Linear and multilinear algebra; matrix theory Definition
Finite-dimensional vector space
Builds on Spanning set
Definition
Let be a field and let be a vector space over . The space is finite-dimensional when it has a finite spanning set. Otherwise is infinite-dimensional.
Remarks
A finite-dimensional space has a finite basis, because every spanning set contains a basis. The spaces and are finite-dimensional. The polynomial ring is infinite-dimensional: finitely many polynomials have a bounded degree, so they do not span . Each subspace of a finite-dimensional space is finite-dimensional.
Reading path
To understand this page from the ground up, read these 6 pages in order. The foundations come first.
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- Vector space definition
- Linear combination definition
- Linear subspace definition
- Linear span definition
- Spanning set definition
- Finite-dimensional vector space definition
Where this leads
28 pages build on this one.
- Nullity
- Rank of a product of matrices
- Dimension
- Dimension of a direct product
- Grassmann's formula
- Dimension of the annihilator
- Geometric multiplicity
- An injective endomorphism of a finite-dimensional space is surjective
- Linearly independent set of the right size is a basis
- Classification of complex quadratic forms
- Dimension of the orthogonal complement
- Rank of a matrix
- Rouché–Capelli theorem
- Rank of a linear map
- Orthogonal complement of the orthogonal complement
- Dimension of the coordinate space
- Dimension of a subspace
- Dimension of a tensor product
- Dimension of a quotient space
- Row rank equals column rank
- Finite-dimensional spaces are isomorphic iff they have equal dimension
- Rank–nullity theorem
- A subspace of full dimension is the whole space
- Dimension of the space of linear maps
- Cochran's theorem
- Dimension of the solution space of a linear recurrence
- Independence of sample mean and sample variance for normal samples
- Classical group
Further reading
About this page
| Tag | 018W3V3TCT |
|---|---|
| URI | m2://018W3V3TCT |
| Kind | definition |
| Writing | W0 stub |
| MSC | 15A03 |
| Defines | Finite-dimensional vector space |
| Uses | Spanning set |
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