# Definition:Standard Ordered Basis

## Definition

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $n$ be a positive integer.

For each $j \in \closedint 1 n$, let $e_j = \tuple {0_R, 0_R, \ldots, 1_R, \cdots, 0_R}$ 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 $\sequence {e_k}_{1 \mathop \le k \mathop \le n} = \tuple {e_1, e_2, \ldots, e_n}$ 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.

Let $\left({\mathbf V, +, \circ}\right)_{\mathbb F}$ be a vector space over $\mathbb F$, as defined by the vector space axioms.

Let the unity of $\mathbb F$ be denoted $1_{\mathbb F}$, and its zero $0_{\mathbb F}$.

Let $\mathbf e_i$ be a vector whose $i$th term is $1_{\mathbb F}$ and with entries $0_{\mathbb F}$ elsewhere.

Then the ordered $n$-tuple $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ is the standard ordered basis of $\mathbf V$.