Page 247 - Matrix Analysis & Applied Linear Algebra

242 Chapter 4 Vector Spaces

n
ξ
and determine the action of T on any u ∈U by using u = j=1 j u j and

T(u j )= m α ij v i to write
i=1

n n n m

T(u)= T ξ j u j = ξ j T(u j )= ξ j α ij v i
j=1 j=1 j=1 i=1 (4.7.3)

= α ij ξ j v i = α ij B ji (u).
i,j i,j
This holds for all u ∈U, so T = α ij B ji , and thus B L spans L(U, V).
i,j
It now makes sense to talk about the coordinates of T ∈L(U, V) with
respect to the basis B L . In fact, the rule for determining these coordinates is
contained in the proof above, where it was demonstrated that T = i,j α ij B ji
in which the coordinates α ij are precisely the scalars in
 
α 1j
m

 α 2j 
T(u j )= α ij v i or, equivalently, [T(u j )] B =  .  , j =1, 2,...,n.
 . 
i=1 .
α mj
This suggests that rather than listing all coordinates α ij in a single column
containing mn entries (as we did with coordinate vectors), it’s more logical to
arrange the α ij ’s as an m × n matrix in which the j th column contains the
coordinates of T(u j ) with respect to B . These ideas are summarized below.

Coordinate Matrix Representations

Let B = {u 1 , u 2 ,..., u n } and B = {v 1 , v 2 ,..., v m } be bases for U
and V, respectively. The coordinate matrix of T ∈L(U, V) with

respect to the pair (B, B ) is defined to be the m × n matrix

[T] BB = [T(u 1 )] B [T(u 2 )] B ··· [T(u n )] B . (4.7.4)

In other words, if T(u j )= α 1j v 1 + α 2j v 2 + ··· + α mj v m , then

   
α 1j α 11 α 12 ··· α 1n
 α 21 α 22 ···
 α 2j  α 2n 
[T(u j )] B =  .  and [T] BB =  . . . .  . (4.7.5)
. . . .
 .   . . . . . 
α mj α m1 α m2 ··· α mn
When T is a linear operator on U, and when there is only one basis
involved, [T] B is used in place of [T] BB to denote the (necessarily
square) coordinate matrix of T with respect to B.