Definition:Ordered Dual Basis

This page is about Ordered Dual Basis in the context of Linear Algebra. For other uses, see Dual.

Definition

Let $R$ be a commutative ring with unity.

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 $G$.

Then there is an ordered basis $\sequence {a'_n}$ of $G^*$ satisfying $\forall i, j \in \closedint 1 n: \map {a'_i} {a_j} = \delta_{i j}$.

This ordered basis $\sequence {a'_n}$ of $G^*$ is called the ordered basis of $G^*$ dual to $\sequence {a_n}$, or the ordered dual basis of $G^*$.

Also see

• Existence of Ordered Dual Basis

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations
• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.): Appendix $\text B$. Review of Tensors