University of Toronto at Scarborough Department of Computer and Mathematical Sciences
Linear algebra II
MATB24 Fall 2010 Section 1.4
29. F F T T F T T T F T
1
Section 1.5
2
Section 4.2
3
Addition: 1) a) x 5 a1 x 4 a2 x 3 a3 x 2 a4 x a5 b) 0 c)
1i j n
(x
1. a) A field is a set F equipped with two binary operations addition and multiplication satisfy the following properties: addition: AA1: a + (b + c) = (a + b) + c AA2: a+b=b+a AA3: 0, s.t. a + 0 = a AA4: a, s.t. a + (a) = 0 multiplication: SS1: a (b c) =
Term Test MATB24 Linear Algebra II
2009
1. (10 points)
. a) Give the definition of a field. b) Let K be a vector space over Z2 with basis cfw_1, t, so K = cfw_a + bt | a, b Z2. It is known that K becomes a field of four elements if we define t2 = 1 + t. W
Lecture 6 Thm: Let V , V be finite-dimensional vector spaces and T : V V is an isomorphism. If B 1 , b2 , ., bn is any basis of V, then (b1 ), T (b2 ), ., T (bn )is a basis b T of V .
6.1 Matrix representation of transformation Review Section 2.3 Thm: Let
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
Midterm Test
MATB24H Linear Algebra II
Examiner: E. Moore
Date: October 25, 2013
Duration: 110 minutes
1. [11 points]
(a) Define what it means for a set V to be a vector spac
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 B24H3
2015/2016
Assignment #5
You are expected to work on this assignment prior to your tutorial in the period October 8
October 21, 2015. You may ask questions about th
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
Midterm Test
MATB24H Linear Algebra II
Examiners: X. Jiang
E. Moore
Date: October 24, 2014
Duration: 110 minutes
1. [10 points]
(a) Define what it means for a set F to be a f
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 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 B24H3
2015/2016
Assignment #11
The Final Examination will take place on December 19, from 2 pm 5 pm
Final Exam Room Assignments
Surname
go to room
A to E
SW 128
F to J
SW
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 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 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 #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 B24H3
2015/2016
Assignment #2
You are expected to work on this assignment prior to your tutorial in the week of September 17 September 23, 2015. You may ask questions abo
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H3
2015/2016
Assignment #4
You are expected to work on this assignment prior to your tutorial in the period October 1
October 7, 2015. You may ask questions about thi
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H3
2015/2016
Assignment #3
You are expected to work on this assignment prior to your tutorial in the week of September 24 September 30, 2015. You may ask questions abo
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
2008 Term Test
MATB24 Linear Algebra II
1. (10 points)
. a) Give the definition of a general vector space.
b) Let V be the set of 2 2 matrices with zero determinant, with the usual matrix addition
and scalar multiplication. Is V a vector space? Why or why
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 B41H
2016/2017
Assignment #7
This assignment is due at
November 11 November 17, 2016.
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Cha
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B41H
2016/2017
Assignment #6
This assignment is due at
November 4 November 10, 2016.
the
start
of
your
tutorial
in
the
period
A. Suggested reading: Marsden & Tromba, Chap
S4:
S3:
S2:
S1:
A4:
A3:
A2:
1
a
(v)
a (b c) a b a c
2
(iv) For each a in and each nonzero b in , there are elements c and d
in such that a c 0 and b d 1
(iii) There are elements 0 and 1 in such that 0 a
a and 1 a
For all u, v, w in n, r, s in , v + w, rv
0 if and only if v = 0.
rv , w
An inner product space is a vector space V together with an
inner product on V .
0 if and only if v = 0.
v , rw
u, v u, w
(iv) v, v t 0 and v, v
(iii) r v, w
(ii) u, v w
Definition 4.1:
An inner product on a (real) vector sp
u v w = u v w
0 and we write w = v
r s v = r v s v
(viii) if 1 is the multiplication identity in F then 1 v
(vii) r ( s v ) = (rs ) v
(vi)
(v) r v w = r v r w
vw
v.
(iv) For each v in V, there exists an element w in V such that
(iii) There exists an elem
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
2016 Fall
MATB24H3F
Assignment #5
You are expected to work on this assignment prior to your tutorial in the week of
October 17, 2016. You may ask questions about this assignm
You are expected to work on this assignment prior to your tutorial in the week of
October 31, 2016. You may ask questions about this assignment in that tutorial.
In your tutorial in the week of November 7 you will be asked to write a quiz
based on this as
v E by > v @E = [r1, r2, , rn].
1
we define the coordinate vector of v relative to E , denoted > v @E or
Definition 3.2:
Let E = (b1, b2, , bn) be an ordered basis for a finite
dimensional vector space V . For v V , with unique representation
v = r1 b1 +
1 .
Addition:
(a + bi) r (c + di) = (a r b) + (c r d)i
1
Geometrically, a complex number can be viewed either as a point or a
vector in the xy-plane, where the x-axis is the real axis and the y-axis is
the imaginary axis.
= cfw_a + bi | a, b
The set of
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
2016 Fall
MATB24H3F
Assignment #4
You are expected to work on this assignment prior to your tutorial in the week of October
3, 2016. You may ask questions about this assignme
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2015/2016
Solutions #4
1. (Fraleigh & Beauregard, pages 226229)
(1) T : F R defined by T (f ) = f (4). For f, g F and R, we have T (f +g) =
(f + g)(4) = f (4) + g(4)
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B24S
Fall 2011
MAT B24
Lectures 15.
1
The Gram-Schmidt Process
In this section we will develop a process that will give us a basis of a subspace
W of Rn with the prope