The Standard Basis
\[V\]
of dimension \[n\]
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 \[\left\{ 1,x,x^2 \right\}\]
\[1 =\mathbf{e}_1 =\begin{pmatrix}1\\0\\0\end{pmatrix},\: x =\mathbf{e}_2 =\begin{pmatrix}0\\1\\0\end{pmatrix},\: x^2 =\mathbf{e}_3 =\begin{pmatrix}0\\0\\1\end{pmatrix} \]
This is the standard basis for
\[\mathbb{R}^2\]
. 
