Standard Ordered Basis is Basis

Theorem
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 $$e_j$$ 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 $$\left \langle {e_n} \right \rangle$$ 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 $$\left\{{e_1, e_2, \ldots, e_n}\right\}$$ is called the standard basis of $$R^n$$.

Proof
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