Mat 3 4 4 - Fall 2016
Worksheet 1
2016/09/14
Throughout this worksheet, G = (V , E) is a finite directed graph, and (a, b) stands for (a , b). All paths are
assumed to be directed paths.
Recall the definition of a vertex basis for G from the textbook. (Go
Solutions to Worksheet 1
September 15, 2016
1
Exercise 1
(a)
R(h) = cfw_h
R(j) = cfw_j, i, h
R(i) = cfw_i, h
R(g) = cfw_g, h
R(f ) = cfw_f, g, i, h, b
R(b) = cfw_b, f, g, i, h
R(a) = cfw_a, b, f, g, i, h
R(c) = R(d) = R(e) = cfw_c, d, e, b, f, i, h, g
(b)
Mat 3 4 4 - Fall 2016
Worksheet 2
2016/09/28
Recall that for a plane graph depiction of a (connected) planar graph G, we define r to be the number of
enclosed regions in that drawing, plus 1 for the "unbounded region" outside the graph.
The letters e and
Mat 3 4 4 - Fall 2016
Worksheet 5
2016/11/23
For the first two questions, youll want to consult the identities in Section 6.2 to help with creating
series expressions for generating functions. (Yes, you will be given these identities on the last quiz.)
E
Mat 3 4 4 - Fall 2016
Worksheet 3
2016/10/26
E x e rc i s e 1 For following graph, each of your group members should pick a different vertex, and then
draw the graph as an undirected tree with that vertex as its "root". Once everyone is done, exchange pap
Mat 3 4 4 - Fall 2016
Worksheet 4
2016/11/9
E x e rc i s e 1 A committee of four people is to be formed from a set of six superheroes, seven supervillains,
and four ordinary citizens. How many ways are there to form the committee if:
(a) Ms. Marvel, aka C
Mat 3 4 4
Q uiz 2 - Part A
Feb 1st, 2017
Individual Work ( 10 Points Available)
Instructions
Please write your Name, Student Number, and Tutorial Number at the top of this page.
Questions
For the following questions, make sure to show as many steps of you
Mat 3 4 4
Quiz 1 - Part B
Jan 18th, 2017
G roup Work ( 10 Points Available)
Instructions
1. Please write your Group Number, Tutorial Number, and list the Names of Group Members Present.
2. Hand-in Only One Copy Per Group.
Questions
For the following quest
Mat 3 4 4
Quiz 2 - Part B
Feb 1st, 2017
G roup Work ( 10 Points Available)
Instructions
1. Please write your Group Number, Tutorial Number, and list the Names of Group Members Present.
2. Hand-in Only One Copy Per Group.
Questions
For the following questi
Mat 3 4 4
Q uiz 1 - Part A
Jan 18th, 2017
Individual Work ( 10 Points Available)
Instructions
Please write your Name, Student Number, and Tutorial Number at the top of this page.
Questions
For the following questions, make sure to show as many steps of yo
Worksheet 2
Mat 3 4 4
Jan 25, 2017
Cube Graphs
In this worksheet, were going to focus on a particular class of graphs called cube graphs. Well use them
as an example to test our understanding of some of the concepts weve covered up to now in the course,
l
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Engineering Strategies and Practices Lecture Notes
September 15 2015
Design Process:
-Involves critical thinking
-Managing information
-Reduces error (due diligence)
Obligation to deeply investigate the problem from multiple perspectives
Finding all relev
Mat224: Linear Algebra II
Due 26 October 2016
Assigment 3 of 6
This assignment is based on material from Chapters 8.1, 8.2 and 8.3 of the textbook. Note that
8.1 was covered in MAT 223 and so it is assumed that you are familiar with the material even
thou
Mat224: Linear Algebra II
Due 21 September 2016
Assigment 1 of 6
1. Show that the following are vector spaces over the specified fields.
(a) Rn over R
(b) R over Q
(c) The set Mm,n (R) of all m-by-n real matrices, over R
(d) cfw_(x, y) R2 |x + y = 0 over
Mat224: Linear Algebra II
Due 9 November 2016
Assigment 4 of 6
This assignment is based on material from Chapters 8.5, 8.6, and 9.1 of the textbook.
1 3
1
1. Consider the matrix A =
. Start with X0 =
, and using the power method, compute
0 2
1
X4 and r3
MAT224 Linear Algebra II
More Practice Problems
1. Exercises 2(d,e,6), 3c of 6.2
2. Example 9 of 6.4.
3. Why is the following statement true?
Given polynomials f1 (x), , fn+2 (x) of degree n, there are r1 , , rn+2 R such that
r1 f1 (x) + r2 f2 (x) + + rn+
University of Toronto Mississauga
MAT224 Test 2
July 27, 2016
No Aids Allowed
Duration: 80 minutes
Instructor: P. Eskandari
Solutions
First Name:
Last Name:
Student Number:
Instructions:
This test has 12 pages, including this front page. The test is out
MAT102F - Introduction to Mathematical Proofs - Fall 2015 - UTM
Quiz 4 (Version A) - November 19, 2015 - SOLUTIONS
1. Consider the following six sets
RQ
,
P (N)
,
(no explanation is needed for this question).
[ , ] Z
,
N cfw_0 , 1
,
(0 , )
,
(3 marks)
cfw
Mathematical and Computational Sciences
University of Toronto Mississauga
MAT232H5Y - Calculus of Several Variables - Summer 2014
Term Test 1 - June 18, 2014
Aids Permitted: None
Time Allotted: 90 minutes
Full Name:
Last
First
UTORid:
Your UofT E-mail:
In
Mathematical and Computational Sciences - University of Toronto Mississauga
MAT232H5Y - Calculus of Several Variables - Summer 2015
Term Test 1 - June 17, 2015
Aids Permitted: None
Time Allotted: 90 minutes
Full Name (IN PEN):
Last
First
Student ID (IN PE
Exercises in Solving Systems of Linear Equations
1. Solve the following system of linear equations. What is the rank of the system?
Verify that the number of parameters in the general solution is equal to the number
of variables minus the rank.
3x1 + 3x2
MAT232H5Y
Term Test 1 - June 2016
Last Name (Family name)
Page 9 of 9
First Name (Given name)
Student #
FORMULA SHEET
Trigonometry
cos(A + B) = cos A cos B sin A sin B
sin(A + B) = sin A cos B + cos A sin B
cos2 =
1 + cos(2)
2
sin
sin
4
6
= cos
= co
MAT232H5Y - Calculus of Several Variables - Summer 2016 - UTM
Problem Set C - TO BE SUBMITTED TO YOUR TA
Due:
The week of May 30, in your tutorial.
This assignment must be submitted to your TA at the beginning of the tutorial on the above date.
Marking