Page 106 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes (d) If  is in W, let X denote the coordinate matrix of  relative to the -basis and X’ the
coordinate matrix of  relative to the ’-basis. Find the 3 × 3 matrix P such that X = PX’ for
every such .
To answer these questions by the first method we form the matrix A with row vectors  ,  ,  ,
1 2 3
find the row-reduced echelon matrix R which is row-equivalent to A and simultaneously perform
the same operations on the identity to obtain the invertible matrix Q such that R = QA:
1 2 2 1 1 0 2 0
0 2 0 1 R = 0 1 0 0
2 0 4 3 0 0 0 1

1 0 0 1 6 6 0
0 1 0 Q = 2 5 1
0 0 1 6 4 4 2
(a) Clearly R has rank 3, so  ,  and  are independent.
1 2 3
(b) Which vectors  = (b , b , b , b ) are in W? We have the basis for W given by  ,  ,  , the row
1 2 3 4 1 2 3
vectors of R. One can see at a glance that the span of  ,  ,  consists of the vectors  for
1 2 3
which b = 2b . For such a  we have
3 1
 = b  + b  + b 
1 1 2 2 4 4
= [b , b , b ]R
1 2 4
= [b b b ]QA
1 2 4
= x  + x  + x a
1 1 2 2 3 3
where x = [b b b ]Q :
i 1 2 4 i
1 2
x 1 b 1 b 2 b 4
3 3
5 2
x 2 b 1 b 2 b 4 ... (1)
6 3
1 1
x 3 b 2 b 4
6 3
(c) The vectors 1 , 2 , 3 are all of the form (y , y , y , y ) with y = 2y and thus they are in W.
1
4
3
1
3
2
One can see at a glance that they are independent.
(d) The matrix P has for its columns
P = ' 
j j
where  = { ,  ,  }. The equations (1) tell us how to find the coordinate matrices for , , .
1 2 3 1 2 3
For example with  = ' we have b = 1, b = 0, b = 2, b = 0, and
1 1 2 3 4
1 2
x = 1  (0)  (0)  1
1
3 3
5 2
x =   (0)  (0)   1
1
2
6 3
1 1
x =  (0)  (0)  0
3
6 3

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