University of Ottawa
Department of Mathematics and Statistics
MAT 1302: Mathematical Methods II
Instructor: Alistair Savage
Final Exam
23 April 2012
Surname
First Name
Student #
Instructions:
(a) You have 3 hours to complete this exam.
(b) This exam consi
University of Ottawa
Department of Mathematics and Statistics
MAT 1302: Mathematical Methods II
Instructor: Alistair Savage
Final Exam
23 April 2012
Surname
First Name
Student #
Instructions:
(a) You have 3 hours to complete this exam.
(b) This exam consi
University of Ottawa
Department of Mathematics and Statistics
MAT 1302A: Mathematical Methods II
Instructor: Alistair Savage
Final Exam
April 2013
Surname
First Name
Student #
Seat #
Instructions:
(a) You have 3 hours to complete this exam.
(b) This exam
MAT 1302  Mathematical Methods II
Alistair Savage
Mathematics and Statistics
University of Ottawa
Winter 2010 Lecture 18
Alistair Savage (uOttawa)
MAT 1302  Mathematical Methods II
Winter 2010 Lecture 18
1 / 30
Announcements
Third Assignment
Solutions n
University of Ottawa
Department of Mathematics and Statistics
MAT 1302: Mathematical Methods II
Instructor: Alistair Savage
Final Exam
23 April 2012
Surname
First Name
Student #
Instructions:
(a) You have 3 hours to complete this exam.
(b) This exam consi
University of Ottawa
Department of Mathematics and Statistics Math 1302A: Mathematical Methods II Instructor: Alistair Savage Final Exam April 15, 2008 Solutions
Surname Student #
First Name
Instructions: (1) You have 180 minutes to complete this exam. (2
University of Ottawa
Department of Mathematics and Statistics
Math 1302A: Mathematical Methods II
Instructor: Alistair Savage
Final Exam April 15, 2008
Surname
First Name
Student #
Instructions:
(1) You have 180 minutes to complete this exam.
(2) This exa
University of Ottawa
Department of Mathematics and Statistics
MAT 1302A: Mathematical Methods II
Instructor: Alistair Savage
Third Midterm Test Solutions White Version
26 March 2013
Surname
First Name
Student #
DGD (14)
Instructions:
(a) You have 80 minut
University of Ottawa
Department of Mathematics and Statistics
MAT 1302A: Mathematical Methods II
Professor: Aziz Khanchi
Final Exam Solutions
April 2010
Surname
First Name
Student #
Seat #
Instructions:
(a) You have 3 hours to complete this exam.
(b) This
MAT 1302 D FINAL EXAM April, 12th 2011
Instructor: Termeh Kousha
NAME
I.D.#
Duration: 3 hours
NO CALCULATORS. NO BOOKS. NO NOTES.
Instructions: This exam consists of 10 multiple choice questions and 8 Long answer questions on 16 pages. The marks for each
1
1.
Find all k for which the following system has infinitely many solutions.
x + 2y +
z=0
x + 3y + 2kz = 0
2x + 3y + kz = 0
A.
B.
C.
D.
E.
F.
0
1/2
3
3
1
1
2.
The coefficient matrix A in a homogeneous system of 12 equations in 16 unknowns
is known to h
What does dimension really mean ?
Intuition
A point is zero dimensional.
A line is one dimensional.
A plane is two dimensional.
z
b
y
x
zero dimensional
one dimensional
two dimensional
Dimension means.
Number of independent parameters.
Number of independ
Review : Powers of a matrix
Given a square matrix A and a positive integer k, we define
Ak = AA A
 cfw_z
k times
Note that the multiplications AA, AAA, . make sense.
Example.
"
1
0
"
1
A3 = (A2 )A =
0
"
1
A4 = (A3 )A =
0
Suppose A =
#
"
#"
1
1 1 1
. Th
Third Midterm Exam
The exam is given on March 24 in class.
You should know your DGD number.
The exam is based on Lectures 116 (including today).
1 / 27
Recall : definition of determinant
What is det A?
Suppose A is an n n matrix.
The determinant of A is a
Review (Matrix Product : Special Case)
Suppose A =
h
a1
a2
i
an is a 1 n matrix and v =
vector.Then the product Av is defind as follows :
Av =
h
a1
a2
i
an
b1
b2
.
.
bn
b1
b2
.
.
bn
is an n 1
i
h
= a b + a b + + a b
1 1
2 2
n n
The result is a 1 1 m
A quick introduction
The focus of this course is on an area of mathematics called Linear
Algebra
The basic idea of linear algebra is to solve systems of linear equations.
systems of linear equations
matrices
Matrices appear in:
Mathematical modelling of e
Review
Linear combinations
Suppose we are given vectors v1 , v2 , ., vp in Rn , and scalars c1 , ., cp .
The vector
w = c1 v1 + c2 v2 + + cp vp
is called a linear combination of v1 , ., vp with weights c1 , ., cp .
