Page 85 - DMTH502_LINEAR_ALGEBRA

Unit 4: Co-ordinates

' Notes
1 1
'
where X 2 , ' 2
X
 
'
n
n
P P ij ...(12)

where P is an n × n matrix whose i,j entry is P since and ' basis are independent sets, X 0 is
ij
only possible if X ' 0 also. Now the transformation matrix P is such that its inverse exists.
Hence multiplying (6) by P we obtain
1
1
X ' P X ...(13)
'
So the new set of co-ordinates x ' ,x ' ,x ' ,...x are related to the old set of co-ordinates
1 2 3 n
x ,x ,...x n of the vector by the relation (13).
1 2

Example 1: From equation (4), P matrix is given by
1 1 0
P 1 1 1
0 1 1

1 1 0
let P 1 1 1 1
0 1 1

'
Thus the new basis ' ' , , ' is given in terms of old basis , , 3 by the matrix
1 2 3 1 2
relation
'
1 1 1 0 1
' 1 1 0
2 2 ...(14)
' 0 1 1
3 3
0 1 1
Now P 1 1 1 1 ...(15)
1 1 0

,
If the co-ordinates of x 1 ,x x 3 in old basis then in the new basis ' they are given by
2
x ' 0 1 1 x
1 1
x ' 2 1 1 1 x 2 ...(16)
x ' 1 1 0 x 3
3

Example 2: Show that the vectors ' 1,1,0,0 , ' 0,0,1,1 , ' 1,0,0,4 , ' 0,0,0,2
1 2 3 4
form a basis for R 4 . Find the co-ordinates of each of the standard basis vectors in the ordered
'
'
basis ' , , , ' .
1 2 3 4
'
'
Solution: To prove that the set ' , ' , , form a basis, we have to show that they are independent.
1 2 3 4
c
c
So let c 1 , , ,c 4 are scalars not all of them zero such that ' s are dependent, then
3
2
i
c ' c ' c ' c ' 0
1 1 2 2 3 3 4 4
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