Linear and multilinear algebra; matrix theory Definition
Linear subspace
Builds on Vector space
Definition
Let be a field and let be a vector space over . A linear subspace of is a subset with these three properties:
- .
- If , then .
- If and , then .
Remarks
With the operations of , a subspace is itself a vector space over . The sets and are subspaces of . The lines through the origin are the one-dimensional subspaces of ; a line that misses the origin is not a subspace. The kernel and the image of a linear map are subspaces.
Reading path
To understand this page from the ground up, read these 2 pages in order. The foundations come first.
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- Vector space definition
- Linear subspace definition
Where this leads
171 pages build on this one.
- Dimension of the dual space
- Linear map of a matrix
- Nullity
- Every vector space has a basis
- Jordan–Chevalley decomposition
- Finite-dimensional vector space
- Reflection
- Rank of a product of matrices
- Determinant of a linear map
- Direct sum of subspaces
- Parseval's identity
- Complementary subspaces
- 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
- Orthogonal complement
- Third isomorphism theorem for vector spaces
- 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
- Second isomorphism theorem for vector spaces
- Spanning set
- Coordinate vector
- Least squares
- 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
- Invariant subspace
- Row space
- QR decomposition
- Rank of a matrix
- Column space
- Orthonormal basis
- Linear span
- Rouché–Capelli theorem
- Quotient map
- 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
- Orthogonal projection is the closest point
- Change of basis formula
- First isomorphism theorem for linear maps
- Trace is the sum of the eigenvalues
- Matrix of a bilinear form
- Existence of a complementary subspace
- Restriction of an endomorphism to an invariant subspace
- Principal axis theorem
- Orthogonal complement of the orthogonal complement
- Orientation of a real vector space
- Orthogonal projection
- Cayley–Hamilton theorem for linear maps
- Dimension of the coordinate space
- Semisimple endomorphism
- Dimension of a subspace
- Annihilator of a subspace
- Dimension of a tensor product
- A linear map is determined by its values on a basis
- Projection
- Sum of subspaces
- Algebraic multiplicity
- Basis of a tensor product
- A space is the direct sum of a subspace and its orthogonal complement
- Change of basis for a bilinear form
- Quotient space
- Dimension of a quotient space
- Row rank equals column rank
- Intersection of subspaces
- Determinant of a family of vectors in a basis
- Finite-dimensional spaces are isomorphic iff they have equal dimension
- Cartan–Dieudonné theorem
- Gram–Schmidt process
- 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
- Gram–Schmidt vectors are orthogonal
- Existence of an orthonormal basis
- Distinct eigenvalues imply diagonalisable
- Matrix of a linear map
- Decomposition into generalised eigenspaces
- Dimension of the space of linear maps
- Universal property of the quotient space
- Canonical isomorphism with the double dual
- Matrix of the dual map is the transpose
- Non-negative least squares
- Derivation of the conjugate gradient method
- Conjugate residual method
- Local regression
- Beurling's theorem
- Hilbert–Schmidt expansion theorem
- Minimal residual method
- Toeplitz operator
- Partial isometry
- Céa's lemma
- Bilinear series of a symmetric kernel
- Ordinary least squares
- QR algorithm
- Smoothing spline
- Lax–Milgram theorem
- Field norm
- Hadamard's inequality in Fredholm theory
- Response surface methodology
- Nonlinear regression
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Generalized minimal residual method
- Minimum total potential energy principle
- Natural transformation
- Hadamard's inequality
- Arnoldi iteration
- Gauss–Newton algorithm
- Hilbert space adjoint
- Lidskii's trace formula for integral operators
- Normal coordinates
- Sequence space
- Hadamard's maximal determinant problem
- Spectrum of the Laplace–Beltrami operator
- Cochran's theorem
- Hilbert projection theorem
- Dimension of the solution space of a linear recurrence
- Ridge regression
- L-curve
- Independence of sample mean and sample variance for normal samples
- Biconjugate gradient stabilized method
- Galerkin method
- Bernstein's theorem on absolutely convergent Fourier series
- Transcendence degree
- Operator monotone function
- Empirical risk minimization
- Isotonic regression
- Projection body
- Mercer's theorem
- Classical group
- Normal basis theorem
- Principal component analysis
- Schauder basis
- Ladyzhenskaya–Babuška–Brezzi condition
- Discriminant of a field extension
- Levenberg–Marquardt algorithm
- Schur product theorem
- Fundamental theorem of space curves
- Riesz representation theorem
- Box–Jenkins method
- Singular values of a compact operator
- Lasso
- Householder transformation
- Numerical methods for linear least squares
- Schmidt's singular value decomposition of a kernel
- Existence of transcendence bases
- Hilbert's Theorem 90
- Three-term recurrence for orthogonal polynomials
- Bareiss algorithm
- Frenet–Serret formulas
Further reading
About this page
| Tag | 0EPTJA5QV4 |
|---|---|
| URI | m2://0EPTJA5QV4 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 15A03 |
| Defines | Linear subspace |
| Uses | Vector space |
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What this page needs for W2
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