Department of Mathematics, University of Toronto
MAT224H1F - Linear Algebra II
Fall 2012
Problem Set 5
Due Tues. Nov 20, 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
University of Toronto
Department of Mathematics
MAT224H1S - Linear Algebra II
Winter 2015
Writing Assignment 3
1. Let A, B Mmn (R).
(a) Prove that rank(A + B) rank(A) + rank(B).
(b) Prove that if rank(A + B) = rank(A) + rank(B), then col(A) col(B) = cfw_0
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Winter 2016
Tutorial Problems 1
1. Consider the subset S = cfw_x4 + x + 1, x3 + x2 + 2x, x3 + x2 + 1, x4 + 2x of P4 (R).
(a) Is 5x4 + 5x3 + 5x2 + 13x + 4 span(S)? If it is, exp
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Winter 2014
Tutorial Problems 4
1. Let = cfw_(3, 5, 2), (4, 1, 1), v3 and = cfw_v1 , (6, 5, 6), v3 be bases for R3 , and that the change of
basis matrix from to is
3 1
5
1 1
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Winter 2015
Tutorial Problems 1
1. A semimagic square is an n n matrix in which every row and every column has the same sum c.
For example, the identity matrix is a semimagic s
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Winter 2012
Problem Set 4:
Due Tues. March 20, 6:10pm sharp. Late assignments will not be accepted - even if its one minute
late!
You may hand in your problem set either to y
University of Toronto
Department of Mathematics
MAT224H1S
Linear Algebra II
Midterm Examination
February 28, 2013
M. El Smaily, S. Uppal, O. Yacobi
Duration: 1 hour 50 minutes
Last Name:
Given Name:
Student Number:
Tutorial Group:
No calculators or other
MAT 224: Some Notes for Writing Assignment 8
Peter Crooks
March 27, 2014
In the writing assignment, A is a 4 4 matrix with entries in a eld F. The idea is
to consider the linear transformation T : F4 F4 dened by
T (x) = Ax.
The matrix A is said to have in
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Winter 2014
Tutorial Problems 10
1. Textbook, Section 6.1, #1.
2 (a) Show that for any linear operator T on a vector space V , the subspaces Ker(T k ) and Im(T k ), k Z+ ,
are
MAT 224 QUIZ 1 (Version A) (SOLUTIONS)
Wednesday, September 16, 2015
Duration: 20 minutes.
Last Name:
Given Name:
Student Number:
Tutorial:
Aids: NO AIDS.
Please note: Solve the following problems, and write up your solutions neatly in the space provided.
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Winter 2011
Solutions to Problem Set 1
1. In the rst class we discussed elds and showed that, in addition to the real numbers, the complex
numbers form a eld. There are of cour
University of Toronto
Department of Mathematics
MAT224H1S
Linear Algebra II
Midterm Exam II
March 21, 2014
P. Crooks, M. Mota, S. Uppal
Duration: 1 hour 50 minutes
Last Name:
Given Name:
Student Number:
Email (@mail.utoronto.ca):
Instructions:
No calcula
Mat 224 Tut 1
Question One
1.1 Prove that A (B C) = A B (A C).
1.2 Let A, B X. Prove that (i) A B X B X A and (ii)
A B A (X B) = .
1.3 If A, B X, we write B c for X B and Ac for X A. Prove that
A B i B Ac = X.
1.4 Prove that A B = A A B.
1.5 Prove that A
University of Toronto
Department of Mathematics
MAT224H1F
Linear Algebra II
Midterm Examination
October 13, 2010
S. Uppal
Duration: 1 hour 20 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 I
Midterm Exam I
February 14, 2014
M. Mota, S. Uppal, F. Vera-Pacheco
Duration: 1 hour 50 minutes
Last Name:
Given Name:
Student Number:
Email (@mail.utoronto.ca):
Instructions:
No
University of Toronto
Department of Mathematics
MAT224H1S
Linear Algebra I
Midterm Exam I
February 14, 2014
M. Mota, S. Uppal, F. Vera-Pacheco
Duration: 1 hour 50 minutes
Last Name:
Given Name:
Student Number:
Email (@mail.utoronto.ca):
Instructions:
No
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.
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Winter 2012
Problem Set 3
Due Tues. Feb. 14, 6:10pm sharp. Late assignments will not be accepted - even if its one minute late!
You may hand in your problem set either to you
Department of Mathematics, University of Toronto
MAT224H1F - Linear Algebra II
Fall 2012
Problem Set 2
Due Tues. Oct 9, 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
MAT224 Assignment 4
Due Tuesday Nov 17 at the beginning of the lecture
Please write your arguments carefully. Neatness will count. Marks may be taken off if your arguments
are not clear or if the presentation of your solutions shows a lack of effort, e.g.
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Summer 2017
Assignment 1 Due Thursday July 13 in tutorial
x
1. Let V = cfw_ 1 | x1 , x2 R, x1 > 0, x2 > 0. Define the operations of vector addition and scalar
x2
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MAT224 Tut3-4 Solution
Tutorial 3
Question 1
Find the basis and dimension for the subspaces.
2a)
W = cfw_x1 + x4 = 0, 3x1 + x2 + x4 = 0 R4
uc
at
io
n
Solving using the augmented matrix, we have
1 0 0 1 0
1 0 0 1 0
3 1 0 1 0
0
1 0 2 0
ey
Thus the basis of
MAT224 Assignment 5
Due Tuesday Dec 8 at the beginning of the lecture
Please write your arguments carefully. Neatness will count. Marks may be taken off if your arguments are
not clear or if the presentation of your solutions shows a lack of effort, e.g.
MAT224 Assignment 3
Due date and how to submit the assignment: You can submit your assignment any afternoon Tuesday Oct
27 to Friday Oct 30 in the Math Aid Centre (in Sydney Smith building). There will be a drop-off box labeled
for MAT224. Leave your assi
MAT224 Tut1-2 Solution
Tutorial 1
uc
at
io
n
Question 1
Let V be a real vector space.
(Note that real means we are allowed to multiply by real numbers.)
By definition of vector space, there exists ~0 V .
Thus V has at least 1 vector.
If V consist of only
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Summer 2017
Assignment 2 Due Thursday July 20 th
a
1. Let W = cfw_
c
b
M22 (R)|a + d = 0
d
0 1
0 1
1
(a) Show that the set = cfw_
,
,
1 0
1 0
0
2 1
(b) Let x =
. Determine
Friday March 17
START: 2:10pm
DURATION: 110 mins
University of Toronto
Department of Mathematics
MIDTERM EXAMINATION II
MAT224H1S
Linear Algebra II
EXAMINERS: K. Koziol, K. Matetski, S. Uppal, Z. Wolske
Last Name (PRINT):
Given Name(s) (PRINT):
Student NU
Friday February 10
START: 2:10pm
DURATION: 110 mins
University of Toronto
Department of Mathematics
MIDTERM EXAMINATION I
MAT224H1S
Linear Algebra II
EXAMINERS: K. Koziol, K. Matetski, S. Uppal, Z. Wolske
Last Name (PRINT):
Given Name(s) (PRINT):
Student
Friday Febrauary 12
START: 12:10
DURATION: 110 mins
University of Toronto
Department of Mathematics
MIDTERM EXAMINATION II
MAT224H1S
Linear Algebra II
EXAMINERS: I. Biborski, A Garcia-Raboso, F. Herzig, K.K. Leung, S. Uppal
Last Name (PRINT):
Given Name(s