-
Matrix representation
Notation : An ordered basis for a finite - dimensionalvector
space V
is a basis for V endowed with a specificorder .
( es . IR'
Ello ) , 1913¥E (9) ill )} as" orderedI Ip , P2 basis )
Definition ' Let V be a finite - dimensional vector space
and
p={ vii. ii. , . . . ,In ) be an ordered basis for V .
Then , tix EV , I ! a. , az ,- .
,an EF sit . I = II
,
ai Ii.
The coordinate vector of I relative to p , denoted asEX]p ,
is the coluqrfn,vector Fine
--
( ha EF's
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Remark " Define a map V → Fn . This map islinear
Its CI3p( HW - [ a
'IyF3p = a -ii3ptCY3p )
Now , suppose V andW are finite - dimensional vector spaces
with ordered bases p - { iii. Iz , . . ,In } and 8=1 WT , . . .
, Tim }
( for V ) ( for W )respectively .
Let T - . V -1W be alinear transformation .
Then for each IEJ Eh ,I aijef csis m such
that
T ( rij ) = ,÷,
aijwi for a En ,A
w
-
Definition : with this notation as above , we call the
matrix
A ( aij ) # em thematrix representation
I s n
of T in the ordered bases Band I
,and
denoted it as A = I Ttp .If V = W and p
-- 8
,then we simply
write [T3§X C TIP
T ( iij ) = it,
aijwi for a En ,a
w
-
T ( Tj ) =,
aijwi for i En ,
. i:o÷:÷::÷④÷÷÷÷÷:÷:* .Hi:D , cab Fiends
-
p= { Ii , Ja , . . . , Tin } for V
j = { WT , Ja ,. .
.
,I'm } for W
:÷::*ina .-n
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Examples : Fn
• Let AEMmxnlH.LA: F " → Fm defined by = LA ( ¥)Et AI
'
Let p,
and 8 be thestandard bases for F
"
and Fm resp .
{ (b:o) , ) . . } nth col of A
first Col . of A
I Latif . . . cafeAf !) - - firstof = ftp.n.ntjo = A
-
• For T - - Pnllk ) → Pm, LIR) defined as Tffcxi ) = fix , .
Let p= { I , X , X ? ..
.
,X
" } be an ordered basis forPnllr )
Let D= { I , X ,X
'
,-
..
,X
" } be an ordered basis forPn-11112)
rn : repin. ."
÷
.
.. .
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Lecture 10:
Recall : . Coordinate representation ofTi EV w - r - t . B
,
Write Ii = a. I i tazvzt . . . tannin { it , , . . . , In }
I E I -
[ Its =/ !!) e Fnordered
• Matrix representation of a T: V → W
p 8
TEE. ,! . .in }
'
II . , . . . , m3
ITI -- ftp.t, . . . . ITme Mmxn
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Example : T : Mzxzllk ) → Mzxz L IR ) defined by :
TCA ) EetAT t 2AT
tram pose
P = { ('
o: ) , ( :! ) , ( o, ;) , ( °o°, )
3 - ordered basis
a m-
- K::) .li : ) .co . :) . c ::Xi Tip
-
-
( ! ! ! ! )
-
Example : T : Pz L IR ) → Mzxz CIR ) defined by :
Tcf ) Eet (Sto ) fu ) )O f ' C o )Consider ordered basis =
p = { I , × , ×' } for PLUR )
a- a ::) . c :: , .ci :3 . c ::B
Tim - - El :: ) . c :'
die ::D
:-C Ttp =/ ! ! 1) E Max ,
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Composition of linear transformations and matrix
multiplication
Thin : Let V and W be twovector spaces
over the same field 't.
And let T : Vs W and U' - W -32 be linear
.my -( it Then the composition UT : V → 2 ① I ⑥→u⑦is linear .a p
8
( ii ) If V , w , Zhaveordered bases L
, p , 8
rnespeclivmely,
P
then : [ UT ] ! = [ U ] ! [ T ) ! E Mmxn
a ←matrix multiplication .Mpxn Mpxm
-
( it Let I'
, 5'
EV and AEF.
Then :
UT Catty ) = U ( atx I -1715'
)) = a UT Cx ) t U Tty )
a
'
. UT is linear .
( ii ) Suppose a-
- { I , .ir , . . . , In }
p = I hi , , ii a , . . ,I'm }
8 = { E.,
Iz,
..
,Ep ]
[ UH = ftp.n-tjjlaih) :{inemmeans : Uco 's , =
,
air Ii
Alka- k isham=
I l: IApkT
'
heth
-
[ TIE = B Eet l bkj ) :mnmeans Thijs z!bkjTika
Mmxnlf ) for I Ey
Then : UT ( Tj ) = Ul ¥,
bkjwk )= ¥
,
beg. Utwk )
=E?bkj⇐aiaEi) - - ¥, '
i
TC is j I - entry of
So,
CUT ] ! = AB = [ U ] ! I TIP AB
-
× pExample : Consider : U : penury → Pm C IR )
defined by i
U Cfcxi )Eet ft × ,Consider T :pmfcuz, →paindefined by :Tcfcx )
) Etf! ftldtLet d and p be standard ordered bases for Pn CIR ) and
Put UR )
respectively,
in :p:÷÷÷.
.
:. " " I÷÷:÷ .
So,
[ UTI,
= CUTIE = Idgaf.li?--EIpn..umIp
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Corollary : Let V and W be finite -dimensional Vector spaces
with ordered basis p and8 respectively .
Let T : V → W be linear . Then : for anyI EV
,we have :
[Tignish
= CTI ; e p-
I Matrixmultiplication
Lin.
Transf.
Proof : Fix rt e V and consider twolinear transformations :
j& p
f : F → V g: F → W
defined bydefined by
flags= art EV g la ) = a Thi )
E W
f and g are linear transformations .Also , 9 = To -5 .
-
Let d = { I } be the standard orderedbasis for F
.
[ Thing = C gaily = C g) I= IT ] } I ftp.CTJpcfc/Dp
To't = E Ttp E utpx p 8
FE, VI. wLEI }
g8 -¥f
F - w
Egg ! -_() - - (&'D!
-
Example : T : Mzxz I IR ) → Muz C IR) defined by :
TCA ) Eef AT -12A .
is -- El ::) , c :: , .ci : , ,c :: , }
uh - - f ! ! ! ! )
iii.'
i÷ : : : in it:L:c :"
