Page 83 - DMTH502_LINEAR_ALGEBRA

Unit 4: Co-ordinates

Notes
th
We shall call x , the i co-ordinate of relative to the basis
i
, , ,...
1 2 3 n
,
If is an other vector having ordered co-ordinates y y ,...y , then
1 2 n
n
y ,y ,...y y ,
1 2 n i i
i 1
n n
then x i i y i i
i 1 i 1

n
x i y i i ...(3)
i 1

th
th
So that the i co-ordinate of in this ordered basis is x i y i . Similarly the i co-ordinate
,
of c is cx . It is clear that every n-tuple z z 2 ,...z is V is the n-tuple of co-ordinates of some
n
n
1
i
vector z in V namely the vector
n
n
z z ...(4)
i i
i 1
4.2 Change of Basis from One Ordered Basis to Another
In a three dimensional space V , we have 1 1,0,0 , 2 0,1,0 , 3 0,0,1 as three independent
3
set of basis vectors. We also know that by taking a certain combination of these i 's we find
another set like

1,1,0 , 1,1,1 and 0,1,1 ...(4A)
1 2 3
which is again independent. The set 1 , , 3 is related to the set , , by the relations
3
2
1
2
1 1 2
and ...(4B)
2 1 2 3
3 2 3

So by taking 1 , 2 , 3 as a new basis of V the vector will have new co-ordinate system
3
x ,x x 3 given by
,
1 2
3
x , ...(5)
i i
i 1

,
,
We can now find a relation between the new co-ordinates x x x and old co-ordinates
3
2
1
x 1 ,x 2 ,...,x of in n dimensional space.
n

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