University of Toronto
Faculty of Arts & Science
MAT224H1S: Linear Algebra II
Midterm Test
July 18, 2016
SOLUTIONS
1
1. Let V and W be vector spaces. [8 marks]
(a) Let v1 , v2 , . . . , vn V . Define what it means for v V to be a linear combination of v1 ,
APM462, Winter 2016 Quiz 1
February 1, 2016, 6:10-7pm
This test has a total of 5 pages (not including the cover page). Each page
contains a question worth 10 points.
Do not write your name on any page other than this one.
Only answers written on the front
APM462H1S, Winter 2016.
Homework 2 solutions
In several questions below, we use the following important fact: a symmetric
n n matrix Q is positive definite if and only if for every k = 1, . . . , n, the k k
matrix formed by the upper left corner of Q has
APM462H1S, Winter 2016.
Homework 3
This covers material related to the Global Convergence Theorem in Section
7.7 of the book of Luenberger and Ye.
Before getting to the homework, here is a remark about the Global Convergence
Theorem:
The theorem may be a
APM462H1S, Winter 2016.
Homework 5
The first two questions are related to Application 2: Penalty methods from
pages 240-1 of the textbook, discussed in the February 29 lecture. That is, the
second question is related to the above material, and the first q
APM462H1S, Winter 2016.
Homework 7
Exercise 1. Assume that Q is a symmetric n n matrix, and consider the
problem
minimize f (x) = xT Qx,
subject to the constraint kxk2 = 1.
a. Explain (in one sentence) why a minimizer exists.
solution. The problem involve
APM462, Winter 2016 Quiz 3
March 28, 2016, 6:10-7pm
This test has 6 pages (including the cover page). Each page after the cover
page contains a single question worth 10 points.
Total: 5 questions, 50 marks.
Do not write your name on any page other than th
APM462H1S, Winter 2016,
Homework 9
Exercises 1 - 3. For each of the following functionals,
a. compute F
u by considering
tions v().
d
ds F [u() + sv()]
for compactly supported func-
b. Also compute F
formula we have for
u by using the general
Rb
a functio
APM462, Winter 2016 Quiz 2
March 7, 2016, 6:10-7pm
This test has 6 pages (including the cover page). Each page after the cover
page contains a single question worth 10 points.
Total: 5 questions, 50 marks.
Do not write your name on any page other than thi
APM462H1S, Winter 2016.
Homework 4
This covers material related to Section 8.1 and 8.2 of the book of Luenberger
and Ye, 4th edition.
(1) Luenberger and Ye, 4th edition, exercise 8 on page 258.
solution:
To show that S is closed at (x, d), we consider seq
APM462H1S, Winter 2016.
Homework 6
Exercise 1. Problem 1 on p. 281 of Luenberger and Ye, 4th edition.
solution: (refer to the book for notation).
We will prove by induction that dTj Qdk = 0 for all j < k.
This is clear if k = 0, since then there is no j <
APM462H1S, Winter 2016.
Homework 8
Exercise 1. Consider the problem
minimize
subject to
and
f (x, y) = 2x2 + y 2
in E 2
g1 (x, y) = 2 x y 0
g2 (x, y) = x y 0.
a. Draw a picture of the set of points that satisfy the constraints. (Note: this is
sometimes ca
APM462: Homework 2
Due date: Tue June 13, in class.
Suggested problems (not to be turned in):
(A) Prove that the sum of two convex functions is convex.
(B) Prove that the function af is convex whenever f is convex and a is a
non-negative real number.
(C)
APM462: Homework 3
Due date: Thursday June 22, in class .
