How to prove a set of vectors does not span a space. Ok, so I'm a bit curios as to how you can prove a set does not span a vector space.
For example, let ${S}$ be the vector set
\begin{bmatrix}
1\\
0\\
0\\
0\\
\end{bmatrix}
\begin{bmatrix}
0\\
1\\
0\\
0\\
\end{bmatrix}
\begin{bmatrix}
0\\
0\\
1\\
0\\
\end{bmatrix}
\begin{bmatrix}
0\\
0\\
0\\
1\\
\end{bmatrix}
So how would you prove that it is not in ${R^3}$.
Thanks!

Plug the vectors in as rows in a matrix, then row-reduce to find a basis for the row space.  Remember the row space of a matrix is the subspace spanned by the initial row vectors.  If you end up with one or more rows of zeros after row-reduction, then that indicates that your initial row vectors were not linearly independent.
