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The Span of a Set of Vectors

The span of a vector space  
\[V\]
  is the set of all linear combinations of elements of  
\[V\]
.
Often we define a vector space by its basis, which is a minimal set of vectors which give rise to the vector space. The span of the basis is the vector space  
\[V\]
. A vector space is well defined, but the basis giving rise to a particular vector space may not be unique.
We may consider the vector space  
\[\mathbb{R}^2\]
  to be the span of the vectors  
\[\left\{ \begin{pmatrix}1\\0\end{pmatrix} , \begin{pmatrix}0\\1\end{pmatrix} \right\}\]
  or the set  
\[\left\{ \begin{pmatrix}1\\1\end{pmatrix} , \begin{pmatrix}1\\-1\end{pmatrix} \right\}\]

There are in general may sets of elements which give rise to the same space. In any set of elements, if the number of linearly independent elements in the set is  
\[n\]
, the sett of all linear combinations of elements of the set will give rise to a vector space  
\[V\]
  of dimension  
\[n\]
. Thus, the vectors  
\[\left\{ \begin{pmatrix}1\\0\\2\end{pmatrix} , \begin{pmatrix}0\\1\\1\end{pmatrix} \right\}\]
  gives rise to a vector space dimension 2 which is a subspace of  
\[\mathbb{R}^2\]

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