Standard basis

In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system. For example, the standard basis for the Euclidean plane is formed by vectors

and the standard basis for three-dimensional space is formed by 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. Each of these vectors is sometimes referred to as the versor of the corresponding Cartesian axis.

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 ${\displaystyle n}$-dimensional Euclidean space, the standard basis consists of n distinct vectors

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