standardbasis

Standard basis, also called the canonical basis, refers to a natural choice of basis in a vector space used to express coordinates of elements. In the real n-dimensional space R^n, the standard basis consists of n vectors e1, e2, ..., en where each e_i has a 1 in the i-th position and 0 in all other positions: e1 = (1, 0, ..., 0), e2 = (0, 1, ..., 0), ..., en = (0, ..., 0, 1). Any vector x in R^n can be written uniquely as x = x1 e1 + x2 e2 + ... + xn en, where x_i is the i-th component of x.

The vectors e_i are linearly independent and span R^n, so they form a basis. They are also orthonormal with respect to the standard inner product, meaning e_i · e_j = δ_{ij}. This orthonormality enables straightforward coordinate computations: the i-th coordinate map that extracts x_i is the linear functional sending x to its i-th component. The identity matrix I_n is the matrix whose columns are e1, ..., en, illustrating that many linear-algebra operations become simple when expressed in the standard basis.

In linear transformations, the matrix representing a map T relative to the standard basis has columns given by T(e_i). Thus, changing basis to a nonstandard one changes the coordinate representation of vectors and linear maps according to a change-of-basis matrix.

Beyond R^n, the notion generalizes to any finite-dimensional vector space equipped with a chosen identification with F^n; the fixed basis in that setting plays the role of a standard basis.