MTH 235 - Homework 10 Sections 5.1, 5.2, 5.4
Due FRIDAY, 4/26 by 4PM
Please answer the questions in the order they are listed, STAPLE the pages together and write clearly in the front of the assignment your NAME, SECTION,
and also the names of students yo
Chapter 4: Summary of Determinants
Here we list the main results and important properties of the determinant.
Basic properties:
1. det : Mnn (F ) F is multi-linear (n-linear) in row. That is, det is a linear function of
each row, while the others are held
Math 235: Linear Algebra
Midterm Exam 2
November 19, 2013
NAME (please print legibly):
Your University ID Number:
Please circle your professors name:
Friedmann
Tucker
The presence of calculators, cell phones, iPods and other electronic devices
at this ex
Math 235: Abstract Algebra
Midterm Exam 1
October 15, 2013
NAME (please print legibly):
Your University ID Number:
Please circle your professors name:
Friedmann
Tucker
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at this e
MTH 235 - Homework 11 Sections 6.1, 6.2, 6.4
Question 1. Let V be an inner product space.
(a) Prove that x + y
2
2
+ xy
2
=2 x
+2 y
2
holds for all x, y V.
Solution:
2
+ xy
2
=
x + y, x + y + x y, x y
=
x+y
x, x + x, y + y, x + y, y + x, x x, y y, x + y,
MTH 235 - Homework 8 Section 3.3, 3.4 & 4.1
Due Thursday, 4/11 by 5PM
Question 1. True of False?
Solution: (a) True Since Ax = 0 for A Mmn always has the trivial solution x = 0 Rn .
(b) False Consider the system of 2 equations in 3 unknowns x z = 0, y z =
MTH 235 - Homework 9 Chapter 4
Due Thursday, 4/18 by 5PM
Question 1. True of False?
(a) A square matrix A is invertible if and only if det(A) = 0. FALSE - a square matrix A is
invertible if and only if det(A) = 0.
(b) A square matrix A has linearly depend
MTH 235 - Homework 7 Section 2.5, 3.1
Due Thursday, 3/28 by 5PM
General advice: Use the notation developed in the course as much as possible to support
your calculations. That way it will be clear if errors correspond to simple numerical slips or
incomple
MTH 235 - Homework 6 Section 2.4-2.5
Due Thursday, 3/21 by 5PM
Question 1. True of False?
Solution: (a) True - by denition of invertibility.
(b) True - since IV IV = IV this follows from the denition of inverse map.
(c) False - If Onn is the n n zero matr
MTH 235 - Homework 5 Sections 2.12.3
Question 1.
Discussion: T is onto i R(T ) = W rank(T ) = dim(W ); T is one-to-one i N (T ) =
cfw_0 nullity(T ) = 0. So it natural to check what the Rank-Nullity Theorem gives us.
Solution: (a) T is onto i R(T ) = W i r
MTH 235 - Homework 4 Sections 2.12.2
Question 1.
Discussion: We know how T acts on (1, 2) and (1, 3), which incidentally form a basis for R2 ,
so we express (1, 0) as a linear combination of the two vectors and use the linearity of T . For
the second part
MTH 235 - Homework 9 Chapter 4
Due Thursday, 4/18 by 5PM
Please answer the questions in the order they are listed, STAPLE the pages together and write clearly in the front of the assignment your NAME, SECTION,
and also the names of students you collaborat
MTH 235 - Homework 7 Section 2.5, 3.1
Due Thursday, 3/28 by 5PM
Please answer the questions in the order they are listed, STAPLE the pages together and write clearly in the front of the assignment
Name
SECTION
Names of students you collaborated with.
Q
MTH 235 - Homework 6 Section 2.4
Due Thursday, 3/21 by 5PM
Please answer the questions in the order they are listed, STAPLE the pages together and write clearly in the front of the assignment
Name
SECTION
Names of students you collaborated with.
