University of Toronto at Scarborough Department of Computer & Mathematical Sciences
MAT B24S Fall 2007
Assignment #5 You are expected to work on this assignment prior to your tutorial in the week of October 15th. You may ask questions about this assignmen
Example 4.3: Determine whether each of the following vector spaces
is an inner product space.
2
a)
with <[x1, x2], [y1, y2]> = 3x1y2K + 2x2y2. (
b) M2(
LAIB
with <A, B> = det(AB).
>
t
Aic
<
=detAB+detAC=
>
1
c) C[0, 1] with f , g
F
2
f. g.
fig
>
=
Dz
<
h
Review

A*=
A
Unitary
FUS
.tU*AU=R
R*=R

FTA
1
Fundi
mental
Theorem
of
calculus
)
R=[
e
R
Corollary 7.19: Every nn real symmetric matrix, has n real
eigenvalues, counted with their algebraic
multiplicities and is diagonalizable by a (real)
orthogonal ma
Proof
Let
Let
dinv=n
cfw_ I
,
,
.
.
,
,bI
dim
w=k
be
basis
a
cfw_
To
,
.
.
.
.
,
Let >U=
view
Theorem 7.2: The orthogonal complement W of a subspace W of
a finite dimensional inner product space V has the
following properties.
1) W is a subspace.
2) dim(
Review
Notations:
Let T : V W be a linear transformation ( ).
(V domain, W codomain)
3.2 Linear Transformation
Definition 3.3: Let V and W be vector spaces (over ). A function
T: V
W is called a linear transformation from V to
W if, for all u, v V and r
(
7.6 Orthogonal linear transformation
0 1 1
1 0 1 . Find an orthogonal matrix P
1 1 0
Example 7.5.2: Let A
Definition 7.10: A linear transformation T : V
V of an inner
product space V is orthogonal if
< T(v), T(w) > = < v, w> ,
for all v, w V .
such that P
Review
.
=
Assume that J is a Jordan form for a nn matrix A. We have
previously seen that similar matrices represent matrix
representations of the same linear transformation but w.r.t different
bases. Hence there is a basis b1 , b 2 , ., b n of n with the
Review
Example 3.2.5: Find ker(T) and range(T) in the example 3.2.4.
in Ex
zu
.
Ker
cfw_ Fe
H=
Mzltcvko
tzbtd )
( feet
.
.
tzctztbtd )x2t
cfw_ [ Tbd ]/T(
=
.
)=(
( [ 9th ]
T
'
(
+
alts
a
)xtcfw_
)=o
ais
cfw_
d=o
c
b
dtb
)x+(c
cider

b
.
Definition 3
Riiew
Properties
n
Definition 5.1: Let v = [v1, v2, , vn], w = [w1, w2, , wn]
.
The Euclidean inner product (complex dot product) of v and w is
v, w
v1w1 v2 w2 . vn wn
n
i 1
i, 2, 1 3i and w
Example 5.2.6: Let v
2. v, u
vi wi .
I
,
Tv )
(
=
=i(
) (
i
I

