples)
1. *Find all invertible matrices 2 by 2 with entries only 0 or 1.

2. Discuss briefly how you determined your response.
3.  *Let Q[sqr(2)] ={x : x = a + b*sqr(2) where a and b are rational numbers}.

4. Verify that Q[sqr(2)] is a field.
5. *Explain why Z4 = {0,1,2,3} with + and * given by "mod 4" arithmetic is not a field.
6. *Suppose a is an element of Z2

7. Show that  a2 + a + 1 is not 0.
8. *Suppose F is a field with exactly 4 elements
1. Show that 1+1 must equal 0.
2. Show that if a is not 0 or 1, then a2 must be a +1.

Find all invertible matrices 3 by 3 matrices with entries only 0 or 1.
9-5 LA: pp 4-10
SO: 4.1,4.2, 4.3
SO: 4. 72, *74, *76
9-8 LA:11-14
SO:4.1,4.5
SO: 4.  77, *78, *80, *81a, 82
LA:: *4, *6

9-12 LA:14-18
SO:4.10
LA: *8,10,*13,*15
SO: 4.  *118, 124

9-15 LA:21-27
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 9-12 is due by 5 pm.

9-19
LA: 21-29
SO: 4.4,  4.7, 4.8
LA: 2:  *1,*2, *5,
SO:  4.  83,84, 86,  89,  *90a,  *91,  92

9-22
LA: 21-29
LA: 2: *8,*9, *12, 13,14, 17
SO:  4:  97 a, 99a,  *101, 103

9-26
LA:  27-34; 38-41
SO: 4.8, 4.9, 4.11

LA : 2: *17
SO: 4.:  104a, 107, 110a, *128, *132

9-29
LA: 38-41
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
Z2.  Prove that V has exactly 8 elements.

Generalize the statement of problem 2 and prove your generalization is correct.
10-3
LA:41-47
SO:  5.4, 5.6,
LA: 3: 3, *4, *5
SO: 5. 56, 60, *70,*71, *72a

10-10
LA: 44-47, 53-57
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, v-P(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.

10-13
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

11-3
LA: 64-71, 76- 83,87-90 179-182
SOS: 9.1,9.2, 9.7(in part)
LA:  4: 1, *2, *4;
LA: 5: *5, *8,
SO:  9. *41a,  46, 49a

11-12
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 11-5 about the polynomial g.

11-19
LA: 186-187,  97-116
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

12-3
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:
T=(
 0 1/2 1/3 1/4 0 2/3 3/4 1/2 0

)
In particular: Why is this a regular Markov Process? If the initial distribution is (x,y,z) with x+y+z = 100, give an estimate for the distribution after 100 steps. Explain how you made your estimate using the theory of regular Markov chains.
SOS:7.  *83, *86
12-10
TBA
SOS:  8. 39, 41, *60, *69
LA: 7. *21
LA: 10 . *20, *21