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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Example $27.6$