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

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sum_{k \mathop = 1}^n \lambda_k e_k\)	\(=\)	\(\displaystyle \lambda_1 \left({1_R, 0_R, 0_R, \ldots, 0_R}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \lambda_2 \left({0_R, 1_R, 0_R, \ldots, 0_R}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \ldots\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(+\)	\(\displaystyle \lambda_n \left({0_R, 0_R, 0_R, \ldots, 1_R}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\lambda_1, \lambda_2, \lambda_3, \ldots, \lambda_n}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Also see

• Engineering Notation

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 27$: Example $27.6$