Existence of Ordered Dual Basis
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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