Page 248 - Matrix Analysis & Applied Linear Algebra

4.7 Linear Transformations 243

Example 4.7.3
Problem: If P is the projector defined in Example 4.7.1 that maps each point
3
v =(x, y, z) ∈ to its orthogonal projection P(v)=(x, y, 0) in the xy -plane,
determine the coordinate matrix [P] B with respect to the basis

      
1 1 1
 
1
2
2
B = u 1 =   , u 2 =   , u 3 =   .
1 2 3
 
Solution: According to (4.7.4), the j th column in [P] B is [P(u j )] B . Therefore,
   
1 1
1
P(u 1 )=   =1u 1 +1u 2 − 1u 3 =⇒ [P(u 1 )] B =  1  ,
0 −1
   
1 0
2
P(u 2 )=   =0u 1 +3u 2 − 2u 3 =⇒ [P(u 2 )] B =  3  ,
0 −2
   
1 0
P(u 3 )=   =0u 1 +3u 2 − 2u 3 =⇒ [P(u 3 )] B =  3  ,
2
0 −2

 
1 0 0
so that [P] B =  1 3 3  .
−1 −2 −2

Example 4.7.4
Problem: Consider the same problem given in Example 4.7.3, but use different
bases—say,

  
   
1 1 1
 
1
0
1
B = u 1 =   , u 2 =   , u 3 =  
0 0 1
 
and
      
−1 0 0
 
1
B = v 1 =  0  , v 2 =   , v 3 =  1  .

0 0 −1
 
For the projector defined by P(x, y, z)=(x, y, 0), determine [P] BB .
Solution: Determine the coordinates of each P(u j ) with respect to B , as