Page 114 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Then T is a linear transformation from F m n into F m n ,
because
T(CA + B) = P(CA+ B)Q
= (CPA + PB)Q

= C PAQ + PBQ
= CT(A) + T(B)

Example 8: The linear transformation preserves the linear combination; that is, if
, , ... are vectors in V and C , C , .... C are scalars, then
1 2 n 1 2 n
( T C 1 1 C 2 2 ... C n n ) C 1 (T ) C T ( 2 ) .... C n (T n ).
2
This follows readily from the definition. For example

( T C 1 1 C 2 2 ) C 1 (T ) C 2 (T 2 )
Theorem 1: Let V be a finite dimensional vector space over the field F and let ( , , ... ) be an
1 2 n
ordered basis for V. Let W be a vector space over the same field F and let , , ... , be a set of
1 2 n
any vectors in W. There is precisely one linear transformation T from V into W such that
T = , i = 1, 2, .... n
i i
Proof: To prove that there is some linear transformation T with T = i, we proceed as follows,
i
given in V, there is a unique n-tuple (x , x , .... x ) such that
1 2 n
= x + x + .... + x
1 i 2 2 n n
For this we define
T = x + x + .... + x .
1 1 2 2 n n
Then T is a well define rule for associating with each vector in V a vector T in W. From the
definition it is clear that T = for each j. To see that T is linear, let
j j
= y + y + .... + y
1 1 2 2 n n
be in V and let C be any scalar. Now

C (Cx 1 y 1 ) 1 (Cx 2 y 2 ) 2 .... (Cx n y n ) n

and so by definition
( T C ) (Cx y ) (Cx y ) ...... (Cx y )
1 1 1 2 2 2 n n n
on the other hand
n n
C (T ) T e x i i y i i
i 1 i 1
n
(Cx i y i ) i
i 1
and thus

( T C ) C (T ) (T )

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