Standard basis

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 $${\displaystyle v_x{\mathbf{e}}_x+v_y{\mathbf{e}}_y+v_z{\mathbf{e}}_z,}$$ the scalars v_x, v_y, v_z being the scalar components of the vector v.

In the n-dimensional Euclidean space ${\displaystyle {\mathbb{R}}^n}$, the standard basis consists of n distinct vectors $${\displaystyle \{{\mathbf{e}}_i:1\le i\le n\},}$$ where e_i denotes the vector with a 1 in the ith coordinate and 0's elsewhere.

Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices M_m×n, the standard basis consists of the m×n–matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices $${\displaystyle \begin{array}{cccc}{\mathbf{e}}_{11}& =\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right),& {\mathbf{e}}_{12}& =\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),\\ {\mathbf{e}}_{21}& =\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),& {\mathbf{e}}_{22}& =\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right).\end{array}}$$

Properties

By definition, the standard basis is a sequence of orthogonal unit vectors. In other words, it is an ordered and orthonormal basis.

However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i.e., $${\displaystyle \begin{array}{cc}{v}_1& =\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)\\ v_2& =\left(\frac{1}{2},\frac{-\sqrt{3}}{2}\right)\end{array}}$$ are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system, so the basis with these vectors does not meet the definition of standard basis.