Tuesday 11/01, lecture notes by Y. Burda
1
Fields
To do linear algebra we only need to do arithmetic operations to numbers. So
instead of numbers we can work with anything we can add, multiply, subtract
and divide (with the usual properties of the operati
University of Toronto
Department of Mathematics
MAT224H1S - Linear Algebra II
Winter 2015
Writing Assignment 1
1. Typically, the axioms of a vector space, or any similar mathematical axiomatic denition, are given as a minimal set so that no one property i
Midterm Solutions
MAT224H1S - Summer 2012
Wednesday, July 25
Question 1.
We have
T (p(x) = p(2)(x3 x) + p(1)(x + 1) = p(2)x3 (p(2) p(1)x + p(1),
so p(x) ker(T ) if and only if
p(2) = 0, p(2) p(1) = 0, p(1) = 0.
i.e.
p(x) = (x 1)(x 2)q(x),
for some q(x) P
Department of Mathematics, University of Toronto
MAT224H1F - Linear Algebra II
Fall 2012
Problem Set 3
Due Tues. Oct 16, 6:10pm sharp, in class . Late assignments will not be accepted - even if its one
minute late!
Be sure to clearly write your name, st
Solutions to Problem Set 1
MAT224H1S - Summer 2012
1. (a) The characteristic polynomial of A is
det(A I) = det
1
3
3
1
= (1 )2 + 9 = 2 2 + 10.
The eigenvalues are its roots:
2 + 4 40
= 1 + 9 = 1 + 3i, 2 = 1 3i.
1 =
2
The set of eigenvectors (u, v) C2 for
Problem Set 2
MAT224H1S - Summer 2012
due July 18, 2012
1. Let T : P2 (R) R3 be the linear transformation of real vector spaces
dened by
T (p(x) = (p(1) + p (0), p (2) p (1), p (2) + p(0),
where p (x) =
d
dx p(x)
and p (x) =
d
dx p
(x).
(a) Let = cfw_1, x
Solutions to Problem Set 2
MAT224H1S - Summer 2012
1. (a) We have
T (1) = (1, 0, 1), T (x) = (2, 1, 1), T (x2 ) = (1, 0, 4),
so that
1
[T ] = 0
1
1
0 .
4
2
1
1
(b) We can row reduce the matrix as follows: subtract the the rst and
second rows from the thir
Solutions to Problem Set 3
MAT224H1S - Summer 2012
Problem 1. If t denotes the common quantity of the equations for the rst
line, we have
t = x 1 = y 2 = z 3.
We can rewrite this in the form
x = 1 + t, y = 2 + t, z = 3 + t,
and put the resulting equations
Solutions to Problem Set 4
MAT224H1S - Summer 2012
Problem 1. The characteristic polynomial of A is:
x
1
2
p(x) = det(A xI) = det 1 1 x 1
2
1
x
= x(x2 + x 1) (x + 2) + 2(1 + 2(x + 1)
= x3 x2 + 6x = x(x2 + x 6) = x(x + 3)(x 2).
So the eigenvalues are 1 =
Solutions to the Suggested Diagonalization
Problems
MAT224H1S - Summer 2012
Question 1. The characteristic polynomial is
2x
0
1
4x
2 = (x 2)(x 4)(x 1),
det 2
0
0
1x
The case F = R: The eigenvalues are 2,4, and 1. Since the characteristic polynomial splits
University of Toronto
Department of Mathematics
MAT224H1S - Linear Algebra II
Winter 2015
Writing Assignment 2
1. Remark 1.3.10 on page 24 of the textbook notes that if W1 and W2 are subspaces of
a vector space V , their union, W1 W2 , need not be a subsp
University of Toronto
Department of Mathematics
MAT224H1S - Linear Algebra II
Winter 2015
Writing Assignment 4
1. Let V and W be vector spaces, and T : V W a linear transformation. Prove that
the following statements are equivalent:
(i) T is injective (on
Another example of a vector space: Kmn - the space of m n matrices
with entries from the eld K . This is a vector space over K with the addition
being the usual matrix addition and scalar multiplication being the entrywise multiplication by scalars: (aij
University of Toronto
Department of Mathematics
MAT224H1F
Linear Algebra II
Midterm Examination
October 24, 2006
S. Uppal
Duration: 1 hour 50 minutes
Last Name:
Given Name:
Student Number:
Tutorial Code:
No calculators or other aids are allowed.
FOR MARKE
University of Toronto
Department of Mathematics
MAT224H1F
Linear Algebra II
Midterm Examination
October 24, 2006
S. Uppal
Duration: 1 hour 50 minutes
Last Name:
Given Name:
Student Number:
Tutorial Code:
No calculators or other aids are allowed.
FOR MARKE
University of Toronto
Department of Mathematics
MAT224H1S
Linear Algebra II
Midterm Examination
March 1, 2007
M. Krishnapur, M. Pinsonnault, S. Uppal
Duration: 1 hour 50 minutes
Last Name:
Given Name:
Student Number:
Tutorial Code:
No calculators or other
University of Toronto
Department of Mathematics
MAT224H1S
Linear Algebra II
Midterm Examination
March 1, 2007
M. Krishnapur, M. Pinsonnault, S. Uppal
Duration: 1 hour 50 minutes
Last Name:
SOLUTIONS
Given Name:
THE
Student Number:
Tutorial Code:
No calcul
Department of Mathematics, University of Toronto
MAT224H1F - Linear Algebra II
Fall 2012
Problem Set 1
Due Tues. Oct 2, 6:10pm sharp, in class . Late assignments will not be accepted - even if its one
minute late!
Be sure to clearly write your name, stu
Department of Mathematics, University of Toronto
MAT224H1F - Linear Algebra II
Fall 2012
Brief Course Description
Welcome to MAT224H1F Linear Algebra II. This sheet answers the most common questions about the
course. Please take a few minutes to read this
University of Toronto
Department of Mathematics
MAT224H1S
Linear Algebra II
Midterm Examination
Feruary 25, 2010
M. Mazin, S. Uppal
Duration: 1 hour 50 minutes
Last Name:
Given Name:
Student Number:
Tutorial Code:
No calculators or other aids are allowed.
ECO208 Fall / Winter 2012
Tutorial 1 October 15, 2012
1.
Consider the simple model of consumer choice developed in class. The consumer
is endowed with h units of time which can be allocated to work and leisure. The budget
s
constraint for the consumer is
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATB24H Linear Algebra II
Examiners: X. Jiang
E. Moore
Date: December 12, 2014
Start Time: 9:00AM
Duration: 3 hours
1. [15 points] Multiple Choice
Each ques