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 ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for these vectors, including {ex, ey, ez}, {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 vx, vy, vz being the scalar components of the vector v.
In -dimensional Euclidean space, the standard basis consists of n distinct vectors
where ei denotes the vector with a 1 in the th coordinate and 0's elsewhere.
Read more about Standard Basis: Properties, Generalizations, Other Usages
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