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 $e_j=\left({ 0_R,0_R,\cdots,1_R,\cdots,0_R}\right)$ be the ordered $n$-tuple of elements of $R$ whose $j$th entry is $1_R$ and all of whose other entries is $0_R$.

Then the ordered $n$-tuple $\left \langle {e_n} \right \rangle = \left({e_1,e_2,\cdots,e_n}\right)$ is called the standard ordered basis of $R^n$.

That this indeed is an ordered basis (on $R$-module $R^n$) is proved in Standard Ordered Basis is Basis.

The corresponding (unordered) set $\left\{{e_1, e_2, \cdots, e_n}\right\}$ is called the standard basis of $R^n$.

In a 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 entry is $1_{\mathbb F}$ and with entries $0_{\mathbb F}$ elsewhere.

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

The corresponding (unordered) set $\left\{{\mathbf e_1, \mathbf e_2, \cdots, \mathbf e_n}\right\}$ is called the standard basis of $\mathbf V$

Again, see Standard Ordered Basis is Basis for the proof that $\left({\mathbf e_i}\right)_{i=1}^n$ is indeed an basis.