Linear transformations thestuffpoint.com

1. Linear transformation
TheStuffPoint.Com
BY: Abu Bakar Soomro

2. Linear transformation
( ),U F ( )V F
:T U V
1 2For all , , , ;u u u U a F 
( ) ( ).a aT Tu u
1 2 1 2( ) ( ) ( ),u uT T Tu u  


1 2 1 2
1 2
( ) ( ) ( )
, , , ;
uaT Tu u u
u u
T
a FbU
b  
  

3. 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 y
T ax ax ax by by by
T ax by ax by ax by
ax 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
, ))
( , ) ( , )
( ( ), ( )) ( ( ), ( ))
( ( ) ( ), ( ) ( ))
( , )
y
a x x x x b y y y y
a x x a x x b y y b y y
a x x b y y a x x b y y
ax by ax by ax by ax by
     
     
      
      
1 2 1 2( ) ( ) ( )T au bu aT u bT u  

4. Matrix of linear transformation
:T U V
( ) ,T Au u u U 
: n m
R RT 
( ) ,m
n
nA RT x x x 
:r
r
c
c
RA T R  

5. 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

6. 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 0
0 1
,
( , , )0
0 0( )1
,
, , .
u u u
u
u
u
B 



1 2 3
1
2
2
*
{ , , },
( , ),
( ).
1 0
0 1,
v v v
v
v
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  

7. 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     

8. 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 9
5 3 2
A
 
   

9. If is linear transformation
given by the matrix ,
find m, n and express T in terms of coordinates.
Q:
: n m
R RT 
6 1
1 2
1 3
 
 
 
  
:
r
r
c
c
R
A
T R


b
6 1
1 2
1 3
A
 
   
  
Solution:
3 2r c  
2 3
: R RT 
.2 3,n m 

10. 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 2
1
1 2
2
1 2
6 1 6
( ) 1 2 2
1 3 3
x x
x
T x x x
x
x x
   
          
        