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
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 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 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 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 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 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 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 55 4/21 DISCUSSION QUESTIONS
1.
(a)
(b)
(c)
(d)
(e)
How many relations are there on a set A = cfw_a, b with two elements?
Of these relations, how many are reexive?
How many are symmetric?
How many are transitive?
How many are equivalence relations?
S
MATH 55 4/14 DISCUSSION QUESTIONS
1. If G(x) is the generating function for the sequence cfw_ak , what is the generating function for
each of the following sequences?
(a) 0, 0, 0, a3 , a4 , a5 , .
(b) a0 , 0, a1 , 0, a2 , 0, .
(c) 0, 0, 0, a0 , a1 , a2 ,
MATH 54 FINAL EXAM PRACTICE QUESTIONS
1. Let M be an m n matrix. In terms of the pivots of M , how do we tell if the matrix equation
M x = b always has at least one solution? At most one solution?
2. T/F: In some cases, it is possible for 4 vectors to spa
MATH 54 4/4 QUIZ
There is only one problem on this quiz.
1. Find all dierential functions y satisfying the following conditions:
y 4y + 4y = 0
y (0) = 1
y (1) = 0
Solution: The general solution to the given homogeneous equation is
ygen = Ae2x + Bxe2x .
Pl
MATH 54 4/11 QUIZ
1. Find a general solution to the following dierential equation:
y y = e2t + tet
1
2
MATH 54 4/11 QUIZ
2. Suppose that yp and yq are two solutions to the dierential equation
y + 2y + y = tan3 (x).
Suppose further that
yp (0) = yp (0) = 1
MATH 54 3/7 QUIZ
You have 20 minutes to complete this quiz. No notes or books are allowed.
1.
Let P2 be (as usual) the vector space of polynomials of degree 2. Let D : P2 P2 be the linear
transformation sending p(t) to p (t).
(a) Find the matrix of D rela
MATH 54 3/14 QUIZ
1. Which of the following are orthogonal matrices? Circle all that apply.
0 1/2
a.
0 1/ 2
b.
0 1
1 0
c.
1 2
2 1
d.
1 0
0 1
Solution: Only b and d are orthogonal matrices. A matrix has to have orthonormal columns to be
an orthogonal matri
MATH 54 MIDTERM 1 REVIEW
1. Make sure you review the denitions of the following terms.
(1) augmented matrix vs. coecient matrix
(2) echelon form and reduced echelon form
(3) pivots, pivot rows, pivot columns
(4) parametric form of solution set to Ax = b
(
MATH 54 2/7/2013 QUIZ
You have 15 minutes to complete the following quiz. Notice that there are problems on each
side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more
sophisticated. It really isnt necessary, though.
MATH 54 2/14/2013 DEFINITIONS QUIZ
1. Below is shown a matrix A, together with the reduced
[A|I].
1 4 2 1 0 0
1
[A|I] = 1 3 3 0 1 0 0
3 6 12 0 0 1
0
echelon form of the augmented matrix
0 6 3 4 0
1 1 1 1 0
0 0 3 6 1
Based on this calculation, answer the
MATH 54 QUIZ 2/28
1. Let P2 be the set of polynomials of degree 2. Find the change-of-basis matrix P from the
CB
basis B = cfw_1, t, t2 to the basis C = cfw_1 2t + t2 , 3 5t + 4t2 , 2t + 3t2 .
Solution: The matrix P has as columns the vectors [bi ]C . To
MATH 54 1/31/2013 QUIZ SOLUTIONS
You have 15 minutes to complete the following quiz. Notice that there are two problems, one on
each side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more
sophisticated. It really isn
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 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 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 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 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 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 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 ),