Page 113 - DMTH502_LINEAR_ALGEBRA

Unit 7: Algebra of Linear Transformation

We now show that Notes

(   b  )H (a  ) H ( . )H ...(3)
b

clearly equation 1 follows from equation (3) by selecting a = 1 = b and equation (2) follows by
choosing b = 0.
7.2 Linear Transformation

Condition (3) is the requirement of linearly and since homomorphism is a mapping we call a
homomorphism a linear transformation.
Thus a linear transformation T from a vector space V to a vector space W, both over the same
field is a mapping of V onto W such that for all , V and for all a, b, F,
( T
(a b )T ( a  Τ ) b  )

Example 1: Identity transformation. If V is any vector space, then the identity
transformation I defined by I = , is linear transformation from V into V.
The zero transformation 0, defined by 0 = 0, is a linear transformation from V into V.

Example 2: If V be the space of polynomial function f from the field F into F, given by
f ( ) C 0 C x C x 2 ...... C x n
x
2
n
1
Let Df ( ) C 1 2C x 3C x 2 ... nC x n 1
x
n
3
2
Then D is a linear transformation from V into V the differentiation transformation.
Example 3: In two dimension space V , the transformation
2
(x, y)T= (x cos – y sin , x sin + y cos ) is a linear transformation
Example 4: In the space V represented geometrically by the plane the transformation
2
(x, y)T = (ax, by)

Example 5: Let R be the field of real numbers and let V be the space of all functions from
into R which are continuous. Define T by
x
dt
x
(Tf )( ) f ( ) .
t
0
Then T is a linear transformation from V into V. The function Tf is not only continuous but has
a continuous first derivative. The linearity of integration is one of its fundamental properties.
Example 6: Let A being a fixed m n matrix with entries in the field F. The function T
defined by T(X) = AX is a linear transformation from F n × 1 into F m × 1 . The function U defined by
m
n
U( ) = A is a linear transformation from F into F .
Example 7: Let P be a fixed m m matrix with entries in the field F and let Q be a fixed
n m matrix over F. Define a function T from the space F m n into itself by T= PAQ.

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