1.12. The Linear Span of a Finite Set of Vectors
Let S  Ai | i  1,
, k
be a set of k vectors in Rn.
A vector X  R n is said to be spanned by S if we can write
k
X    i Ai
i 1
Definition
The set of all vectors spanned by S is called the linear span L  S  of S .
Thus, L  S  is just the set of all possible linear combinations of the vectors in S.
If L  S   R n , we say S span the whole space.
Example 1
Let S  A1 , then L  S    A1 |   R
Example 2
Since
k
 A  O
i 1
i
i  0  i
if
i
(aa)
therefore, every nonempty set spans O.
Eq(aa) is also called the trivial representation of O.
More interesting are the non-trivial representations of O as discussed in the following.
Definition
Let S  Ai | i  1,
, k
be a set of k vectors in Rn that spans a vector X.
We say S spans X uniquely if
k
k
i 1
i 1
X   i Ai    i Ai

i   i
Theorem 1.7
S spans L  S  uniquely iff S spans O uniquely.
i
(1.10)
Proof
Given any 2 linear combinations of X  L  S  , e.g.,
X  i Ai   i Ai
i
i
we have
 
i
 i  Ai  O
i
Now, if S spans O uniquely, we must have i  i  oi  0 for each i.
Hence, S spans X uniquely.