Definition:Standard Ordered Basis

Definition
Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $n$ be a positive integer.

For each $j \in \left[{1 \,.\,.\, n}\right]$, let $e_j = \left({0_R, 0_R, \ldots, 1_R, \cdots, 0_R}\right)$ be the ordered $n$-tuple of elements of $R$ whose $j$th term is $1_R$ and all of whose other entries is $0_R$.

Then the ordered $n$-tuple $\left \langle {e_k} \right \rangle_{1 \mathop \le k \mathop \le n} = \left({e_1, e_2, \ldots, e_n}\right)$ is called the standard ordered basis (of the $R$-module $R^n$).

Vector Space
The concept of a standard ordered basis is often found in the context of vector spaces.

Also see

 * Standard Ordered Basis is Basis


 * Definition:Standard Basis