ACTIVE LEARNING ASSIGNMENT

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

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 UFor More Than Two Linear Transformation:

(T3.T2.T1) (u) = T3(T2(T1(u)))

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]

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

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)

KERNEL AND 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

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)

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) x = t 1/2 y 1

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

• (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) 1 = 1 . 1 (T2.T1) T1 T2

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

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

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

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