Existence of Ordered Dual Basis

Definition

Let $R$ be a commutative ring with unity whose unity is $1_R$.

Let $\struct {G, +_G, \circ}_R$ be an $n$-dimensional module over $R$.

Let $\sequence {a_n}$ be an ordered basis of $G$.

Let $G^*$ be the algebraic dual of $G$.

Let $\sequence {a'_n}$ be the ordered dual basis of $G^*$.

This ordered dual basis $\sequence {a'_n}$ is guaranteed to exist.

Proof

From Basis for $R$-Module $R$, $\set {1_R}$ is a basis of the $R$-module $R$.

Hence from Basis for Set of Linear Transformations, this ordered dual basis as defined exists.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations