standard basis

In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as $\mathbb^n$ or $\mathbb^n$) is the set of vectors whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane $\mathbb^2$ formed by the pairs of real numbers, the standard basis is formed by the vectors
:$\mathbf_x = (1,0),\quad \mathbf_y = (0,1).$
Similarly, the standard basis for the three-dimensional space $\mathbb^3$ is formed by vectors
:$\mathbf_x = (1,0,0),\quad \mathbf_y = (0,1,0),\quad \mathbf_z=(0,0,1).$

Here the vector e_''x'' points in the ''x'' direction, the vector e_''y'' points in the ''y'' direction, and the vector e_''z'' points in the ''z'' direction. There are several common notations for standard-basis vectors, including , , , and . These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors).

These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as
:$v_x\,\mathbf_x + v_y\,\mathbf_y + v_z\,\mathbf_z,$
the scalars $v_x$, $v_y$, $v_z$ being the scalar components of the vector v.

In the - dimensional Euclidean space $\mathbb R^n$, the standard basis consists of ''n'' distinct vectors
:$\,$
where e_''i'' denotes the vector with a 1 in the th coordinate