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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
.
www.notesolution.com
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.
www.notesolution.com
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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Description
MAT224H1b.doc Linear Transformations E XAMPLES AND E LEMENTARY P ROPERTIES Definition If V and W are two vector spaces, a funT :V W is called a linear transformation if it satisfies the following axioms: T1: T v +V v1)= T V +W T v1)v v1V T V T v 1 . T2: T r V v = rW T v ,r R,vV . Example a+bc a+b+c Define T : 2 M 22, andv = a+bx+cx P T2v = ( ) 1 . Show that T is linear. 2 a+b+c ab+c T 1 T v v 1)= T a a 1)+(b b 1) + c c 1) 2) . = 1 a a 1 + b b 1 c c)1( a a 1 + b b 1 + c c(1 ) ( ) 2 (a a 1 +)b b 1 + c c 1 a a 1 b b 1 + c c(1 ) ( ) 2 2 T v + T v1 = T a+bx +x )+T a1+ b1x+c 1 ) = 1 a + + + c + 1 a1+ b1 c1 a1+ b1+ c1. 2 + + c a +c 2 a1+ b1+c1 a1 b1+ c1 1 (a+a 1 +) +b 1 c+) 1 a +)1 + b+b1 + c+ 1 ) ( ) = 2 (a+a +) +b + c+) ( a+a ) (+b + )+( ) ( ) 1 1 1 1 1 1 T 2 1 ra rb rc ra rb rc T r v )T ra rbx rcx 2)= 2 ra rb rc ra rb rc . =r 1 a + + + c = T(v) 2 + + c a +c Therefore, T is linear. Example The following are linear transformations: 2 D : n P n1 where D p xn= p x n ex: D x +3x = 2x +3 . x I;Pn Pn+1 where I p n = ) ap n dy .( ) Theorem Let T :V W be a linear transformation. 1) T 0) = 0 . V W 2) T v = T v ,vV . n n 3) T a v = a T v( ). i1 i i i1 i i Page 1 of 15 www.notesolution.com MAT224H1b.doc Theorem Let T :V W and S :V W be two linear transformations. Suppose that V =span v , ,v }. If 1 n T vi= S v ii , then T = S . n n n Proof: Let v = aiviV . So T v = T aivi = aiT v i , and ( ) i=1 i 1 i=1 n n S(v)= S a v = a S v . Thus,(T ))= S v). i i i i i=1 =1 Theorem Let V and W be vector spaces, ande1, ,en} a basis oV . Given any vectow 1, ,wnW , there exits a unique linear transformatioT :V W satisfyingT e )= w ,i . In fact, the actioTis as follows: i i n n Given v = a iv iV , thenT v = aiT vi .( ) i1 i=1 Example 1 0 2 0 1 Find a linear transformatioT: P2 M 22 such thatT(1+ x =)0 0 , T x + x )= 1 0 , and 2 0 0 T 1+x )= . 0 1 2 2 (1+ x, x + x , 1+ x } is a basis Pf2 a + bx+ cx2 =c (1+ x)+c x( +x 2)+c 1+ x2) 1 2 3 a +bc c1= a c1 c3 = 0 2 2 a +b+c . (a c1 c3 + b c 1 c2 x + c c 2 c3x = 0 b c)1 c2 = 0 c2= c c c = 0 2 2 3 a b+c c3= 2 T v )=T c1(1+ x)+c 2( +x 2)+c3 1+ x2))= c1 1 + x)+c2 x + x2)+ c3T(1+ x2) 1 0 1 0 0 a+ c 1 0 + +b c 0 1 a b c 0 0 = c10 0 +c2 1 0+ c30 1 = 0 0 + 1 0 + 0 1 . 2 2 2 a +bc a +b+c T( )= 2 2 a +b+c a b+c 2 2 K ERNEL AND IMAGE OF A L INEAR T RANSFORMATION Definition Let T :V W be a linear transformation. Then: kerT = vV T v = 0 . Page 2 of 15 www.notesolution.com
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