G.H.PATEL COLLEGE OF ENGINEERING & TECHNOLOGYANAND

Chapter : 4Linear Transformations

2110015__150110111041(Shreyans Patel) 150110111042(Smit Patel)

150110111043(Piyush Kabra)150110111044(Hardik Ramani)150110111045(Shivam Roy)

GENERAL LINEAR TRANSFORMATIONS

INTRODUCTION :- Linear Transformation is a function from one vector

space to another vector space satisfying certain conditions. In particular, a linear transformation from Rn to Rm is know as the Euclidean linear transformation . Linear transformation have important applications in physics, engineering and various branches of mathematics.

Introduction to Linear Transformations Function T that maps a vector space V into a vector space W:

spacevector :, ,: mapping WVWVT

V: the domain of T

W: the codomain of T

DEFINITION :-

Let V and W be two vectors spaces. Then a function T : V W is called a linear transformation from V to W if for all u, U Ɛ V and all scalars k,

T(u + v) = T(u) T(v); T(ku) = kT(u). If V = W, the linear transformation T: V V is called a linear

operator on V. 

PROPERTIES OF LINEAR TRANSFORMATION :-

Let T : V W be a linear transformation. Then T(0) = o T(-v) = -T(u) for all u Ɛ V T(u-v) = T(u) – T(v) for all u, u Ɛ V T(k1v1 + k2v2+ ….. +knvn) = k1T(v1) + k2T(v2) + …..

+knT(vn), Where v1,v2,….vn Ɛ V and k1, k2, …. Kn are scalars.

Standard Linear Transformations

Matrix Transformation: let T : Rn Rm be a linear transformation. Then there always exists an m × n matrix A such that

T(x) = Ax This transformation is called the matrix transformation or the

Euclidean linear transformation. Here A is called the standard matrix for T. It is denoted by [T].

For example, T : R3 R2 defined by T(x,y,z) = (x = y-z, 2y = 3z, 3x+2y+5z) is a matrix transformation.

ZERO TRANSFORMATION Let V and W be vector spaces.

The mapping T : V W defined by

T(u) = 0 for all u Ɛ V

Is called the zero transformation. It is easy to verify that T is a linear transformation.

IDENTITY TRANSFORMATION Let V be any vector space.

The mapping I : V V defined by

I(u) = u for all u Ɛ V

Is called the identity operator on V. it is for the reader to verify that I is linear.

Linear transformation from images of basic vectors

A linear transformation is completely determined by the images of any set of basis vectors. Let T : V W be a linear transformation and {v1,v2,……vn} can be any basis for V. Then the image T(v) of any vector u Ɛ V can be calculated using the following steps.

STEP 1: Express u as a linear combination of the basis vectors v1,v2,……,vn,say

V = k1v1 + k2v2+ ….. +knvn.

STEP 2: Apply the linear transformation T on v as T(v) = T(k1v1 + k2v2+ ….. +knvn) T(v) = k1T( v1)+ k2 T(v2)+ ….. +knT(vn)

Composition of linear Transformations Let T1 : U V and T2 : V W be linear transformation. Then the composition

of T2 with T1 denoted by T2 with T1 is the linear transformation defined by,

(T2 O T1)(u) = T2(T1(u)), where u Ɛ U.

Suppose that T1 : Rn Rm and T2 : Rm RK are linear transformation. Then there exist matrics A and B of order m × n and k × m respectively such that T1(x) = Ax and T2 (x) = Bx

Thus A = [T1] and B = [T2].Now,

(T2 0 T1)(x) = T2 T1(x) = T2 (Ax) = B(Ax) (BA)(x) = ([T1][T2])(x)

So we have T2 0 T1 = [T2] [T1]

Similarly, for three such linear transformations

T3 0 T2 0 T1 = [T2] [T1][T3]

Ex 1: (A function from R2 into R2 )22: RRT

)2,(),( 212121 vvvvvvT

221 ),( Rvv v

(a) Find the image of v=(-1,2). (b) Find the preimage of w=(-1,11)

Sol:)3 ,3())2(21 ,21()2 ,1()(

)2 ,1( )(

TT

av

v

)11 ,1()( )( wvTb

11 2 1

21

21

vvvv

4 ,3 21 vv Thus {(3, 4)} is the preimage of w=(-1, 11).

