Standard Basis

In mathematics, the standard basis (also called natural basis or canonical basis) for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system. For example, the standard basis for the Euclidean plane are the vectors

and the standard basis for three-dimensional space are the vectors

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 these vectors, including {e_x, e_y, e_z}, {e₁, e₂, e₃}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as 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

the scalars v_x, v_y, v_z being the scalar components of the vector v.

In -dimensional Euclidean space, the standard basis consists of n distinct vectors

where e_i denotes the vector with a 1 in the th coordinate and 0's elsewhere.