Basis for Finite Submodule of Function Space

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

Let $$A$$ be a set.

For each $$a \in A$$, let $$f_a: A \to R$$ be defined as $$\forall x \in A: f_a \left({x}\right) = \begin{cases} 1 & : x = a \\ 0 & : x \ne a \end{cases} $$.

Then $$B = \left\{{f_a: a \in A}\right\}$$ is a basis of the Finite Submodule of Function Space $$R^{\left({A}\right)}$$.

Proof
Let $$\left \langle {a_n} \right \rangle$$ be a sequence of distinct terms of $$A$$.

Let $$\left \langle {\lambda_n} \right \rangle$$ a sequence of scalars.

Then: $$\sum_{k=1}^n \lambda_k f_{a_k}$$ is the mapping whose value at $$a_k$$ is $$\lambda_k$$ and whose value at any $$x$$ not in $$\left\{{a_1, a_2, \ldots, a_n}\right\}$$ is zero.

Hence $$B$$ is a linearly independent generator of $$R^{\left({A}\right)}$$.

If $$A = \left[{1 \,. \, . \, n}\right]$$, then $$B$$ is the standard basis of $$R^n$$.