Linear transformation

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BY: Abu Bakar Soomro

Linear transformation( ),U F ( )V F

:T U V

1 2For all , , , ;u u u U a F

( ) ( ).a aT Tu u1 2 1 2( ) ( ) ( ),u uT T Tu u

1 2 1 2

1 2

( ) ( ) ( ) , , , ;uaT Tu u u

u uT

a FbUb

3 2: R RT

1 2 11 3 32( , , ) ( , )x x xxT xx x 1 2 1 2 3 1 2 3

1 2 3 1 2 3

1 1 2 2 3 3

1 1 2 2 1 1 3 3

1 2 1 2 3 1

( ) ( ( , , ) ( , , )) (( , , ) ( , , )) ( , , )

( , )( ) ( ) ( , , ) ( ,

T au bu T a x x x b y y yT ax ax ax by by byT ax by ax by ax byax by ax by ax by ax by

aT u bT u aT x x x bT y y

2 3

1 2 1 3 1 2 1 3

1 2 1 3 1 2 1 3

1 2 1 2 1 3 1 3

1 1 2 2 1 1 3 3

, ))( , ) ( , )

( ( ), ( )) ( ( ), ( ))( ( ) ( ), ( ) ( ))( , )

ya x x x x b y y y ya x x a x x b y y b y ya x x b y y a x x b y yax by ax by ax by ax by

1 2 1 2( ) ( ) ( )T au bu aT u bT u

Matrix of linear transformation

:T U V

( ) ,T Au u u U

: n mR RT

( ) ,mn

nA RT x x x

:rr

ccRA T R

Q:

2

3( ),( )

R

R

F

F

1 2 3 1 21 2 3 33 4 9 5 3 2( , , ) ( , )x x x x xx xT x x

3 2: RRT

Find the matrix of linear transformation with respect to standard bases for the vector spaces

Standard basis for the vector space is:3( )R F 2( )R F

Standard basis for the vector space is:

1

*1 1 2 3

2

3

{ , , },( , , )1 0 00 1

,( , , )00 0( )1

,, , .

u u uuuu

B

1 2 3

1

2

2

* { , , },( , ),( ).1 00 1,

v v vvv

B

1 2 3 1 21 2 3 33 4 9 5 3 2( , , ) ( , )x x x x xx xT x x

1 1 0( ) ( , , ) 3( 50 , )T Tu

2 0 1 0( ) ( , 3, ) ( , )4uT T

3 0 0 1( ) ( , 2, ) ( , )9uT T

21 1( ) ( , , ) 3 5( , ) ( , ) (1 0 01 0 )0 ,1aT u b aT v bv

( , ) ( , )3 5 0 0( , ) ( , )a b a b

( , ) ( , )3 5a b

11 2( ) ( , , ) ( ,3 5) ,1 0 0 3 5T v vTu

12 2( ) ( , , ) ( , )4 30 1 0 ,4 3u vT T v

3 1 29( ) ( , , ) ( 20 0 , ) 9 2 .1 ( )v vuT T

11 2( ) ,3 5vuT v

2 1 24( ) ,3u v vT

13 29 2( ) ( ) .T v vu

Hence, the matrix of transformation is

3 4 95 3 2

A

If is linear transformation given by the matrix ,

find m, n and express T in terms of coordinates.

Q:

: n mR RT 6 11 21 3

:

r

r

c

cR

A

T R

6 11 21 3

A

Solution:

3 2r c 2 3: R RT

.2 3,n m

1 2 1 2 1 2 1 2( , ) (6 , 2 , 3 )T x x x x x x x x

1 2( ) , ( , )T x Ax x x x

1 21

1 22

1 2

6 1 6( ) 1 2 2

1 3 3

x xx

T x x xx

x x