Math 110
PROFESSOR KENNETH A. RIBET
C
Final Exam December 12, 2002
12:303:30 PM
The scalar field F will be the field of real numbers unless otherwise specified. Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, P
MATH 110 SUPPLEMENTARY MATERIAL: POLYNOMIALS
If you are interested in working over an arbitrary eld, then Denitions 2.11 and 2.12 in
your book arent quite the right denitions for the vector space of polynomials over F. In this
handout well give a denition
MATH 110 MIDTERM 2
JULY 31, 2014
NAME:
You are allowed one 2-sided sheet of notes on this midterm. Please clear your desk of
all materials except for this sheet of notes, writing utensils, and scratch paper. You have 65
minutes to complete the test. Answe
MATH 110 HWK 1 SOLUTIONS
Solve the following problems. Prove all assertions. For each problem, you may use any of the results
in Chapter 1 and 2.A, the Fields supplementary reading handout, or previously solved homework problems
without proof.
1. Let F be
MATH 110 HWK 6: DUE WEDNESDAY, JULY 30, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2, 3, 4, 5, 6, any of the handouts on the website, or previously solved homework problems with
MATH 110 HWK 2: DUE THURSDAY, JULY 3, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2 and 3.A, the Fields and Polynomials supplementary reading handouts, or previously solved
homew
MATH 110 HWK 4: DUE WEDNESDAY, JULY 16, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2, 3, 4, 5.A, any of the handouts on the website, or previously solved homework problems witho
MATH 110 HWK 5: DUE WEDNESDAY, JULY 23, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2, 3, 4, 5, any of the handouts on the website, or previously solved homework problems without
MATH 110 HWK 3 SOLUTIONS
Solve the following problems. Prove all assertions. For each problem, you may use any of the results in
Chapters 1, 2 and 3.A-D, the Fields and Polynomials supplementary reading handouts, or previously solved
homework problems wit
MATH 110 HWK 7: DUE WEDNESDAY, AUGUST 6, 2014
Solve the following problems. Prove all assertions. For each problem, you may use any
of the results in Chapters 1, 2, 3, 4, 5, 6, 7.A-7.B, any of the handouts on the website, or
previously solved homework pro
MATH 110 MIDTERM 2 REVIEW PROBLEMS
The midterm will cover Section 3.E, and Chapters 5-6. Below are given some midterm-level
problems to give you an idea of what to expect.
1. Determine whether the following statements are true or false.
(a) If a linear op
MIDTERM 1 REVIEW SOLUTIONS
1.Make sure you know the denitions of the following terms.
eld
vector space over F
subspace
sum and direct sum
linear combination
span, linear dependence vs. linear independence
nite vs. innite dimensional vector space
b
Math 110 Midterm Exam
Professor K. A. Ribet October 31, 2002
Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Please write yo
MATH 110 CHAPTER 1 AND 2 PRACTICE PROBLEMS: SOLUTIONS
Below are some brief answers to the practice problems for Chapters 1 and 2. These solutions are only meant for you to check if you are on the right track. Several of the answers
would not receive full
MATH 110 FINAL REVIEW PROBLEMS
The nal will cover the entire class, but at least half of the questions will cover Chapters
7 and 8. Here are some sample problems to give you an idea of the type of coverage.
1. Determine whether the following statements ar
MATH 110 SUPPLEMENTARY MATERIAL: FIELDS
1. The definition of a field
Before we introduce abstract vector spaces, were going to introduce the notion of a eld.
Basically, a eld is a set of objects where weve dened how to add, subtract, multiply, and
divide
MATH 110 CHAPTER 5 REVIEW PROBLEMS
1. Let M be the matrix
4
0
M =
0
0
2
1
0
0
2
7
6
4
3
0
.
0
0
Find 6 distinct M -invariant subspaces of R4 .
Solution: In addition to cfw_0 and R4 , the spans of the following lists of vectors are M invariant: (v1 ),
MATH 110 SUPPLEMENTARY MATERIAL: SUMMARY OF THE RELATION
BETWEEN LINEAR MAPS AND MATRICES
In this handout I will summarize the propositions relating linear maps between nitedimensional matrices to matrices. There will be no proofs or examples in this hand
MATH 110 SUPPLEMENTARY MATERIAL: CHANGE OF BASIS
Let V and W be nite-dimensional vector spaces over a eld F, and T : V W a
linear transformation. In this handout we will study what matrices can occur as MC (T ), for
B
varying bases B of V and C of W . Fir
MATH 110 MIDTERM 1
JULY 10, 2014
NAME:
You are allowed one 2-sided sheet of notes on this midterm. Please clear your desk of
all materials except for this sheet of notes, writing utensils, and scratch paper. You have 65
minutes to complete the test. Answe
MATH 110 QUIZ 5
AUGUST 7, 2014
NAME:
No notes or books are allowed on the following quiz. Please justify all of your answers
unless indicated otherwise.
1. Consider C3 with the standard inner product. Determine whether each of the following linear operato