Page 135 - DMTH502_LINEAR_ALGEBRA

Unit 9: Representation of Transformations by Matrices

(i) Show that T is linear Notes
(ii) Find the matrix M represented by T w.r.t. basis (1,0) and (0,1)

9.2 Illustrative Examples

Example 4: Let T be the linear transformation defined by

T x ,x x ,0 .
1 2
The matrix of T in the standard basis 1,0 , 0,1
1 2
1 0
is T 0 0

Let ' be the ordered basis for R given by ' 1,1 , ' 2 2,1 .
2
1
P
Then ' , ' 2 , so that matrix is
1 1 2 1
1 2 1 2
P and P 1
1 1 1 1

1
Thus T P T P
'
1 2 1 0 1 2
1 1 0 0 1 1

1 2 1 2
1 1 0 0

1 2
1 2

We can easily check that this is correct because

T ' 1,0 ' '
1 1 2
T ' 2 2,0 2 ' 1 2 ' 2 .

Example 5: Let V be the space of polynomial functions from R into R which have ‘degree’
less than or equal to 3. As in Example 3, let D be the differentiation operator on V, and let
 = f f 2 , , f 4
f
,
3
1
be the ordered basis for V defined by f x x i -1 . Let t be a real number and define g x x t i 1 ,
1 1
that is
g f
1 1

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