Page 82 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Now we want to introduce co-ordinates in the vector space V analogous to the natural
co-ordinates x of the vector
i
x 1 ,x 2 ...x n

n
in the space F . The co-ordinates in the three dimensional space F are x, y, z co-ordinates. So the
3
co-ordinates of a vector in V relative to the basis will be the scalars which serve to express
as a linear combination of the vectors in the basis. If the vectors in the basis are 1 , 2 , 3 ,... n
then the vector is expressible in terms of its co-ordinates as well as in terms of the vectors of
basis as follows

n
x ,x ,x ,...x x
1 2 3 n i i ...(1)
i 1
y
For another vector having co-ordinates y 1 , ,...y we have
2
n
n
,
y ,y y ,...y y .
1 2 3 n i i
i 1
writing
x ,x x ,...x
,
1 2 3 n
the vector has a unique expression as a linear combination of the standard basis vectors (1), and
th
the i co-ordinates x of is the coefficient of i in the expression (1). By this way of ‘natural’
i
ordering of the vectors in the standard basis i.e. by writing as the first vector, as the second
1
2
vector etc. we define the order of the co-ordinates of the vector also. So we have the definition:
Definition: If V is a finite-dimensional space, the ordered basis for V is a finite sequence of basis
vectors 1 , 2 , ,... n which is a linearly independent set and spans V. So we just say that
3
, , ,... ...(2)
1 2 3 n
is an ordered basis for V. Now suppose V is a finite dimensional vector space over the field F and
(2) is an ordered basis for V, there is a unique n-tuple x 1 ,x 2 ,...x of scalars such that:
n
n
x .
i i
i 1
The n-tuple is unique, because if we also have

n
z
i i
i 1
n
then x i z i i 0
i 1
Since i for each i, is an independent set, so

x z 0
i i
or x z
i i

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