Definition:Standard Basis

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Definition

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\sequence {e_k}_{1 \mathop \le k \mathop \le n}$ be the standard ordered basis of the $R$-module $R^n$.


The corresponding (unordered) set $\set {e_1, e_2, \ldots, e_n}$ is called the standard basis of $R^n$.


Vector Space

The concept of a standard basis is often found in the context of vector spaces.


Let $\left({\mathbf V, +, \circ}\right)_{\mathbb F}$ be a vector space over $\mathbb F$.

Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\mathbf V$.


The corresponding (unordered) set $\left\{{\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right\}$ is called the standard basis of $\mathbf V$


Also see


Sources