Page 115 - DMTH502_LINEAR_ALGEBRA

Unit 7: Algebra of Linear Transformation

If U is a linear transformation from V into W with U = , i = 1, 2, ... n, then for the vector = Notes
i i
n
x
i i , we have
i 1
n
U U x i i
i 1
n
x (U )
i i
i 1
n
x i i
i 1
So that U is exactly the rule T which we defined above. This shows that the linear transformation
T with T = is unique.
i i
Relations and operations of Linear Transformations
1. Two linear transformations T and T from v to w are said to be equal if and only if
1 2
T 1 T 2 for all . v
2. The sum T T of linear transformation from v to w are defined, respectively, by
1 2
T 1 T 2 ) T 1 T 2 for all . v
3. The scalar multiple C  T of linear transformations from v to w are defined as
1
c  T 1 ) c T 1 ), for all , v c F .
(
Special Linear Transformation
(a) The zero linear transformation Z is defined from v to w by

Z =  for every v
(b) Negative transformation (–T) from v to w, is defined by
(–T) = – T for every v
(c) Identity linear transformation I from v to v is defined by
I = for every v

(d) Product transformation T  T .
1 2
Let v, w and y be vector spaces over the field F; let T be a linear transformation from v to w and
1
T be a linear transformation from w to y. Then the product transformation T  T is the
2 1 2
mapping from v to y defined by
(T  T ) = ( T )T for every v.
1 2 1 2
Thus for every T we have

T Z = T
T –T = Z
T  I = I  T + T.

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