Page 184 - A Course in Linear Algebra with Applications

168 Chapter Six: Linear Transformations

of introducing linear transformations, given that they can be
described by matrices. The answer is that there are situations
where the functional nature of a linear transformation is a
decided advantage. In addition there is the fact that a given
linear transformation can be represented by a host of different
matrices, depending on which ordered bases are used. The
real object of interest is the linear transformation, not the
representing matrix, which is dependent on the choice of bases.
Example 6.2.8
Define T : P n + i(R) P n (R) by the rule T(f) = /', the
n
derivative. Let us use the standard bases B = {1, x, x ,..., x }
2
and C = {l,ai, x , ...,x n_1 } for the two vector spaces. Here
l % 1 l
T{x ) = ix ~ , so [T(x )]c is the vector whose ith entry is i
and whose other entries are zero. Therefore T is represented
by the n x (n + 1) matrix

/ 0 1 0 °\
0 0 2
0
0 0 0
A 0

0 0 0 n
Vo o o 0 /

For example,
2
( \
-1
6
3
A 0
0
V 07
V 0/
which corresponds to the differentiation

2
2
T(2 - x + 3x ) = (2-x + 3x )' = 6x - 1.