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# Vcla.ppt COMPOSITION OF LINEAR TRANSFORMATION KERNEL AND RANGE OF LINEAR TRANSFORMATION INVERSE OF LINEAR TRANSFORMATION

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# Vcla.ppt COMPOSITION OF LINEAR TRANSFORMATION KERNEL AND RANGE OF LINEAR TRANSFORMATION INVERSE OF LINEAR TRANSFORMATION

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

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

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### Vcla.ppt COMPOSITION OF LINEAR TRANSFORMATION KERNEL AND RANGE OF LINEAR TRANSFORMATION INVERSE OF LINEAR TRANSFORMATION

1. 1. ACTIVE LEARNING ASSIGNMENT TOPIC: COMPOSITION OF LINEAR TRANSFORMATION KERNEL AND RANGE OF LINEAR TRANSFORMATION INVERSE OF LINEAR TRANSFORMATION
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 11. • (ii) T = 2 -1 -8 4 R1/2 = 1 -1/2 -8 4
12. 12. • (ii) T = 2 -1 -8 4 R1/2 = 1 -1/2 -8 4 R2+8R1 = 1 -1/2 0 0
13. 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. 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. 15. ONE TO ONE TRANSFORMATION T 1 5 2 6 3 7 V W
16. 16. T IS NOT ONE TO ONE TRANSFORMATION T 1 5 2 3 7 V W
17. 17. T IS ON TO TRANSFORMATION T 1 5 2 3 7 V W
18. 18. T ISN’T ON TO TRANSFORMATION T 1 5 2 6 3 7 V W
19. 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. 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. 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. 22. 1 1 = 3/10 -1/10 T1 T2 1/10 3/10