## Topics for "partnership" presentationSO = Schaum's Outline Linear Algebra LA = The Keys to Linear Algebra

No credit for answers without neatly organized work!
Optional Problems will be graded for bonus assignment points.
Assignment Number
Date Due
Problems
* indicates the problem will be collected
** indicates problem to be presented by partnerships
Optional
#0
8-24
Preface to course:
LA: Preface to the Instructor!
LA: A.1; A.3 ; A.4

#1
8-29
Review
SO: 1.1- 1.4
SO: 2.1- 2.9
SO: 3.1- 3.9
LA: 1.1.1-1.2.5
LA:  2.2-2.4
SO 1. 41(a-d), *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 (a-e)
LA: 2.4: 5, 16, 20 (a,b), 21b,  22a

#2
8-31
Complex Numbers
LA: 2.1;
SO: 1.7
SO: p 112 notations.
LA: 2.1:8
SO 1.  66, *68

9-7(changed from 9-2)
Fields (Definition and examples)
LA: pp 175-178
LA: Appendix B.2-B.4
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.
Summary 9-12
*Partnership Summary #1 of work till 9-9
1 page 2 sides or 2 pages 1 side
One submission per partnership.

#3
9-16 (Changed 9-12)

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
• Prove: a*0 = 0.
• Prove: (-1)v = -v.
• Prove: If v+z = w + z then v=w. [cancellation]
• Prove: -v is unique.

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
9-21
SO:4.1,4.5
SO:4.10
SO: 4.  77, *78, */**80, *81b, 82

#5
9-23
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
10-3 (Changed 9-28)
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
10-5

*Partnership Summary #2 of work till 9-30
1 page 2 sides or 2 pages 1 side
One submission per partnership.

#8
10-5
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
Z2.  Prove that V has exactly 8 elements.
Generalize the statement of problem 2 and prove your generalization is correct.
#9
10-14
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
10-19
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
10-19

*Partnership Summary #3 of work till 10-14
1 page 2 sides or 2 pages 1 side
One submission per partnership.

#11
10-21
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, v-P(v) is in N(P).

MIDTERM EXAMINATION
Self-scheduled
Between Monday, 10-24, 3:00pm and Tuesday, 10-25 8pm .

This Exam will cover material through Assignment 11.

#12
11-2
SOS: 5.2, 5.7, 6.2,6.5
LA: 5.3-5.5
SO: 5. */**87, 88 (should refer to 87);  6. 7, 8,11,*39b, */**68, */**69

Summary #4
11-4

*Partnership Summary #4 of work till 10-31
1 page 2 sides or 2 pages 1 side
One submission per partnership.

#13
11-9
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
12-2
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 11-5 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:
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

TBA
SOS:  8. 39, 41, *60, *69

## Topics for "partnership" presentation:

• Linear Programming :
• Simplex Method
• Duality
• Linear Codes [Error Correcting Codes]
• Computer Graphing
• Homogeneous coordinates
• Projective geometry