Definition:Basis of Vector Space
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Definition
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
A basis of a vector space can also be referred to as a basis for a vector space.
Some sources refer to it as an algebraic basis.
A basis of a vector space over a subfield of $\C$ may also be known as a Hamel basis, for Georg Karl Wilhelm Hamel, to contrast with Schauder basis.
Also see
- Equivalence of Definitions of Basis of Vector Space
- Definition:Dimension of Vector Space
- Bases of Vector Space have Equal Cardinality
- Results about bases of vector spaces can be found here.
Generalizations
Linguistic Note
The plural of basis is bases.
This is properly pronounced bay-seez, not bay-siz.
Sources
- 1992: Frederick W. Byron, Jr. and Robert W. Fuller: Mathematics of Classical and Quantum Physics ... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.2$ The Resolution of a Vector into Components
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): basis (plural bases)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): basis (plural bases)