Linear and multilinear algebra; matrix theory Definition
Spanning set
Builds on Linear span
Definition
Let be a field and let be a vector space over . A subset is a spanning set of , or spans , when , where is the linear span. Equivalently, each vector of is a linear combination of finitely many vectors of .
Remarks
The vectors span , and so do . A linearly independent spanning set is a basis. Every spanning set contains a basis. A vector space with a finite spanning set is finite-dimensional.
Reading path
To understand this page from the ground up, read these 5 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
Where this leads
98 pages build on this one.
- Dimension of the dual space
- Linear map of a matrix
- Nullity
- Every vector space has a basis
- Finite-dimensional vector space
- Rank of a product of matrices
- Determinant of a linear map
- Parseval's identity
- Eigenvalues are the roots of the characteristic polynomial
- A matrix is diagonalisable iff it has a basis of eigenvectors
- Isomorphism with the coordinate space
- Dimension
- Matrix of a composition is the product of matrices
- Dimension of a direct product
- Grassmann's formula
- Spectral theorem
- Every spanning set contains a basis
- Dimension of the annihilator
- Geometric multiplicity
- Existence of an eigenvalue over an algebraically closed field
- Steinitz exchange lemma
- An injective endomorphism of a finite-dimensional space is surjective
- Linearly independent set of the right size is a basis
- Coordinates in an orthonormal basis
- Coordinate vector
- Characteristic polynomial of a linear map
- Determinant is the product of the eigenvalues
- Classification of complex quadratic forms
- Matrix of the adjoint is the conjugate transpose
- Spectral theorem for Hermitian matrices
- Dimension of the orthogonal complement
- Rank of a matrix
- Orthonormal basis
- Rouché–Capelli theorem
- Rank of a linear map
- Determinant of a linear map is the determinant of its matrix
- Signature of a quadratic form
- Change of basis matrix
- Every linearly independent set extends to a basis
- Change of basis formula
- Trace is the sum of the eigenvalues
- Matrix of a bilinear form
- Principal axis theorem
- Orthogonal complement of the orthogonal complement
- Orientation of a real vector space
- Cayley–Hamilton theorem for linear maps
- Dimension of the coordinate space
- Dimension of a subspace
- Dimension of a tensor product
- A linear map is determined by its values on a basis
- Algebraic multiplicity
- Basis of a tensor product
- Change of basis for a bilinear form
- Dimension of a quotient space
- Row rank equals column rank
- Determinant of a family of vectors in a basis
- Finite-dimensional spaces are isomorphic iff they have equal dimension
- Basis
- Rank–nullity theorem
- Dual basis
- Trace of a linear map is the trace of its matrix
- Sylvester's law of inertia
- A subspace of full dimension is the whole space
- Dimension theorem for vector spaces
- Existence of an orthonormal basis
- Distinct eigenvalues imply diagonalisable
- Matrix of a linear map
- Dimension of the space of linear maps
- Canonical isomorphism with the double dual
- Matrix of the dual map is the transpose
- Hilbert–Schmidt expansion theorem
- Bilinear series of a symmetric kernel
- Field norm
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Natural transformation
- Lidskii's trace formula for integral operators
- Normal coordinates
- Spectrum of the Laplace–Beltrami operator
- Cochran's theorem
- Dimension of the solution space of a linear recurrence
- Independence of sample mean and sample variance for normal samples
- Bernstein's theorem on absolutely convergent Fourier series
- Transcendence degree
- Operator monotone function
- Mercer's theorem
- Classical group
- Normal basis theorem
- Principal component analysis
- Schauder basis
- Discriminant of a field extension
- Schur product theorem
- Fundamental theorem of space curves
- Singular values of a compact operator
- Schmidt's singular value decomposition of a kernel
- Existence of transcendence bases
- Hilbert's Theorem 90
- Frenet–Serret formulas
Further reading
- Web https://en.wikipedia.org/wiki/Linear_span, lead section
About this page
| Tag | 0BJKVF90Z6 |
|---|---|
| URI | m2://0BJKVF90Z6 |
| Kind | definition |
| Writing | W0 stub |
| MSC | 15A03 |
| Defines | Spanning set |
| Uses | Linear span |
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