Definition:Basis (Linear Algebra)
Contents
Module
Let $R$ be a ring with unity.
Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module.
Definition 1
A basis of $G$ is a linearly independent subset of $G$ which is a generator for $G$.
Definition 2
Let $\mathcal B = \family {b_i}_{i \mathop \in I}$ be a family of elements of $M$.
Let $\Psi: R^{\paren I} \to M$ be the homomorphism given by Universal Property of Free Module on Set.
Then $\mathcal B$ is a basis if and only if $\Psi$ is an isomorphism.
Vector Space
Let $K$ be a division ring.
Let $\struct {G, +_G, \circ}_R$ be a vector space over $K$.
Definition 1
A basis of $G$ is a linearly independent subset of $G$ which is a generator for $G$.
Definition 2
A basis is a maximal linearly independent subset of $G$.
Also known as
The phrase basis for $G$ can also be seen instead of basis of $G$.
Also see
- Equivalence of Definitions of Basis (Linear Algebra)
- Vector Space has Basis
- Definition:Ordered Basis
- Definition:Dimension (Linear Algebra)
- Expression of Vector as Linear Combination from Basis is Unique
- Definition:Change of Basis Matrix
Linguistic Note
The plural of basis is bases.
This is properly pronounced bay-seez, not bay-siz.
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