Questi
Math 235: Linear Algebra
Midterm Exam 1
October 21, 2014
NAB/IE (please print legibly): 30 IM IU‘V'S
Your University ID Number:
Please circle your professor’s name: Bobkova Friedmann
o The presence of calculators, cell phones, iPods and other electronic d
Math 235: Linear Algebra
Midterm Exam 2
November 20, 2014
NAME (please print legibly): “’15? P“ r i? ’71 QM“
Your University ID Number: g wig 53
Please circle your professor’s name: Bobkova Friedmann
o The presence of calculators, cell phones, iPods and o
Math 235: Linear Algebra
Midterm Exam 1
October 15, 2013
NAME (please print legibly):
Your University ID Number:
Please circle your professors name:
Friedmann
Tucker
The presence of calculators, cell phones, iPods and other electronic devices
at this exa
Math 235: Linear Algebra
Final Exam
May 5, 2014
NAME (please print legibly):
Student ID Number:
CIRCLE YOUR INSTRUCTOR:
Fatima Mahmood
Geordie Richards
Read all instructions and all problems carefully.
This is a closed-book and closed-notes exam. No pap
MTH 235 - Homework 5 Section 2.4
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. True of False? Fo
MTH 235 - Homework 5 Sections 2.22.3
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. In each of th
MTH 235 - Homework 11 Sections 5.1, 5.2, 5.4
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment your COURSE ID NUMBER and
also the names of students you collaborated with.
Q
MTH 235 - Homework 7 Sections 2.5, 3.1
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. True of Fal
MTH 235 - Homework 8 Sections 3.2 - 3.4
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. True of Fa
MTH 235 - Homework 10 Chapter 4
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment your COURSE ID NUMBER and
also the names of students you collaborated with.
Question 1. Tr
MTH 235 - Homework 9 Sections 4.1 - 4.3
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. You are gi
MTH 235 - Homework 1 Sections 1.21.3
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and write clearly in the front of the assignment
Name
Section
Names of students you collaborated with.
Question 1. For what values
MTH 235 - Homework 4 Section 2.1
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and write clearly in the front of the assignment
COURSE ID NUMBER
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Question 1. Suppose that T :
MTH 235 - Homework 3 Section 1.6
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and write clearly in the front of the assignment
COURSE ID NUMBER
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Question 1. W = cfw_(x1 , . .
PSC/IR 265
Problem Set Solutions
Testing the Robustness of the Bargaining Model of War
Solutions
1) Draws. Before, all wars ended in complete victory or complete
defeat. Suppose instead that R wins and takes the entire good with
probability pR , G wins an
MTH 235 - Homework 2 Sections 1.41.5
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and write clearly in the front of the assignment
Name
Section
Names of students you collaborated with.
Question 1. You are given a
In Homework 14, Part II. A (d), we observed that the eigenvalue with eigenvector ~v is real: hT(~v ), ~v i =
h~v , ~v i = h~v , ~v i, and that hT(~v ), ~v i = hT(~v ), ~v i. So, now suppose the field is complex. Then hT(~v ), ~v i =
h~v , ~v i = h~v , ~v
And many more if we rearrange the order of the Jordan Blocks, but these are the possible Jordan canonical
forms.
(c) If ker(A) and ker(A 2I) are both 2-dimensional, what is the Jordan canonical form of A?
Using Rank-Nullity Theorem, we use dim(ker(T I)d+1
Because 1 and 2 are distinct, 1 6= 2 but 1 h~v1 , ~v2 i = 2 h~v1 , ~v2 i. The statement can only be true if
h~v1 , ~v2 i = 0. Therefore, ~v1 and ~v2 are orthogonal.
(e) Show that V has an orthonormal basis consisting of eigenvectors of T
We use the Gram-S
(b) If J is a matrix in Jordan canonical form, show that J is similar to its transpose.
Now let Bi be the corresponding matrix to Ji such that Pi1 Ji Pi = JiT , with Ji being the corresponding
Jordan blocks to J the Jordan canonical form.
P1
J1
P2
J2
P =
MTH 235: Linear Algebra
Midterm 2
April 4, 2017
NAME (please print legibly):
Your University ID Number:
Circle your instructor:
Evan Dummit
Carl Mc Tague
The presence of electronic devices (including calculators), books, or formula
cards/sheets at this e