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!

You simply have to check how many linearly independent vectors you have. If you have 3, you span $\mathbb{R}^3$, if you have four - they span $\mathbb{R}^4$. If you have $n$ vectors you can be sure they can't span more than $\mathbb{R}^n$.
