5 oh; h'o v1.5
MAS 3105 Test 2 Matrix and Linear Algebra Fall 2014
5 1. (a) Show that if A is an invertible matrix, then detA‘1 = dei A.
l O (b) Using part (a), ﬁnd the detA“ where
—3—21—4
130—3
A‘ —3 4—2 8
3—404
If using row reduction, specify the effe
Abdominal cavity starts as high as the 4th intercostal space
Upper and lower parts of the abdomen are enclosed by two bony rings (the lower
margin of the rib cage superiorly, and the pelvis inferiorly)
The flexible lumbar vertebrae are located in the mid
Some Thoughts on How to Understand Proofs and to Write Them
Most people find it difficult to learn what constitutes a valid proof. Often ability in this
area comes about by some form of osmosis in which eventually it either sinks in or
it doesnt! I believ
General Structure of a Theorem
The purpose of a theorem is to present a mathematical truth. Of course, just stating that something is
true does not make it so. Mathematics history is littered with theorems that later were discovered to be
false. What that
MAS 3106 Linear Algebra
Problem Set 13 Not to be handed in
1.
Page 336, Problem 1
2.
Page 336, Problem 3
3.
Page 336, Problem 5
4.
Page 337, Problem 6 (parts (b), (c), (d), and (e) of the theorem)
5.
Page 337, Problem 12
6.
Page 338, Problem 17
7.
Page 35
MAS 3106 Linear Algebra
Problem Set 12 Due 4/14/11
1.
Page 279, Problem 1abcdefg
2.
Page 280, Problem 5 (just state the indicated result; do not prove it)
3.
Page 280, Problem 8
4.
Page 281, Problem 10
5.
Page 281, Problem 12
6.
Page 281, Problem 13
7.
Pa
MAS 3106 Linear Algebra
Problem Set 11 Due 4/7/11
1.
Page 256, Problem 1
2.
Page 257, Problem 3b
3.
Page 257, Problem 4f
4.
Page 257, Problem 8 (For c, just state the result; dont prove it)
5.
Page 258, Problem 11 (for part c, use parts a and b, not the m
MAS 3106 Linear Algebra
Problem Set 10 Due 3/31/11
1.
Page 228, Problem 1
2.
Page 228, Problem 5
3.
Page 229, Problem 10
4.
Page 229, Problem 11
5.
Page 229, Problem 12
6.
Page 229, Problem 13
7.
Page 229, Problem 15
8.
Page 229, Problem 18 (Type 3 only)
MAS 3106 Linear Algebra
Problem Set 9 Due 3/24/11
1.
Page 220, Problem 1
2.
Page 221, Problem 2, 3, 4
3.
Page 221, Problem 9
4.
Page 221, Problem 12
5.
Page 222, Problem 13
6.
Page 222, Problem 21
7.
Page 222, Problem 25
8.
Page 222, Problem 26
9.
Page 22
MAS 3106 Linear Algebra
Problem Set 8 Due 3/10/11
1.
Page 179, Problem 1
2.
Page 179, Problem 2g, 3g (find a different solution to the nonhomogeneous system from the one
given in the answer on page 576)
3.
Page 180, Problem 4b
4.
Page 180, Problem 7ad
5.
MAS 3106 Linear Algebra
Problem Set 7 Due 3/3/11
1.
Page 151, Problem 1
2.
Page 151, Problem 2
3.
Page 151, Problem 3c
4.
Page 151, Problem 7 (only for Type 3 column operation)
5.
Page 152, Problem 9
6.
Page 165, Problem 1
7.
Page 165, Problem 2f
8.
Page
MAS 3106 Linear Algebra
Problem Set 6 Due 2/24/11
Although they are not assigned, please look at problems 4 and 10 on Page 107 and problems 17
and 20 on page 108. They may be needed later in the course.
1.
Page 106, Problem 1
2.
Page 106, Problem 2df
3.
P
MAS 3106 Linear Algebra
Problem Set 5 Due 2/17/11
1.
Page 84, Problem 1
2.
Page 84, Problem 3
3.
Page 84, Problem 5ab
4.
Page 85, Problem 8
5.
Page 85, Problem 11 (read page 77 before Problem 28)
6.
Page 86, Problem 16 (assume T is onto)
7.
Page 96, Probl
MAS 3106 Linear Algebra
Problem Set 4 Due 2/10/11
1.
Page 74, Problem 1
2.
Page 75, Problem 6 (read before Problem 2)
3.
Page 75, Problem 9
4.
Page 75, Problem 10
5.
Page 75, Problem 12
6.
Page 75, Problem 13
7.
Page 75, Problem 14
8.
Page 76, Problem 18
MAS 3106 Linear Algebra
Problem Set 3 Due 2/3/11
1.
Page 40, Problem 1
2.
Page 41, Problem 3
3.
Page 41, Problem 8
4.
Page 41, Problem 9
5.
Page 42, Problem 13a
6.
Page 42, Problem 16
7.
Page 53, Problem 1
8.
Page 54, Problem 3ce
9.
Page 54, Problem 7
10.
MAS 3106 Linear Algebra
Problem Set 2 Due 1/27/11
1.
Page 19, Problem 1
2.
Page 20, Problem 5
3.
Page 20, Problem 8ae
4.
Page 21, Problem 13
5.
Page 21, Problem 19
6.
Page 21, Problem 20
7.
Page 22, Problem 24 (read definitions before Problem 23)
8.
Page
Problem Set 1 Due 1/20/11
1.
Page 6, Problem 7
2.
Page 553, Example 3. Show this is a field.
3.
Show Z 6 is not a field. What is required to show this?
4.
In a field F, define subtraction as on Page 554. If a, b, c F , prove a(b c ) ab ac
5.
Page 12, Prob
MAS 3106 Linear Algebra Spring 2011
Course Information Sheet and Syllabus
Text: Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence, Linear Algebra (Fourth Edition)
Optional reading: Daniel Solow, How to Read and Do Proofs (Fourth Edition)
Prere
Proving Theorems Using the Principle of Mathematical Induction
The Principle of Mathematical Induction deals with the problem of proving an infinite number of statements.
Obviously it is impossible to prove each one separately, so we need a method of prov
Structure and Proof of If and Only If Theorems
Form of the theorem statement
General hypotheses that hold throughout all parts of the theorem
Conclusion: A if and only if B (where A and B are statements)
This is really a concise way of stating two theorem
Theorem Proofs for Several Equivalent Conditions
In this type of situation the theorem statement is of the following form:
Hypotheses
Conclusion: Then the following are equivalent:
(a)
(b)
(c)
where the number of statements (a), (b), (c), is usually at le
Theorem Proofs Using the Method of Contradiction
As always, in one form or another, a theorem being proved boils down to the
following structure:
Hypotheses
Conclusion: Then Statement A
where A is the statement to be proven using the hypotheses and previo