Conclusions:
(1) In a finitely generated vector space, every independent set
of vectors in V can be enlarged, if necessary, to a basis.
(2)
If dimV = k, every independent set of k vectors in V is a
basis for V and every set of k vectors that span V is a b
Example 7.8.2: Determine whether or not the matrix
1 1
i
not
is
A
1
1
hermitiau
1
i
1
Eigenvalues:
matrix
is unitarily diagonalizable.
i
Eigenvectors:
In fact,
AA*
3
1 2i
1 2i
3
1 2i
1 2i
1 2i
1 2i
3
reali
(
We
Is
are
.
this
P
H
.
M
H
.
?
b/c
NO
T
<
A* A
Review
Theorem 7.8: Let A Mn( ), then the following are equivalent.
1) The rows of A form an orthonormal basis for n.
2) The columns of A form an orthonormal basis for n.
3) A is orthogonal. (A 1 = AT )
ai
aT=o
1 1 1 1 1
1 1
2 1 1
1 1 1
1
Eu
E
ai
at
is an
Review
Example 7.10.1: Let T:
cfw_ J
basis
for
,
,
I. is
3
:
3
defined by T(z) = Az where
FJEC
i
1 1
orthogonal
E=
A
1
i
1 .
Ex
1 1
i
an
C3
/Vt=E
Exz
,
,
7.11 Projection matrix
Unique
3
@ Ex
.
Ex
Theorem 7.28: If A Mm,n( ) has rank k, then the n n matrix
University of Toronto at Scarborough Department of Computer & Mathematical Sciences
MAT B24S Fall 2007
Solution 6: Question 3 1) To break a code given the message: UVZR.YH VOYYMNTOSIJXVV We get the numerical equivalents and form vectors with 2 coordinates
University of Toronto at Scarborough Department of Computer & Mathematical Sciences
MAT B24S Fall 2007
Assignment #6 You are expected to work on this assignment prior to your tutorial in the week of October 22nd. You may ask questions about this assignmen
University of Toronto at Scarborough Department of Computer & Mathematical Sciences
MAT B24S Fall 2007
Assignment #7 You may ask questions about this assignment during the week of October 29th during the math aid hours or oce hour. In the week of November
University of Toronto at Scarborough Department of Computer & Mathematical Sciences
MAT B24S Fall 2007
Assignment #8 You are expected to work on this assignment prior to your tutorial in the week of November 5th . You may ask questions about this assignme
University of Toronto at Scarborough Department of Computer & Mathematical Sciences
MAT B24S Fall 2007
Assignment #9 You are expected to work on this assignment prior to your tutorial in the week of November 12th. You may ask questions about this assignme
University of Toronto at Scarborough Department of Computer & Mathematical Sciences
MAT B24S Fall 2007
Assignment #10 You are expected to work on this assignment prior to your tutorial in the week of November 19th . You may ask questions about this assign
University of Toronto at Scarborough Department of Computer & Mathematical Sciences
MAT B24S Fall 2007
Assignment #10 You are expected to work on this assignment prior to your tutorial in the week of November 26th . You may ask questions about this assign
Review
#E ;
:;L
a
15*11=1

a
[
,tx*lm
mm
If A Mn( ) is similar to J, a Jordan form matrix, we say that J
is a Jordan canonical form for A.
IIiraI
Definition 9.4: Let V be a finite dimensional vector space and let
T: V V be linear. If is a basis for V suc
Definition 6.1: Let a1, a2, , an be n independent vectors in k,
n k . The nbox in k determined by a1, a2, , an
is the set of all vectors x satisfying
x 1a1 2 a2 . n an
for 0
Theorem 6.2: The volume of the nbox in k determined by
independent vectors a1, a2
5 Complex number
Example 5.1.1: Let z1
1
3i and z 2
3 i.
31+32=(1+53)+(53+1)
5.1 Algebra of complex numbers
,

.cz
32=(153)
t
( if

1) i
A complex number is an ordered pair of real numbers, denoted
ii.
z = a + bi
where
a is the real part of z,
b is the
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H3
2015/2016
Assignment #1
You are expected to work on this assignment prior to your tutorial in the week of September 10 September 16, 2015. You may ask questions abo
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT B24H
2013/2014
Term Test Solutions
1. (a) From the lecture notes we have
Definition: A vector space V over a field F consists of a nonempty set
of objects, called vectors
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MATB24H3F
2017 Fall
Assignment #1
You are expected to work on this assignment prior to your tutorial in the week of
September 11, 2017. You may ask questions about this assig
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
Welcome to
MATB24
Linear Algebra II
Fall 2017
Course information
Course textbook:
Linear AlgebraVolume 2, 0201526751, by Fraleigh & Beauregard
Or:
Linear Algebra by Fraleigh & Beauregard, 3rd edition
Course webpage:
Blackboard
Prerequisite:
MATA22H3, MAT