Spanning set

Spanning set
Tag	`0BJKVF90Z6`
URI	`m2://0BJKVF90Z6`
Kind	definition
Writing	W0 stub
MSC	15A03
Defines	Spanning set
Uses	Linear span

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Definition

Let $F$ be a field and let $V$ be a vector space over $F$ . A subset $S \subseteq V$ is a spanning set of $V$ , or spans $V$ , when $\operatorname{span}(S) = V$ , where $\operatorname{span}$ is the linear span. Equivalently, each vector of $V$ is a linear combination of finitely many vectors of $S$ .

Remarks

The vectors $(1, 0), (0, 1), (1, 1)$ span $\R^2$ , and so do $(1, 0), (0, 1)$ . 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.

Explanations

L1 human

A spanning set is a set of building blocks from which we can make every vector of the space. It may contain more vectors than we need. Removing the extra vectors leaves a basis.

What this node needs for W1

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What this needs

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Vector space definition
Linear combination definition
Linear subspace definition
Linear span definition
Spanning set definition

What depends on this

70 nodes use this node, directly or through other nodes.

Dimension of the dual space 00889YD15K
Linear map of a matrix 009892PZ30
Nullity 00K4FK4NBH
Every vector space has a basis 00MS9VMQ9Z
Finite-dimensional vector space 018W3V3TCT
Rank of a product of matrices 02FDWY55AS
Determinant of a linear map 03F5KP94K0
Parseval's identity 047RFCNVSK
Eigenvalues are the roots of the characteristic polynomial 053QVHF6N7
A matrix is diagonalisable iff it has a basis of eigenvectors 056SW1E0AC
Isomorphism with the coordinate space 059THXBRBB
Dimension 05G2G0P224
Matrix of a composition is the product of matrices 05Q1B7MH0M
Dimension of a direct product 06HN47VPTH
Grassmann's formula 06HTJNCMPA
Spectral theorem 06J4PXRG22
Every spanning set contains a basis 06MRNP4BNT
Dimension of the annihilator 06Q1EFVH0Q
Geometric multiplicity 06YSJVR356
Existence of an eigenvalue over an algebraically closed field 08CJB2NQ92
Steinitz exchange lemma 08ZDRN6H41
An injective endomorphism of a finite-dimensional space is surjective 0912EEEVCT
Linearly independent set of the right size is a basis 096V7H13H9
Coordinates in an orthonormal basis 09DZ8CD1AK
Coordinate vector 0BMVBHTWV2
Characteristic polynomial of a linear map 0BZ81AJNWF
Determinant is the product of the eigenvalues 0C2F40MEBC
Classification of complex quadratic forms 0CEX9BRXRZ
Matrix of the adjoint is the conjugate transpose 0DBTE6HVBD
Spectral theorem for Hermitian matrices 0DGZBQWZWN
Dimension of the orthogonal complement 0E2NYPVSP9
Rank of a matrix 0EWQEYRVA8
Orthonormal basis 0FWJ0VW561
Rouché–Capelli theorem 0HV1SKWPMS
Rank of a linear map 0JGB7A6GTS
Determinant of a linear map is the determinant of its matrix 0KMN32XGPF
Signature of a quadratic form 0KQZYGEJTF
Change of basis matrix 0KWXYZPJKN
Every linearly independent set extends to a basis 0M4SXE5WW3
Change of basis formula 0MQXVZEBMT
Trace is the sum of the eigenvalues 0NMXDCNNKR
Matrix of a bilinear form 0P24YW565F
Principal axis theorem 0PR5P9Z4KM
Orthogonal complement of the orthogonal complement 0PXN9G4MX1
Orientation of a real vector space 0Q8BZXVT1Z
Cayley–Hamilton theorem for linear maps 0QEM68Y0DE
Dimension of the coordinate space 0QK7FPE6VE
Dimension of a subspace 0QW084460E
Dimension of a tensor product 0R84PF4JV7
A linear map is determined by its values on a basis 0R8X9XA83A
Algebraic multiplicity 0S7A0EMPF4
Basis of a tensor product 0SH5VT8QYA
Change of basis for a bilinear form 0SPYAGE7T0
Dimension of a quotient space 0SXDQ6TN00
Row rank equals column rank 0TZAG60ZYT
Determinant of a family of vectors in a basis 0V94H674MB
Finite-dimensional spaces are isomorphic iff they have equal dimension 0VJBSK15SN
Basis 0VTQTPQGAV
Rank–nullity theorem 0W4FSW1TVB
Dual basis 0W6M3PSD08
Trace of a linear map is the trace of its matrix 0WCPH5Z7FA
Sylvester's law of inertia 0WFED3K3DQ
A subspace of full dimension is the whole space 0WGAXEK99E
Dimension theorem for vector spaces 0WK0Y8RH1M
Existence of an orthonormal basis 0Y85GYEGSF
Distinct eigenvalues imply diagonalisable 0Y95Z7MT02
Matrix of a linear map 0YYMY95FWS
Dimension of the space of linear maps 0Z9N1ER0W0
Canonical isomorphism with the double dual 0ZJM9D7NPS
Matrix of the dual map is the transpose 0ZPB3180AY