Ex 2: (Verifying a linear transformation T from R2 into R2)

Pf:

)2,(),( 212121 vvvvvvT

number realany : ,in vector : ),( ),,( 22121 cRvvuu vu

),(),(),( :addition(1)Vector

22112121 vuvuvvuu vu

)()()2,()2,(

))2()2(),()(())(2)(),()((

),()(

21212121

21212121

22112211

2211

vu

vu

TTvvvvuuuu

vvuuvvuuvuvuvuvu

vuvuTT

),(),( tionmultiplicaScalar )2(

2121 cucuuucc u

Therefore, T is a linear transformation.

Ex 3: (Functions that are not linear transformations)

xxfa sin)()(

2)()( xxfb

1)()( xxfc

)sin()sin()sin( 2121 xxxx )sin()sin()sin( 3232

22

21

221 )( xxxx

222 21)21(

1)( 2121 xxxxf2)1()1()()( 212121 xxxxxfxf

)()()( 2121 xfxfxxf

nnsformatiolinear tra not is sin)( xxf

nnsformatio tra linearnot is )( 2xxf

nnsformatiolinear tra not is 1)( xxf

Notes: Two uses of the term “linear”.

(1) is called a linear function because its graph is a line.

1)( xxf

(2) is not a linear transformation from a vector space R into R because it preserves neither vector addition nor scalar multiplication.

1)( xxf

Ex 4: (Linear transformations and bases)Let be a linear transformation such that 33: RRT

Sol:)1,0,0(2)0,1,0(3)0,0,1(2)2,3,2(

)0,7,7( )1,3,0(2)2,5,1(3)4,1,2(2 )1,0,0(2)0,1,0(3)0,0,1(2)2,3,2(

TTTTT (T is a L.T.)

Find T(2, 3, -2).

Applications of Linear Operators

1. Reflection with respect to x-axis:?

For example, the reflection for the triangle with vertices is

The plot is given below.

2. Reflection with respect to y=-x :

Thus, the reflection for the triangle with vertices (-1,4),(3,1),(2,6)is

The plot is given below

3. Rotation: Counterclockwise

For example, as

Thus, the rotation for the triangle with vertices is

Rotation: Counterclockwise

The plot is given below.

Rotation: Counterclockwise

Thus, the rotation for the triangle with vertices (0,0),(0,1),(-1,1) is

=L 0 -11 0

00

00

00

L

00

01 = 0 -

11 0

01

-1 0

Rotation: Counterclockwise

The plot is given below.

L -1 1 = -1

1 = -1-1

(-1,1) (0,1) (1,1)

(0,0)

(-1,-1) (0,-1)

(1,0)

Rotation: Counterclockwise

Thus, the rotation for the triangle with vertices (0,0),(-1,0),(-1,-1) is

=L 0 -11 0

00

00

00

L

00

-1 0 = 0 -

11 0

-1 0

0-1

Rotation: Counterclockwise

The plot is given below.

L -1-1 = -1

-1 = 1-1

(-1,1) (0,1) (1,1)

(0,0)

(-1,-1) (0,-1)(1,-1)

(1,0)

Rotation: Counterclockwise

Rotation clockwise

For example, as =180

Thus, the rotation for the triangle with vertices (0,0),(-1,-1),(0,-1) is

A 0 1-1 0

Cos180 -Sin180Sin 180 Cos180

Rotation clockwise

=L 0 1-1 0

00

00

00

L

00

-1-1 = 0 1

-1 0

-1-1

0-1

=L 0 1-1 0

0-1

0-1

00

(-1,-1)

(0,0)

(0,-1)

(-1,1) (0,1)

Rotation clockwise

Shear in the x-direction:

For example, as ,

Thus, the shear for the rectangle with vertices in the x-direction is

Shear in the x-direction:

The plot is given below.

THANKS