Standard_Ordered_Basis_is_Basis

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 $e_j$ 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 $\sequence {e_n}$ is an ordered basis of the $R$-module $R^n$. This ordered basis is called the '''standard ordered basis of $R^n$'''. The corresponding set $\set {e_1, e_2, \ldots, e_n}$ is called the '''standard basis of $R^n$'''.