Page 249 - Matrix Analysis & Applied Linear Algebra

244 Chapter 4 Vector Spaces

shown below:
   
1 −1
0
P(u 1 )=   = −1v 1 +0v 2 +0v 3 =⇒ [P(u 1 )] B =  0  ,
0 0
   
1 −1
P(u 2 )=   = −1v 1 +1v 2 +0v 3 =⇒ [P(u 2 )] B =  1  ,
1
0 0
   
1 −1
P(u 3 )=   = −1v 1 +1v 2 +0v 3 =⇒ [P(u 3 )] B =  1  .
1
0 0
−1 −1 −1

Therefore, according to (4.7.4), [P] BB = 0 1 1 .
0 0 0
At the heart of linear algebra is the realization that the theory of finite-
dimensional linear transformations is essentially the same as the theory of ma-
trices. This is due primarily to the fundamental fact that the action of a linear
transformation T on a vector u is precisely matrix multiplication between the
coordinates of T and the coordinates of u.

Action as Matrix Multiplication

Let T ∈L(U, V), and let B and B be bases for U and V, respectively.
For each u ∈U, the action of T on u is given by matrix multiplication
between their coordinates in the sense that
[T(u)] B =[T] BB [u] B . (4.7.6)

n

Proof. Let B = {u 1 , u 2 ,..., u n } and B = {v 1 , v 2 ,..., v m } . If u = j=1 j u j
ξ
m

and T(u j )= α ij v i , then
i=1
   
ξ 1 α 11 α 12 ··· α 1n
 α 21 α 22 ···
 ξ 2  α 2n 
[u] B =  .  and [T] BB =  . . . .  ,
.
 .   . . . . . . . . 
ξ n α m1 α m2 ··· α mn
so, according to (4.7.3),
m n

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