The Standard Basis

The elements of a vector space  
  of dimension  
  can be written as vectors with n components. The standard basis for the vector space consists of elements of the form  
\[\mathbf{e}_i=\begin{pmatrix}0\\0\\ \vdots \\1\\ \vdots \\0\\0 \end{pmatrix} \]
  or equivalent with a 1 in the ith row and all other entries equal to zero. For example, the vector space of all polynomials of degree 2 may be written with respect to the standard basis  
\[\left\{ 1,x,x^2 \right\}\]
We may assign  
\[1 =\mathbf{e}_1 =\begin{pmatrix}1\\0\\0\end{pmatrix},\: x =\mathbf{e}_2 =\begin{pmatrix}0\\1\\0\end{pmatrix},\: x^2 =\mathbf{e}_3 =\begin{pmatrix}0\\0\\1\end{pmatrix} \]

This is the standard basis for