Definition:Standard Ordered Basis/Vector Space

Definition

Let $\struct {\mathbf V, +, \circ}_{\mathbb F}$ be a vector space over a field $\mathbb F$, as defined by the vector space axioms.

Let the unity of $\mathbb F$ be denoted $1_{\mathbb F}$, and its zero $0_{\mathbb F}$.

Let $\mathbf e_i$ be a vector whose $i$th term is $1_{\mathbb F}$ and with entries $0_{\mathbb F}$ elsewhere.

Then the ordered $n$-tuple $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $\mathbf V$.

Also see

• Standard Ordered Basis is Basis

• Definition:Standard Basis
• Definition:Engineering Notation

Sources

• 1995: John B. Fraleigh and Raymond A. Beauregard: Linear Algebra (3rd ed.): $\S 3.3$