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ACTIVE LEARNING ASSIGNMENT
TOPIC:
COMPOSITION OF LINEAR TRANSFORMATION
KERNEL AND RANGE OF LINEAR
TRANSFORMATION
INVERSE O...
COMPOSITION OF LINEAR
TRANSFORMATION
For Two Linear Transformation:
Let T1 &T2 Be A Linear Transformation. The Application...
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...
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...
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...
KERNELAND RANGE OF LINEAR
TRANSFORMATION
Rank & Nullity Of Linear Transformation:
• The Rank Of T Is Denoted By rank(T) .
...
Continue….
• Dimension Theorem:
• If T:V W Is A Linear Transformation From A Finite
Dimensional Vector Space V To A Vector...
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
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) ...
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) ...
• (ii) T = 2 -1
-8 4
R1/2
= 1 -1/2
-8 4
• (ii) T = 2 -1
-8 4
R1/2
= 1 -1/2
-8 4
R2+8R1
= 1 -1/2
0 0
• (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]
• (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]
ONE TO ONE TRANSFORMATION
T
1 5
2 6
3 7
V W
T IS NOT ONE TO ONE
TRANSFORMATION
T
1 5
2
3 7
V W
T IS ON TO TRANSFORMATION
T
1 5
2
3 7
V W
T ISN’T ON TO TRANSFORMATION
T
1 5
2 6
3 7
V W
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)...
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...
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 1 = 3/10 -1/10
T1 T2 1/10 3/10
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

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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
  • UsmanAbubakar27

    Jul. 15, 2021
  • DeepMehta46

    Apr. 9, 2018
  • DaresAvinOmar

    Jul. 21, 2017
  • MihirRathod9

    May. 13, 2017
  • PriyanshiPatel7

    May. 8, 2017

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

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