COMPOSITION OF LINEAR TRANSFORMATION KERNEL AND RANGE OF LINEAR TRANSFORMATION INVERSE OF LINEAR TRANSFORMATION

1. ACTIVE LEARNING ASSIGNMENT
TOPIC:
COMPOSITION OF LINEAR TRANSFORMATION
KERNEL AND RANGE OF LINEAR
TRANSFORMATION
INVERSE OF LINEAR TRANSFORMATION

2. COMPOSITION OF LINEAR
TRANSFORMATION
For Two Linear Transformation:
Let T1 &T2 Be A Linear Transformation. The Application Of
T1 Followed By T2 Produces A Transformation From U To W.
This Is Called The Composition Of T2 With T1 & Is Denoted
By ‘T2.T1’
(T2.T1) (u) = T2(T1(u))
U= Vector in U
For More Than Two Linear Transformation:
(T3.T2.T1) (u) = T3(T2(T1(u)))

3. Example
T1(x , y)=(2x,3y) Find Domain & Codomain Of
T2(x , y)=(x-y , x+y) (T2.T1)
Solution:-
• Domain & Codomain Of (T2.T1):- [T2][T1]

4. Example
T1(x , y)=(2x,3y) Find Domain & Codomain Of
T2(x , y)=(x-y , x+y) (T2.T1)
Solution:-
• Domain & Codomain Of (T2.T1):- [T2][T1]
= 1 -1 2 0 = 2 -3
1 1 0 3 2 -3

5. Example
T1(x , y)=(2x,3y) Find Domain & Codomain Of
T2(x , y)=(x-y , x+y) (T2.T1)
Solution:-
• Domain & Codomain Of (T2.T1):- [T2][T1]
= 1 -1 2 0 = 2 -3
1 1 0 3 2 -3
(T2.T1)(x , y) = (2x-3y , 2x+3y)

6. KERNELAND RANGE OF LINEAR
TRANSFORMATION
Rank & Nullity Of Linear Transformation:
• The Rank Of T Is Denoted By rank(T) .
• The Nullity Of T Is The Dimension Of The Kernel Of T &
Is Denoted By Nullity(T).
• Theorem 1 :-
• Nullity(Ta) = Nullity(a) ; Rank(Ta) = rank(a)
• We Can Conclude That,
• Ker(T) = Basic For The Null Space
• R(T) = Basic For The Column Space

7. Continue….
• Dimension Theorem:
• If T:V W Is A Linear Transformation From A Finite
Dimensional Vector Space V To A Vector Space W Then,
Rank(T) + Nullity(T) = Dim(V)

8. Example:-
T(x , y) = (2x+y , -8x+4y)
Find The Ker(T) & R(T).
Solution:-
2x- y = 0
-8x+ 4y = 0 2x- y = 0
y = t x = t/2

9. Example:-
T(x , y) = (2x+y , -8x+4y)
Find The Ker(T) & R(T).
Solution:-
2x- y = 0
-8x+ 4y = 0 2x- y = 0
y = t x = t/2
(i) x = t 1/2
y 1

10. Example:-
T(x , y) = (2x+y , -8x+4y)
Find The Ker(T) & R(T).
Solution:-
2x- y = 0
-8x+ 4y = 0 2x- y = 0
y = t x = t/2
(i) x = t 1/2
y 1
ker(T) = 1/2
1

11. • (ii) T = 2 -1
-8 4
R1/2
= 1 -1/2
-8 4

12. • (ii) T = 2 -1
-8 4
R1/2
= 1 -1/2
-8 4
R2+8R1
= 1 -1/2
0 0

13. • (ii) T = 2 -1
-8 4
R1/2
= 1 -1/2
-8 4
R2+8R1
= 1 -1/2
0 0
Basic For R(T) = Basic For Column Space Of [T]

14. • (ii) T = 2 -1
-8 4
2
R1/2 =
= 1 -1/2 -8
-8 4
R2+8R1
= 1 -1/2
0 0
Basic For R(T) = Basic For Column Space Of [T]

15. ONE TO ONE TRANSFORMATION
T
1 5
2 6
3 7
V W

16. T IS NOT ONE TO ONE
TRANSFORMATION
T
1 5
2
3 7
V W

17. T IS ON TO TRANSFORMATION
T
1 5
2
3 7
V W

18. T ISN’T ON TO TRANSFORMATION
T
1 5
2 6
3 7
V W

19. INVERSE OF LINEAR
TRANSFORMATION
If T1 : U V & T2 : V W Are One
To One Transformation Then ,
(i) T2.T1 Is One To One.
(ii) 1 = 1 . 1
(T2.T1) T1 T2

20. EXAMPLE
• [T1] = 1 1 [T2] = 2 1
1 -1 1 -2
Verify The Inverse Of (T2.T1)
Solution:-
(T2.T1) = 3 1 1 = 3/10 -1/10
-1 3 T2.T1 1/10 3/10

21. EXAMPLE
• [T1] = 1 1 [T2] = 2 1
1 -1 1 -2
Verify The Inverse Of (T2.T1)
Solution:-
(T2.T1) = 3 1 1 = 3/10 -1/10
-1 3 T2.T1 1/10 3/10
1 = 1/2 1/2 1 = 2/5 1/5
T1 1/2 -1/2 T2 1/5 -2/5

22. 1 1 = 3/10 -1/10
T1 T2 1/10 3/10