Spancfw_v1 , ., vp
The set of all line
Matrix Multiplication
We want to define the product of two matrices.
We have defined the product of an m n matrix and an n 1 matrix.
If A is an m n matrix and B is an n p matrix, then the product AB is
defined and is an m p matrix.
We write
B = [ b1 b2 b
Midterm
The exam is given on February 3 in class.
You should arrive on time. No extra time will be given.
Calculators, formula sheets, and electronic devices of any kind
are not allowed.
You should know your DGD number. You need to write it on the
exam.
B
Subspaces of Rn
A set H of vectors in Rn is called a subspace if :
1
The zero vector 0 is in H.
2
For every two vectors u and v in H, the sum u + v is also in H.
3
For each vector u in H and every scalar c, the vector cu is also in H.
Basis of a subspace
Recall : definition of determinant
What is det A?
Suppose A is an n n matrix.
The determinant of A is a number which can be calculated by cofactor
expansion across a row or down a column.
The main property of det A is that A is invertible if and only if d
Recall : Eigenvalues and Eigenvectors
Let us start with a simple example.
"
0
Problem. Consider the matrix A =
2
do to vectors in R2 ?
#
1
. What does multiplication by A
3
x 7 Ax
"
0
2
"
0
2
"
0
2
"
0
2
Ax
4
3
x
2
Ax
1
Ax
x x
Ax
0
1
2
x
#" # " #
1
1 2
Recall : Systems of Linear Equations
A system of linear equations is a collection of equations in
variables x1 , ., xn of the form
a1 x1 + + an xn = b.
Example.
x1
x1
+x2
x2
x3
+x3
+2x3
=1
= 1
=0
We are interested in finding the solution set of a system o
Complex numbers
A complex number is a number written in the form
a + bi
where :
a and b are ordinary real numbers,
i is the imaginary number satisfying i2 = 1.
Let z = a + bi. Then :
a is called the real part of z and is represented by Re z
b is called th
Abbas Ghandour
Leaving Boys Behind
In the study Leaving Boys Behind: Gender Disparities in High Academic
Achievement, the study tries to explain how males are being left behind in the
academic rise, in which females are growing in. The study measures the
Abbas Ghandour
Research  Islamic Food
d)
Dates Prophet Muhammed used to eat dates
 Sunna A good thing to do
Slaughter Kill by an incision in the neck
 Least painful way to kill the animal
Pork Pigs = dirty animals
 Bacteria filled food = not good
 Ma
1
Vous commencez votre vie esprant vivre une existence remplisse doptions de rves et
daventures. Malheureusement pour les personnes atteintes dune maladie telle que linfirmit
motrice crbrale la naissance ces rves sont crass. Dans le livre La Traille, par
Abbas Ghandour
LOrdinateur formidable
Cette annonce publicitaire est pour vendre le Lenovo Y50, aussi connu sous le nom
LOrdinateur formidable . Je vends mon Lenovo Y50 pour $700. Les propritaires auxquels je
le vends doivent aimer la technologie de qual
Sewol Ferry Disaster
Abbas Ghandour
Mr. Moon
HSB4M
September 15, 2015
1. Malcolm Gladwell argues that the Korean tradition to respect your elders may have
caused the ferry to tip. The teaching of how to control planes in Korea nowadays is very
different c
MAT1302F Mathematical Methods II
Lecture 21
Aaron Christie
9 April 2015
1
Review Session
This class will be a just be composed of working through a few topics from
the class including problems of the kind you might see on the final. Theres
not enough tim
Chacun de tes trois scnarios doit tre bien expliqu par des phrases compltes comme dans lexemple
en utilisant le tableau cidessous :
NOM DE LLVE : _Abbas Ghandour_
Dure
de
cette
scne :
1
minut
e
premi
er
sketch
Thmes et script pour l'explication du thme
L
1
Two highly sought after politicians, which are highly educated have used their lives
participating in global politics. Janice Stein and Allan Gotlieb have shared their very different,
valued opinions on Canadas new diplomacy in the 21st century. They ha
Abbas Ghandour
BIBLIOGRAPHIE
Centers for Disease Control and Prevention. Childhood Obesity Facts. Centers for
Disease Control and Prevention. Centers for Disease Control and Prevention,
http:/www.cdc.gov/obesity/data/childhood.html. 3 septembre 2014. Web.
Abbas Ghandour
Statistiques  Obsit

1.
2.
3.
4.
5.
27% des personnes dessous le niveau de pauvret sont obse compar au 21% qui sont en
haut de le niveau de pauvret (Smoking, obesity connected 6)
50% des femmes afroamricains qui sont pauvres, sont obse (
The Science of Art: Guernica
Abbas Ghandour
Mr. Moon
HSB4M
October 8, 2015
Vilayanur Ramachandran is an evolutionary neuroscientist in our era. Ramachandran has
created Eight Laws of Artistic Experience, which will be used to break down the masterpiece
Gu