## The Standard Basis

The elements of a vector space
$V$
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
$\left\{ 1,x,x^2 \right\}$
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
$\mathbb{R}^2$
.