University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Solutions #1
+012
012
0012
0000
1. Computing the tables we have
and
. From the tables we
1120
1012
2201
2021
see that Z3 is closed under both addition and

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2014/2015
Term Test Solutions
1. (a) From the lecture notes we have
Definition: A field F is a set on which two operations + and (addition
and multiplication) are de

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H3
2015/2016
Assignment #8
You are expected to work on this assignment prior to your tutorial in the period November 5 November 11, 2015. You may ask questions about t

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24F
2013/2014
Assignment #8
You are expected to work on this assignment prior to your tutorial in the week of November 4, 2013. You may ask questions about this assignme

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24F
2013/2014
Assignment #10
You are expected to work on this assignment prior to your tutorial in the week of November 18, 2013. You may ask questions about this assign

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24F
2013/2014
Assignment #11
You are expected to work on this assignment prior to your tutorial in the week of November 25, 2013. You may ask questions about this assign

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Solutions #1
+ 0 1 2
0 1 2
0 0 1 2
0 0 0 0
1. Computing the tables we have
and
. From the tables we
1 1 2 0
1 0 1 2
2 2 0 1
2 0 2 1
see that Z3 is closed

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Solutions #10
1. (Fraleigh & Beauregard, pages 368 369)
1
1
(2) To project onto sp ([1, 1, 2]), we put A = 1 so A A = [1 1 2] 1 =
2
2
1
[6] and (A A)1 =
.

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Solutions #11
1. (Fraleigh & Beauregard, pages 406 407)
1
1
(2) Since T
=
1
1
and T
1
1
=
5
3
,
T
std
B
=
1 5
.
1 3
1 1
B
B
std
std 1
std
, T B = T B = I s

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATB24H Linear Algebra II
Examiner: E. Moore
1. [9 points]
Date: December 13, 2013
Duration: 3 hours
Let V be a vector space over a eld F .
(a) Dene what it

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H3
2015/2016
Assignment #10
You are expected to work on this assignment prior to your tutorial in the period November 19 November 25, 2015. You may ask questions about

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2015/2016
Assignment #10
The Final Examination will take place on December 16, from 2 pm 5 pm
Final Exam Room Assignments
Surname
go to room
A to D
IC 220
F to SH
IC

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H3
2015/2016
Assignment #6
You are expected to work on this assignment prior to your tutorial in the period October 22 October 28, 2015. You may ask questions about th

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H3
2015/2016
Assignment #7
You are expected to work on this assignment prior to your tutorial in the period October 29 November 4, 2015. You may ask questions about th

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H3
2015/2016
Assignment #9
You are expected to work on this assignment prior to your tutorial in the period November 12 November 18, 2015. You may ask questions about

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24F
2013/2014
Assignment #9
You are expected to work on this assignment prior to your tutorial in the week of November 11, 2013. You may ask questions about this assignm

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24F
2013/2014
Assignment #2
You are expected to work on this assignment prior to your tutorial in the week of September 16, 2013. You may ask questions about this assign

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24F
2013/2014
Assignment #7
There are no tutorials in the week of October 28 November 1.
In your tutorial in the week of November 4 November 8 you will be asked to write

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Solutions #2
1. (a) In Z7 , 11 = 1, 21 = 4, 31 = 5, 41 = 2, 51 = 3, 61 = 6.
(b) In Z7 , x2 = 2 if x = 3 or x = 4.
2. (Fraleigh & Beauregard, pages 202203)

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Solutions #3
1. (Fraleigh & Beauregard, pages 211212)
(2) To nd [x1 , x2 ], the coordinate vector of [2, 4] in R2 relative to the ordered basis,
0
1
2
2
([

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Solutions #6
1. (Fraleigh & Beauregard, pages 463464)
(2) (a) z + w = (2 + 3i) + (5 i) = 7 + 2i; z w = (2 + 3i)(5 i) = 13 + 13i.
(b) z + w = (1 + 2i) + (2

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Solutions #4
1. (Fraleigh & Beauregard, pages 226229)
(1) T : F R dened by T (f ) = f (4). For f, g F and R, we have T (f + g ) =
(f + g )(4) = f (4) + g (

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Solutions #5
1. (Fraleigh & Beauregard, pages 226229)
3
1 2
1
2 1
.
=
4 1
2
1
2
0
(16) Since the j th column of A is T (bj )
B
, A= T
4 1 1
7
1
2 2
2
0

Example 1.1: The set of all the real numbers 1R, under the usual
operations of addition and multiplication.
111:3 Fl-0M
Examplel .2: The rational numbers
Q=cfw_a/b1a,binl,b0
where Z is the set of integers.
(11$ owlldd Ct Ct
' 7" ' ' '21).
/\ 9 QJQJJ ea Qs

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24F
2013/2014
Assignment #1
You are expected to work on this assignment prior to your tutorial in the week of September 9, 2013. You may ask questions about this assignm

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24F
2013/2014
Assignment #6
Term Test Room Assignments
Friday, 25 October, 7:009:00 pm
Surname
go to room
A to C
D to K
L to Z
IC 230
IC 220
IC 130
You are expected to w

University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24F
2013/2014
Assignment #4
You are expected to work on this assignment prior to your tutorial in the week of September 30, 2013. You may ask questions about this assign