3. 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.
4. Introduction to Linear Transformations
Function T that maps a vector space V into a vector space W:
V: the domain of T
W: the codomain of T
5. 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.
6. 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.
7. 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.
8. 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.
9. 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)
10. 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)
11. So we have
T2 0 T1 = [T2] [T1]
Similarly, for three such linear transformations
T3 0 T2 0 T1 = [T2] [T1][T3]
12. Ex 1: (A function from R2 into R2 )
(a) Find the image of v=(-1,2). (b) Find the preimage of w=(-1,11)
Sol:
Thus {(3, 4)} is the preimage of w=(-1, 11).
13. Ex 2: (Verifying a linear transformation T from R2 into R2)
Pf:
16. Notes: Two uses of the term “linear”.
(1) is called a linear function because its graph is a line.
(2) is not a linear transformation from a vector space R into
R because it preserves neither vector addition nor scalar multiplication.
17. Ex 4: (Linear transformations and bases)
Let be a linear transformation such that
Sol:
(T is a L.T.)
Find T(2, 3, -2).
28. 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 -Sin180
Sin 180 Cos180
