Page 250 - Matrix Analysis & Applied Linear Algebra

4.7 Linear Transformations 245

In other words, the coordinates of T(u) with respect to B are the terms
n

j=1 α ij ξ j for i =1, 2,...,m, and therefore
 
   
j α 1j ξ j α 11 α 12 ··· α 1n ξ 1

j  α 21 α 22 α 2n   ξ 2 
 α 2j ξ j  ···
.  . . . . .   . 

  = =[T] BB [u] B .

. . . .   . 
[T(u)] B = 
 .  . . . . .

j α mj ξ j α m1 α m2 ··· α mn ξ n
Example 4.7.5

Problem: Show how the action of the operator D p(t) = dp/dt on the space
P 3 of polynomials of degree three or less is given by matrix multiplication.
2 3
Solution: The coordinate matrix of D with respect to the basis B = {1,t,t ,t }
is
0100
 
 .
 0020 
0003
[D] B = 
0000
2
2
3
If p = p(t)= α 0 + α 1 t + α 2 t + α 3 t , then D(p)= α 1 +2α 2 t +3α 3 t so that
   
α 0 α 1
and  .
 α 1   2α 2 
 [D(p)] B = 
[p] B = 
α 2 3α 3
0
α 3
The action of D is accomplished by means of matrix multiplication because
   0100   
α 1 α 0
 =[D] B [p] B .
 2α 2   0020   α 1 
[D(p)] B = 
 =   
3α 3 0003 α 2
0 0000 α 3
For T ∈L(U, V) and L ∈L(V, W), the composition of L with T is

defined to be the function C : U→ W such that C(x)= L T(x) , and this
composition, denoted by C = LT, is also a linear transformation because

C(αx + y)= L T(αx + y) = L αT(x)+ T(y)

= αL T(x) + L T(y) = αC(x)+ C(y).

Consequently, if B, B , and B are bases for U, V, and W, respectively,
then C must have a coordinate matrix representation with respect to (B, B ),

so it’s only natural to ask how [C] BB is related to [L] B B and [T] BB . Re-
call that the motivation behind the definition of matrix multiplication given on
p. 93 was based on the need to represent the composition of two linear trans-
formations, so it should be no surprise to discover that [C] BB =[L] B B [T] BB .
This, along with the other properties given below, makes it clear that studying
linear transformations on finite-dimensional spaces amounts to studying matrix
algebra.