Page 84 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes To find the relation, it is more convenient to use the matrix of relative to the order basis
x
1
x 2
X
 ...(6)
x
n
,
rather than the n-tuple x x 2 ,...x n of co-ordinates.
1
This notation will be particularly useful as we now proceed to describe what happens to the
co-ordinates of a vector as we change from one ordered basis to another.

Suppose that we are dealing with a space V which is n dimensional and that the basis is
changed to a new basis ' i.e.

, , ,... , ' ' , ' , ' ,... ' . ...(7)
1 2 3 n 1 2 3 n
Let there be unique scalars P such that
ij

n
' P j 1,2,...n ...(8)
ij ij i
i 1
'
Let x ' ,x ' ,x ' ,...x be the co-ordinates of a given vector in the basis ' , then
1 2 3 n
x ' ' x ' ' ...x ' '
1 1 2 2 n n
n
x ' j ' j
j 1
n n
x ' P
j ij i
j 1 i 1
n n
or i P x ' j ...(8A)
ij
j 1 j 1

Putting

n
x P x ' ...(9)
i ij j
j 1
We have

n
x i i ...(10)
i 1

where now x denotes the i co-ordinate of the vector in the old .
th
i
In matrix form equation (9) becomes

X PX ' ...(11)

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