Page 125 - DMTH502_LINEAR_ALGEBRA

Unit 8: Isomorphism

Notes
,
If vector is V, let T be the n-tuple x x 2 ...x of co-ordinates of relative to the ordered
1
n
basis , i.e. the n-tuple such that
x x ... x .
1 1 2 2 n n
,
given in V, there is a unique n-tuple x x 2 ...x of scalars. Thus n-tuple is unique, because if we
n
1
also have
n
z d i
i
i 1
n
then x i z d i 0
i
i 1
and the linear independence of the , tells us that x i z i 0 for each i. We call the ith co-ordinate
of relative to the ordered basis
, ,...
1 2 n
Let another vector be given by

n
y
i i
i 1
n
then x i y i i
i 1
that the ith co-ordinate of in this ordered basis is x i y i . Similarly the ith co-ordinate
,
of c is c i . One should note that every n-tuple x x 2 ,...x n in F is the n-tuple of co-ordinates
n
1
of some vector in V. Thus, there is a one-one correspondence between the set of all vectors in V
n
and the set of all n-tuples in F .
For many purposes one often regards isomorphic vector spaces as being the same, although the
vectors and operations in the spaces may be quite different, that is, one often identifies isomorphic
spaces. Let us denote the space of linear transformation from V into W by L(V,W) over the same
field F.

A Few Comments and Theorems

Suppose T is an isomorphism of V onto W. If S is a subset of V, then we have the following
theorem:
Theorem 2: Let T be a linear transformation from V into W. Then T is non-singular if and only if
T carries each linearly independent subset of V onto a linearly independent sub-set of W.
Proof: First suppose that T is non-singular. Let S be a linearly independent subset of V.
If 1 , 2 ,... n are vectors in S, then the vectors T 1 ,T 2 ,...T k are linearly independent, for if

c T c T ... c T 0
1 1 2 2 k k

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