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Spanning Sets

A spanning set for a vector space is a set of elements of the space such that every element of the vector space can be expressed in terms of any element of the spanning set. Spanning sets are usually not unique an it is not even the case that given any vector in the vector space there is a unique combination of elements of the spanning set that returns the vector.
Example: Take the vector space as the set of linear polynomials of degree 1.
A possible spanning set is  
\[\{1,x \}\]
.
With respect to this spanning set  
\[2-x=2(1)+(-1)(x)\]
  which we may write
\[\begin{pmatrix}2\\-1\end{pmatrix}\]
.
Another possible spanning set is
\[ \{1-x,1+x \}\]
.
In terms of this spanning set  
\[2-x=(1-x)+(1+x)-\frac{3}{2}(1-x) + \frac{1}{2}(1+x)\]
  which we may write as the vector  
\[\begin{pmatrix}3/2\\1/2\end{pmatrix}\]

Every vector space has an associated dimension  
\[n\]
  and every set of elements has an order  
\[m\]
. There are four possible cases.
If  
\[m  the set of elements does not span the vector space. If  
\[m=nn\]
  the set of elements forms a basis for the vector space if and only if the set of elements of  
\[S\]
  are linearly independent. If the set of elemtns of  
\[S\]
  are linearly dependent then we can express at least one of the elements of  
\[S\]
  in terms of other elemts, so throw out this element to get a small set with  
\[n>m\]
  so  
\[S\]
  is not a spanning set.
There are four possible cases.
If  
\[m>n\]
  the set of elements is not linearly independent and may or may not span the vector space. We can reduce the siz of  
\[S\]
  by expressing vectors in terms of other vectors, throwing out the vectors so expressed until we reduce  
\[S\]
  to a linearly independent set with order 
\[s\]
. If  
\[s  the elemts of S do not span the vector space. If  
\[s=n\]
  the elemts of S spans the vector space.
\[s>n\]
  is not possible.