Page 131 - DMTH502_LINEAR_ALGEBRA

Unit 9: Representation of Transformations by Matrices

one space into another the matrix depends upon two ordered bases, one for V and one for W). In Notes
order that we shall not forget this dependence, we shall use the notation

T 
for the matrix of the linear operator T in the ordered basis  . The manner in which this matrix
and the ordered basis describe T is that for each in V

T T .
  

Example 1: Let V be the space of n×1 column matrices over the field F; let W be the space
of m × 1 matrices over F; and let A be a fixed m × n matrix over F. Let T be the linear transformation
of V into W defined by T(X) = AX. Let be the ordered basis for V analogous to the standard
th
basis in F , i.e., the i vector in  in the n × 1 matrix X with a 1 in row i and all other entries 0. Let
n
i
'  be the corresponding ordered basis for W, i.e. the jth vector in ' is the m×1 matrix Y with a

i

1 in row j and all other entries 0. Then the matrix of T relative to the pair  , ' is the matrix A
itself. This is clear because the matrix AX is the j column of A.
th
j
2
Example 2: Let F be a field and let T be the operator of F defined by
T x 1 ,x 2 x 1 ,0 .
It is easy to see that T is a linear operator in F . Let  be the standard ordered basis for
2
F 2 , 1 , 2 . Now
T T 1,0 1,0 1 0
1 1 2
T 2 T 0,1 0,0 0 1 0 2

so the matrix of T in the ordered basis  is

1 0
T .
 0 0

Example 3: Let V be the space of all polynomial functions from R into R of the form

f x c c x c x 2 c x 3
0 1 2 3
that is, the space of polynomial functions of degree three or less. The differentiation operator D
maps V into V, since D is ‘degree’ decreasing’. Let  be the ordered basis for V consisting of the
four functions , , ,f f f f defined by f x x i 1 . Then
1 2 3 4 i
Df x 0, Df 0 f 0 f 0 f 0 f
1 1 1 2 3 4
Df 2 x 1, Df 2 1 f 1 0 f 2 0 f 3 0 f 4

x
Df 3 x 2 , Df 3 0 f 1 2 f 2 0 f 3 0 f 4
2
Df x 3 , Df 0 f 0 f 3 f 0 f
x
4 4 1 2 3 4

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