Given a set of vectors
,
the vector 
is said to
have a representation
in terms of V
If
,
the representation is called a trivial representation .
Saying that a representation of b in terms of V exists
is equivalent to saying
that 
.
Examples
,
the vector 
has a representation
in terms of V
since 
.
There are other representations of b in terms
of V including 
and 
.
Note that 
and 
do not have a representation
in terms of V.
,
the vector 
has a representation
in terms of V and this representation
is unique. Note that (0,1,0) does not have a representation
in terms of V .
With a given set of vectors, representation of a particular vector may or may not be possible and when possible, may or may not be unique, as we have seen in the above examples. The following theorem shows that when we have a representation of a vector in terms of an orthogonal set, that representation is unique.
Proof
Let and 
be a given orthogonal set in 
.
Assume b has a representation in terms of V.
Suppose 
.
Then for an arbitrary i such that 
,
uniqueness of the representation.