Suggested problems (not to be turned in):
p
(A) Let d(x, y) = |x y| = (x1 y1 )2 + + (xn yn )2 be the usual
metric on Rn , and let C Rn be a convex set. Fix > 0. Prove that
the set cfw_x Rn | d(x, C
Assignment 2
APM462 Nonlinear Optimization Summer 2017
Tomas Kojar
June 21, 2017
Question 1
(a) (2/10 marks) Expanding out f (a), we see that
Z 1
Z 1
2
f (a) =
(g(x) pa (x) dx =
(g(x) (x, a)2 dx
0
Z0 1
=
(x, a)(x, a) 2g(x)(x, a) + g(x)2 dx
Z0 1
=
(a, xxT
Assignment 3
APM462 Nonlinear Optimization Summer 2017
Tomas Kojar
July 5, 2017
Question 1
(a) 2e) The second order condition is given by (v, 2 (f h)v)x=x > 0 for nonzero
vector v Tx M . In our case we see
2 (f h) =
2 6
0
0
2 2
When = 1/3, i.e. v Tx M
8
APM462: Homework 1
Due at the beginning of class1
Thursday, June 1, 2017
Suggested problems (not to be turned in):
(1) Prove the 1st order Taylor approximation for functions on Rn . You
can use the 1st order Mean Value Theorem.
(2) Prove the 2nd order Tay
APM462: Homework 5
Due date: Thr, Aug 10 in class.
The following questions are to be handed in:
(1) Let u () be an extremal (e.g. minimizer) of the functional F [u()] =
Rb
0
a L(x, u(x), u (x)dx. Find the Euler-Lagrange equation satisfied
by u () where L(
Assignment 1
APM462 Nonlinear Optimization Summer 2017
Tomas Kojar
June 4, 2017
Question 1
(a)
We compute fs gradient and set it equal to the zero vector (0,0)
(0, 0) = f = (4x + y, 2y + x 1)
4x + y
=0
1 4
x0 = (x, y) = ( , )
2y + x 1 = 0
7 7
satisfies
Assignment 4
APM462 Nonlinear Optimization Summer 2017
Tomas Kojar
July 30, 2017
Question 1
(a) We have
1 4 1/2
x
x (5, 0) x.
2
1/2 2
(0.1)
and so x = Q1 (5, 0) (1.3, 0.3).
(b) Q is positive definite with eigenvalues 0.5(6 + 5) and so f is convex.
(c) We
APM462: Homework 4
Due date: Thr, July 27 in class.
(1) Let f (x, y) = 2x2 + y 2 + 21 xy 5x.
(a) Find an unconstrained local minimum point x for f .
(b) Why is x actually a global minimum point?
(c) Using the method of steepest descent with intial point x
The Axiom of Choice
Contents
1 Motivation
2
2 The Axiom of Choice
2
3 Two powerful equivalents of AC
4
4 Zorns Lemma
5
5 Using Zorns Lemma
6
6 More equivalences of AC
11
7 Consequences of the Axiom of Choice
12
8 A look at the world without choice
14
1
1
Arbitrary products
Contents
1 Motivation
2
2 Background and preliminary definitions
2
3 The space RN
box
4
4 The space RN
prod
6
5 The uniform topology on RN , and completeness
12
6 Summary of results about RN
14
7 Arbitrary products
14
8 A final note on
Metric spaces and metrizability
1
Motivation
By this point in the course, I hope this section should not need much in the way of motivation.
From the very beginning, we have talked about Rnusual and how relatively easy it is to prove
things about it due t
Things You Should Know
This is a slightly modified version of a document authored by Micheal Pawliuk. This list of concepts is
used here with his permission.
1
Basic Set Theory
I will assume students are familiar with all of these terms and symbols. Pleas
Urysohns metrization theorem
1
Motivation
By this point in the course, I hope that once you see the statement of Urysohns metrization
theorem you dont feel that it needs much motivating. Having studied metric spaces in detail
and having convinced ourselve
Urysohns Lemma
1
Motivation
Urysohns Lemma (it should really be called Urysohns Theorem) is an important tool in topology. It will be a crucial tool for proving Urysohns metrization theorem later in the course,
which is a theorem that provides conditions
Tychonoff s theorem, and properties related
to compactness.
1
Motivation
In this section we will prove Tychonoffs theorem. Before we can do that, we will discuss a
different characterization of compactness in terms of filters and ultrafilters. We will hav
Compactness
1
Motivation
While metrizability is an analysts favourite topological property, compactness is surely a topologists favourite topological property. Metric spaces have many nice properties, like being first
countable, very separative, and so on