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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