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Summary notes 2


Department
Mathematics
Course Code
MAT224H1
Professor
Sean Uppal

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MAT224H1b.doc
Page 1 of 15
Linear Transformations
EXAMPLES AND ELEMENTARY PROPERTIES
Definition
If V and W are two vector spaces, a function WVT
: is called a linear transformation if it satisfies the
following axioms:
T1:
(
)
(
)
(
)
(
)
(
)
WvTVTVvvvTVTvvTWV
+
=
+
1111 ,,,, .
T2:
(
)
(
)
VvrvTrvrTWV
=
,, R.
Example
Define 222
:MPT
, and
( )
+++
+++
=++= cbacba
cbacba
vTPcxbxav2
1
2
2. Show that T is linear.
T1:
(
)
(
)
(
)
(
)
(
)
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
+++++++++
+++++++++
=
+++++=+
111111
111111
2
1111
2
1
ccbbaaccbbaa
ccbbaaccbbaa
xccxbbaaTvvT
.
(
)
(
)
(
)
(
)
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
+++++++++
+++++++++
=
+++
+++
+
+++
+++
=
+++++=+
111111
111111
111111
111111
2
111
2
1
2
1
2
1
2
1
ccbbaaccbbaa
ccbbaaccbbaa
cbacba
cbacba
cbacba
cbacba
xcxbaTcxbxaTvTvT
.
T2:
( )
(
)
( )
vTr
cbacba
cbacba
r
rcrbrarcrbra
rcrbrarcrbra
rcxrbxraTvrT
=
+++
+++
=
+++
+++
=++=
2
1
2
1
2
.
Therefore, T is linear.
Example
The following are linear transformations:
1
:
nn PPD where
( )( ) ( )( )
=xpxpDnn ex:
(
)
323
2+=+ xxxD.
1
;+
nn PPI where
( )( ) ( )
=x
ann dyypxpI.
Theorem
Let WVT
: be a linear transformation.
1)
(
)
WV
T00
=
.
2)
(
)
(
)
VvvTvT
=
, .
3)
( )
==
=
n
i
ii
n
i
ii vTavaT
11
.
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MAT224H1b.doc
Page 2 of 15
Theorem
Let WVT
: and WVS
: be two linear transformations. Suppose that
{
}
n
vvV,,span1
=
. If
(
)
(
)
ivSvTii
=
, , then ST
=
.
Proof: Let Vvav
n
i
ii =
=1
. So
( ) ( )
==
=
=
n
i
ii
n
i
ii vTavaTvT
11
, and
( ) ( )
==
=
=
n
i
ii
n
i
ii vSavaSvS
11
. Thus,
(
)
(
)
vSvT
=
.
Theorem
Let V and W be vector spaces, and
{
}
n
ee ,,
1 a basis of V. Given any vector Www n
,,
1, there exits a
unique linear transformation WVT
: satisfying
(
)
iweTii
=
, . In fact, the action of T is as follows:
Given Vvav
n
i
ii =
=1
, then
( ) ( )
=
=
n
i
ii vTavT
1
.
Example
Find a linear transformation 222
:MPT
such that
( )
=+ 00
01
1xT,
(
)
=+ 01
10
2
xxT, and
(
)
=+ 10
00
12
xT.
(
)
(
)
(
)
{
}
22 1,,1xxxx +++ is a basis of P2.
(
)
(
)
(
)
( ) ( ) ( )
+
=
++
=
+
=
=
=
=
=++
+++++=++
2
2
2
0
0
0
0
11
3
2
1
32
21
31
2
322131
2
3
2
21
2
cba
c
cba
c
cba
c
ccc
ccb
cca
xcccxccbcca
xcxxcxccxbxa
.
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
( )
+++
+++
=
+
+
++
+
+
=
+
+
=
+++++=+++++=
22
22
10
00
2
01
10
2
00
01
2
10
00
01
10
00
01
1111
321
2
3
2
21
2
3
2
21
cbacba
cbacba
vT
cbacbacba
ccc
xTcxxTcxTcxcxxcxcTvT
.
KERNEL AND IMAGE OF A LINEAR TRANSFORMATION
Definition
Let WVT
: be a linear transformation. Then:
(
)
{
}
0|ker
=
=
vTVvT.
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MAT224H1b.doc
Page 3 of 15
(
)
{
}
VvvTT
=
|im.
Theorem
If WVT
: is a linear transformation, then Tker is a subspace of V, and
T
im
is a subspace of W.
Definition
(
)
(
)
TT kerdimnullity
=
.
(
)
(
)
TT imdimrank
=
.
Example
Given an m×n matrix A, show that ATAcolim
=
(so ATArankrank
=
), where
(
)
AXXTT mn
A=|:RR .
Write
[
]
n
CCA
1
=
where Ci are columns, and
[
]
R=i
T
nxxxX,
1.
Then
[
]
[
]
ACxCxxxCCAXA nn
T
nn colim1111 =+=== .
ONE-TO-ONE AND ONTO TRANSFORMATION
Definition
Let WVT
: be a linear transformation. Then:
T is said to be onto if WT
=
im.
T is said to be one-to-one if
(
)
(
)
11 vvvTvT
=
=
(each vector in W corresponds to only one
element in V).
Theorem
If WVT
: is a linear transformation, then T is one-to-one if and only if 0ker
=
T.
Proof:
Want: T is one-to-one 0ker
=
T.
Let Tvker
.
(
)
(
)
000
=
=
=
vTvT because T is one-to-one. So 0ker
=
T.
Want: 0ker
=
T T is one-to-one.
Let
(
)
(
)
(
)
0
11
=
=
vvTvTvT.
But since 0ker
=
T, 11 0vvvv
=
=
. So T is one-to-one.
Example
Given
(
)
(
)
xyxyxyxTT ,,,|:32 +=RR , show that T is one-to-one but not onto.
Want: T is one-to-one.
(
)
(
)
{
}
0,|,ker
=
=
yxTyxT.
( ) ( )
=
=
=
=
=+
=+= 0
0
0
0
0
0,,, y
x
x
yx
yx
xyxyxyxT. So 0ker
=
T.
Since 0ker
=
T, T is one-to-one.
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