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
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
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 #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
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
#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
Denition: A eld F is a set on which two operations + and
(addition and multiplication) are dened, so that, for each x, y in
F , there are unique elements x + y and x y in F (closed under +
and ) for which the following conditions hold for all elements a,
Elementary Properties of vector spaces.
Proposition: Cancellation Law for Vector Addition.
If u, v, w V , a vector space, are such that u + w = v + w, then
u = v.
Corollary 1: The zero vector (condition iii) is unique.
Corollary 2: The inverse of a vector
Complex number z = a + bi, a, b R, i =
1.
a = real part of z
b = imaginary part of z.
Benet: Fundamental Theorem of Algebra (FTA): Every
polynomial equation has a solution in the complex numbers.
C = cfw_a + bi  a, b R is the set of complex numbers.
The
Denition: Let V and W be vector spaces (over F ). A function
T : V W is called a linear tansformation from V to W if,
for all u, v V and F , we have
(i) T (u + v) = T (u) + T (v)
(preservation of vector addition)
(ii) T ( v) = T (v)
(preservation of scala