Page 118 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Find
(i) DJ p(x)
(ii) Is JD = DJ?
2
3. Which of the following functions T from R into R are linear transformations?
2
(i) T(x , x ) = (1 + x , x );
1 2 1 2
(ii) T(x , x ) = (x , x );
1 2 2 1
(iii) T(x , x ) = (x , x );
2
1 2 1 2
(iv) T(x , x ) = (x – x , 0).
1 2 1 2
7.3 Algebra of Linear Transformation

In the study of linear transformation from v into w it is of fundamental importance that the set
of these transformations inherits a natural vector space structure.

Theorem 4: Let v and w be vector spaces over the field F. Let T and U be linear transformations
from v into w. The function (T + U) defined by
(T + U) ( ) = T + U

is a linear transformation from v into w. IF c is any element of F, the function (cT) defined by
(CT) ( ) = C(T )
is a linear transformation from v into w. The set of all linear transformations from v into w,
together with the addition and scalar multiplication defined above is a vector space over the
field F.
Proof: Suppose T and U are linear transformations from v into w and that we define (T + U) as
above. Then
(T + U) (C + ) = T(C + ) + U (C + )
= C(T ) + T + C (U )+ U

= C(T + U ) + (T + U )
= C(T + U) + (T + U)
which shows that T + U is a linear transformation.
Similarly
(CT) (d + ) = C[T(d + ]

= C[d(T ) + T ]
= Cd(T ) + C (T )
= d[c(T )] + c(T )

= d[(CT) ] + C (T )
which shows that (CT) is a linear transformation. One must directly check that the vector addition
and scalar multiplication are also satisfied along the above set of linear transformations of v
into w.
We shall denote the space of linear transformations from v into w by L(v, w). It is to be understood
that L(v, w) is defined only when v and w are vector spaces over the same field F.

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